Exploring High-dimensional Hilbert spaces by Quantum Optics · that quantum systems, prepared by a number of diﬀerent approaches, represent a great resource for several tasks, ﬁrst

Exploring High-dimensional Hilbert spaces byQuantum Optics

Scuola di Dottorato in Fisica

Dottorato di Ricerca in Fisica – XXV Ciclo

Candidate

Andrea ChiuriID number 697117

Thesis Advisor

Prof. Paolo Mataloni

A thesis submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

December 2012

Thesis not yet defended

Andrea Chiuri. Exploring High-dimensional Hilbert spaces by Quantum Optics.Ph.D. thesis. Sapienza – University of Rome© December 2012

version: 7 December 2012

email: [email protected]

Contents

Introduction vii

1 Multiqubit photonic state 11.1 MultiDOF states and hybrid entanglement . . . . . . . . . . . . . . . 11.2 Hyperentangled quantum states . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Hyperentanglement . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Hyperentangled/multiDOF photon states: experimental realizations 2

1.3.1 Entanglement in a single degree of freedom . . . . . . . . . . 31.3.2 Hyperentanglement in different degrees of freedom . . . . . . 7

1.4 Hyperentanglement for quantum information . . . . . . . . . . . . . 81.4.1 Quantum nonlocality tests . . . . . . . . . . . . . . . . . . . . 81.4.2 Bell state analysis and dense coding . . . . . . . . . . . . . . 81.4.3 Quantum computing . . . . . . . . . . . . . . . . . . . . . . . 9

1.5 Hyperentanglement source . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Multipartite photonic quantum states 132.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Hyperentangled Mixed Phased Dicke States . . . . . . . . . . . . . . 13

2.2.1 4-qubit hyperentangled Phased Dicke states . . . . . . . . . . 132.2.2 Structural Entanglement Witness . . . . . . . . . . . . . . . . 162.2.3 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.4 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Characterization of the engineered multiDOF Dicke states . . . . . . 212.3.1 Tomographic characterization . . . . . . . . . . . . . . . . . . 212.3.2 Quantum and classical correlations in a tripartite system . . 252.3.3 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . 29

2.4 Quantum Networking via Dicke states . . . . . . . . . . . . . . . . . 312.4.1 1→3 QTC and ODT . . . . . . . . . . . . . . . . . . . . . . . 322.4.2 Description of the protocols. . . . . . . . . . . . . . . . . . . 332.4.3 Experimental implementations of 1→3 QTC. . . . . . . . . . 352.4.4 Experimental implementations of ODT . . . . . . . . . . . . 38

2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Non-Markovianity 413.1 Introduction: Open quantum system . . . . . . . . . . . . . . . . . . 41

3.1.1 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1.2 Evolution of Open Quantum Systems . . . . . . . . . . . . . 42

iii

3.1.3 Markovian Dynamics . . . . . . . . . . . . . . . . . . . . . . . 443.1.4 Non-Markovian Dynamics . . . . . . . . . . . . . . . . . . . . 463.1.5 Signatures of Non-Markovianity . . . . . . . . . . . . . . . . . 47

3.2 Experimental quantum simulation of (non-) Markovian dynamics . . 493.2.1 The Experiment: non-Markovian dynamics . . . . . . . . . . 503.2.2 Experimental results: non-Markovian dynamics . . . . . . . . 533.2.3 The Experiment: Markovian dynamics . . . . . . . . . . . . . 55

3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Noisy Channels 574.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Experimental implementation of a PC and a DC . . . . . . . . . . . 574.3 Optimal estimation of quantum noise in Pauli channels . . . . . . . . 58

4.3.1 The Experiment: Anisotropic Noise . . . . . . . . . . . . . . 594.3.2 The Experiment: Isotropic Noise . . . . . . . . . . . . . . . . 614.3.3 Ancillary Assisted Quantum Process Tomography . . . . . . 62

4.4 Entanglement assisted capacity for the depolarizing channel . . . . . 644.4.1 General scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4.2 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 684.4.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Other Experiments: brief review 735.1 Extremal quantum correlations of 2-qubit states . . . . . . . . . . . 73

5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.1.2 Quantum Discord and AMID . . . . . . . . . . . . . . . . . . 745.1.3 Resource-state generation . . . . . . . . . . . . . . . . . . . . 75

5.2 Fully nonlocal quantum correlations . . . . . . . . . . . . . . . . . . 775.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2.2 The inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 785.2.3 Experimental realization . . . . . . . . . . . . . . . . . . . . . 805.2.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 81

5.3 Linear optics C-Phase gate . . . . . . . . . . . . . . . . . . . . . . . 82

A Decoherence introduced in the Phased Dicke states 87

B On the Quantum Protocols 91B.1 Optimal quantum cloning machine . . . . . . . . . . . . . . . . . . . 91B.2 Quantum Telecloning Protocol . . . . . . . . . . . . . . . . . . . . . 92B.3 QTC 1 → 2 via |ψT C⟩ . . . . . . . . . . . . . . . . . . . . . . . . . . 93B.4 Phase-Covariant QTC 1 → 3 via Dicke State . . . . . . . . . . . . . 95B.5 General QTC 1 → 3 via Dicke State . . . . . . . . . . . . . . . . . . 96B.6 ODT protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

C On the non-Markovian dynamics 99C.1 Controlled-rotation gates from quantum Ising models . . . . . . . . . 99C.2 Experimental system-ancilla density matrices . . . . . . . . . . . . . 101

D Peres-Mermin proof of the KS theorem 103

iv

Bibliography 105

v

Introduction

Quantum Mechanics (QM) has played a fondamental role in the physics of the pastcentury because of its novelties and paradoxes. Several new ideas were introducedsuch as the probabilistic description of the nature, instead of a deterministic one, thequantization of the light, the dualism wave-particle, and other innovative concepts.These events motivated a strong debate in the community, started with the famouspaper written by Einstein, Podolsky and Rosen (EPR) [1] where the authors, thatdidn’t believe to a non-deterministic description of nature, aimed to show that QMwas an incomplete theory. Nowadays the validity of the QM is so widely recognizedthat quantum systems, prepared by a number of different approaches, represent agreat resource for several tasks, first of all Quantum Information (QI) processes andprotocols. QI, introduced in the late 80’s by the merging of classical informationand quantum physics, is a new scientific field with the potential to revolutionizemany areas of science and technology. Its main goal is to understand the quantumnature of information and to learn how to use quantum mechanical principles forcompletely new applications that would be impossible by classical physics.

QI usually deals with quantum bits, or “qubits”, i.e. 2-dimensional quantumsystems that do not possess in general the definite values of 0 or 1 of classical bits,but rather are in a so-called “coherent superposition”, |ϕ⟩ = α|0⟩+ β|1⟩, of the twoorthogonal basis states |0⟩, |1⟩. Such a state reveals unusual properties, especiallywhen dealing with composite systems. The computational power of a quantumcomputer or other QI application increase with the number of the involved qubits.In the present PhD thesis I will describe how to generate and manipulate multi-qubit states encoded in different degrees of freedom (DOFs) of two photons. Indeed,besides photons, qubits can be in principle realized in different ways, for instance byusing trapped ions [2, 3, 4, 5, 6, 7], neutral atoms in interaction with optical cavities[8, 9, 10, 11, 12, 13], superconducting circuits [10, 14, 15, 16, 17], semiconductorquantum dots [18, 19, 20] and also by the nuclear magnetic resonance effect [21].

The most distinctive feature of quantum physics is the possibility of entanglingdifferent qubits. First recognized by Erwin Schroedinger as “the characteristic traitof quantum mechanics” [22], quantum entanglement represents the key resourcefor modern quantum information. It derives from “subtle” nonlocal correlationsbetween the parts of a quantum system and combines three basic structural elementsof quantum theory, i.e. the superposition principle, the quantum non-separabilityproperty and the exponential scaling of the state space with the number of partitions.Two systems A and B are entangled if the (pure) state of the total system |ψ⟩AB

is not separable, i.e. if it cannot be written as a product of two states belongingto A and B:

|ψ⟩AB = |χ⟩A ⊗ |φ⟩B

vii

In the case of mixed states ρAB of the composite system A⊗B, the previous relationgeneralizes to ρAB =

∑k pkρ

Ak ⊗ ρB

k , where pk are probabilities and ρAk ’s (ρB

k ’s) aregeneric density matrix of the system A (B).

This unique resource can be used to perform computational and cryptographictasks otherwise impossible with classical systems. An entangled state shared bytwo or more separated parties is a valuable resource for fundamental quantumprotocols[23], such as quantum teleportation [24], quantum dense coding [25], quan-tum computing [26] and quantum cryptography [27]. By using entangled states wecan deeply investigate the nonlocal properties of the quantum world. In fact, anentangled system exhibits correlations between two (or more) parties, which cannotbe reproduced by a classical theory or, in general, by a Local Hidden VariablesTheory (LHVT).

Quantum optics represents an excellent experimental test bench for various novelconcepts introduced within the framework of quantum information theory. Quan-tum states of photons, generally produced by the spontaneous parametric downconversion (SPDC) process [28], may be easily and accurately manipulated usinglinear and nonlinear optical devices and measured by efficient single-photon detec-tors. Linear optics is at the forefront of current experimental quantum informationprocessing, providing settings, operations and states of outstanding quality, control-lability and testability [23, 29]. Among many other achievements, the first test-beddemonstration of protocols for quantum teleportation [30, 31, 32], measurement-based quantum computation [33, 34, 35, 36] and quantum-empowered communi-cation [37] have been provided by using linear optics setups. Very recently, someseminal proposals and demonstrations of the suitability of bulk- as well as integrated-optics settings have been given in view of the controllable simulation of quantumphenomena, including the properties of frustrated spin systems [38], the statisticsof two-particle (correlated) quantum walks [39, 40], simple quantum games [41] andchemical processes [42].

In the standard conditions of SPDC activated by a continuous wave laser pumpbeam, no more than one photon pair is generated time by time. This correspondsto operate with qubits belonging to a 2 × 2 Hilbert space. On the other hand,many quantum information tasks and fundamental tests of quantum mechanics,such as the simulation of properties of quantum systems, the realization of quantumalgorithms with increasing complexity, or the investigation of the quantum worldat a mesoscopic level, deal with a large number of qubits. In order to take fulladvantage of the possibilities offered by quantum mechanics, more qubits must beadded to quantum states. For example, the possibility to manage a large numberof qubits allows to achieve a stronger violation of Bell inequalities and to increasethe computational power of a quantum processor.

Two approaches may be envisaged to increase the number of qubits. The first oneconsists of increasing the number of entangled particles [8, 9, 23, 43, 44, 45, 46, 47].In this way, multi-qubit entangled states are created by distributing the qubitsbetween the particles so that each particle carries one qubit. This is the way bywhich four-qubit graph states with atoms [8] and photons [43, 44, 45], and six-qubitgraph states with atoms [9] and photons [46] were realized.

A second strategy consists of encoding more than one qubit in each particle, byexploiting different degrees of freedom (DOFs) of the photon [48, 49, 50, 51, 52]. Thishas been used to create two-photon four- and six-qubit graph states [49, 50, 53, 54]

viii

and up to five-photon ten-qubit graph states [51]. The entanglement of two particlesin different DOFs corresponds to so-called hyperentangled (HE) state [55]. Byexploiting several DOFs of a pair of correlated photons, it has been possible toengineer also three-qubit quantum states based on a hybrid approach[56]. In thiscase two qubits are encoded in two different DOFs of one photon while only onequbit has been encoded in a sigle DOF of the other particle.

Multiqubit quantum state can be used also to explore (i.e. to simulate) complexphenomena that are inaccessible through a standard, classical computer [57, 58, 59].Some interesting steps have been performed in this direction: simple condensedmatter and chemical processes have been implemented on controllable quantumsimulators [38, 42, 60]. The relativistic motion and scattering of a particle in thepresence of a linear potential has been demonstrated in a trapped-ion quantumsimulator [3, 6, 61] that opens up the possibility to the study of quantum fieldtheories [4, 5] while the perspective of a universal digital quantum simulators hasbeen recently studied in [7].

The work I carried out during my PhD scholarship is based on the extensiveuse of multi-qubit multi-DOFs photonic quantum states. By exploiting quantumphotonic techniques it was possible to engineer several families of quantum statesspanning Hilbert spaces of different size. The high flexibility of the realized ex-perimental setups, described in this thesis, allowed to carry out several studies indifferent fields of quantum physics. I have considered also another important issuerelated to a typical aspect of quantum optics experiments dealing with quantumstates: the quantum noise which is unavoidably present in any realistic implemen-tation of quantum tasks. This study represents a further remarkable part of thepresent thesis because the performance and the optimisation of quantum tasks quiteoften depend on the level of noise affecting quantum states.

The thesis is organized as follows:

1. The introduction (Chap. 1) deals with the engineering of multi-qubit multi-DOFs photonic quantum states, with particular attention given to hyperentan-gled states. In this section it will be explained how to explore several dimen-sionalities of the Hilbert space by linear optics.

2. Chap. 2 is dedicated to four-qubit Phased and symmetric Dicke states. Thischapter contains the description of the source allowing to engineer this class ofquantum states and the experimental characterization of their entanglementstructure. The last part of the chapter regards the first experimental real-ization of two quantum protocols, namely Quantum Telecloning (QTC) andOpen Destination Teleportation (ODT), based on Dicke states. The experi-ments described in this chapter involve four- and five- qubits.

3. In Chap. 3 I describe an experiment of quantum simulation. In particular,the evolution of a particular physical system interacting with the environmenthas been simulated. Depending on the simulated interaction, it was possibleto simulate both Markovian and Non-Markovian dyanmics in the evolution ofthe considered system. In this experiment a three-quibit multi-DOF photonicstate was exploited.

4. Chap. 4 describes two experiments involving the quantum noisy channels,

ix

precisely the Pauli channels. These experiments are based on two-qubit two-photon states.

5. Other experiments performed during my PhD thesis work are briefly describedin Chap. 5.

x

Chapter 1

Multiqubit photonic state

1.1 MultiDOF states and hybrid entanglement

The use of different DOFs opens the possibility to create general multiqubit state,i.e. multiDOF states. These states involve more that one degree of freedom and,in general, they are not hyperentangled in the sense of the definition which willbe given in Section 1.2.1. The difference between a hyperentangled (HE) and ageneral multiDOF state is based on the different amount of entanglement existingbetween the particles, which is maximized in HE states. As a simple example, byusing a polarization entangled photon pairs and entangling the path DOF with thepolarization of a single photon, a 3-qubit entangled state is generated, however, theentanglement between the two particles is not increased. On the contrary, in aHE states, the amount of entanglement between the two particles grows with thenumber of independent DOFs added to the state.

Another term, so-called hybrid entanglement, is also used in the literature torefer to the entanglement existing between two different degrees of freedom of twoparticles [62, 63, 64].

Different experiments were performed with multiDOF states, I refer here to twoimportant examples. In the first case a ten-qubit entangled state was engineeredby entangling the path and the polarization of five photons initially prepared intoa Greenberger, Horne, Zeilinger (GHZ) polarization state [51]. The multiDOF ap-proach was essential even in the teleportation experiment [31], in fact in this ex-periment the used resource for teleporting an unknown qubit was represented by ahybrid entangled state of two photons.

1.2 Hyperentangled quantum states

1.2.1 Hyperentanglement

A hyperentangled (HE) state can be defined as follows:

|HE⟩ = |Bell⟩1 ⊗ |Bell⟩2 ⊗ |Bell⟩3... (1.1)

where each term corresponds to one of the four Bell states encoded in one DOF oftwo particles.

1

2 1. Multiqubit photonic state

Bell states represent the simplest examples of entangled states. In the compu-tational basis they are expressed as:

|Φ±⟩ = 1√2

(|0⟩A|0⟩B ± |1⟩A|1⟩B) , |Ψ±⟩ = 1√2

(|0⟩A|1⟩B ± |1⟩A|0⟩B) (1.2)

and represent an entangled basis for a two-qubit system.In the framework of quantum optics, two-photon Bell states have been realized

by different approaches. The two qubits may be encoded in a particular DOF ofthe particles, such as polarization [65], momentum based on linear [66], orbital [67],or transverse [68] spatial modes, energy-time [69, 70, 71] and time-bin [72, 73].

A more formal definition of HE state is the following. Let’s consider two photonsA and B and n independent DOFs aj and bj, with j = 1, ...n. Each DOFspans a 2-dimensional Hilbert space (i.e. it is equivalent to a qubit) with basis|0⟩aj

, |1⟩aj (|0⟩bj

, |1⟩bj) for particle A (B). In this way, each particle encodes

exactly n qubits. A state |φ⟩ is separable in the hyperentangled sense if it satisfiesthe following condition:

∃j such that |φ⟩ = |φ1⟩ajI |φ2⟩bjJ (1.3)

where I,J represents a generic bi-partition of the set Tj ≡ a1, b1, · · · , an, bn\aj , bj,so that I ∪ J = Tj and I ∩ J = ∅.

Definition: A (mixed) state is hyperentangled in n degrees of freedom if it isseparately entangled in each of them and cannot be written as a mixture of statessatisfying Eq. (1.3).

In order to experimentally detect hyperentanglement we can develop the samemethod used for entanglement which corresponds to measure a (hyper-)entanglementwitness. A witness W is a hermitian operator whose expectation value is non nega-tive for any separable state, while it is negative for an entangled state. By measuringa negative value of ⟨W ⟩, the presence of entanglement is demonstrated by only fewlocal measurements. The method can be generalized for HE states [74].

In the following Chapters I present the experimental measurement of severalfamilies of entanglement witness used to characterize the presence of entanglementin different multiqubit states.

1.3 Hyperentangled/multiDOF photon states: experi-mental realizations

The techniques commonly used to generate entangled photons exploit the sponta-neous parametric down conversion process, as said.

When an intense pump laser beam (p) shines a nonlinear birefringent crystal,pairs of photons, referred as idler (i) and signal (s), are probabilistically generatedfrom the crystal. The probability of emitting pairs of photons is maximized whenthe following conditions are satisfied:

phase matching: kp = ki + ks , energy matching: ωp = ωi + ωs . (1.4)

Phase matching is usually obtained by exploiting the birefringence of the nonlinearcrystal. More precisely, a SPDC two-photon state may be expressed as

N∫

d2ksd2kidωsdωsAp(ks + ki, ωs + ωi)sinc(∆kzL

2)|ks, ωs⟩|ki, ωi⟩ . (1.5)

1.3 Hyperentangled/multiDOF photon states: experimental realizations 3

Figure 1.1. a) Type I SPDC source. In the degenerate case, i.e. ωi = ωs, the two emittedphotons belong to the external surface of a single emission cone. b) Type II SPDC source.The two degenerate photons are emitted over two different emission cones corresponding tothe two orthogonal polarization.

In the previous equations ki and ks are the transverse momentum coordinates,|k, ω⟩ = a†(k, ω)|0⟩, Ap(k, ω) is the pump profile in the momentum-frequencyspace, N is normalization constant and L is the crystal length. ∆kz representsthe (longitudinal) phase mismatch ∆kz(ki,ks, ωi, ωs) = kpz(ki + ks, ωi + ωs) −ksz(ks, ωs) − kiz(ki, ωi). The longitudinal component of momentum is given by

kz(k, ω) =√[

n(ω)ωc

]2− k2. In usual conditions, the pump wavefunction is assumed

to be a Gaussian function Ap(k, ω) = C0e−

w20

4 k2pe−

τ2p4 (ω−ωp)2 , with τp and w0 re-

spectively representing the coherence time and the beam waist of the pump laserbeam. The phase matching condition is satisfied when ∆kz = 0. Two kinds of phasematching are commonly adopted, depending on the extraordinary (e) or ordinary(o) polarization of the pump and of the SPDC photons:

Type-I: e→ o+ o , Type-II: e→ e+ o . (1.6)

In the first case, assuming degenerate generated photons, ωi = ωs, the phase match-ing is satisfied for all the wavevectors ki and ks belonging to the external surface ofa single emission cone [See Fig. 1.1a)]. With Type-II phase matching, the two de-generate photons are emitted over two different, mutually crossing, emission cones[See Fig. 1.1b)].

1.3.1 Entanglement in a single degree of freedom

Polarization entanglement

Let’s now describe the polarization entanglement based on Type-I phase matching.Typically, two main setups are used in experiments with Type-I phase matching,the so called “2-crystal” source [75] and the one realized in the laboratory of Rome[76, 77, 78]. The former adopts two identical crystals with orthogonal optical axes,shined by a laser beam passing through the crystals. The second source is based ona single Type-I crystal excited by a double passage of the laser beam after reflectionon a spherical mirror [See Sec. 1.5 for details]. Both sources typically generate thepolarization entangled state |Φ±⟩ with |0⟩ → |H⟩ and |1⟩ → |V ⟩ (see Fig. 1.2a) andFig. 1.2b) for details). In the case of Type-II phase matching, two orthogonally


Figure 1.2. a) Polarization entanglement source based on two Type-I crystals with or-thogonal optical axes. In this case the pump beam is polarized along the diagonal direction.Each pair of correlated downconverted photons, emitted over opposite directions of the conesurface, are polarization-entangled. b) Polarization entanglement source based on the dou-ble passage of the pump beam through a Type-I crystal. Here polarization entanglementis realized by spatial and temporal superposition of photon pair emissions occurring withequal probability, back and forth, from a single Type-I crystal under double excitation of avertically polarized UV CW pump beam. A suitable rotation of the polarization on one ofthe two possible emission directions of the photons is then applied. c) Polarization entan-glement source based on a single Type-II crystals. The extraordinary |V ⟩ photons belongto the upper cone, while the ordinary |H⟩ photons belong to the lower one. The photonsemitted along the two cone intersections can be H or V polarized but, since a Type-II crys-tal is employed, if the first photon is H − polarized the second one must be V − polarized(and viceversa). In this way it is possible to generate the Bell states |Ψ±⟩.

polarized photons are emitted over two different cones. The |Ψ±⟩ state can begenerated [65] along the directions of intersection of the two cones (see Fig. 1.2c)).

Path entanglement

Let’s consider a Type-I crystal and a Gaussian pump profile with a relatively largevalue of τp. By assuming monochromatic SPDC photons (satisfying ωi + ωs = ωp),the 2-photon state may be written as

|Ψ⟩ ∝∫

d2ksd2kie−

w20

4 (ks+ki)2sinc(∆kzL

2)|ks⟩|ki⟩. (1.7)

This wavefunction shows that the two correlated photons are emitted over oppo-site directions of the cone surface. The different events corresponding to differentemission directions are coherent because of the transverse coherence of the pump


profile. By selecting different pairs of correlated emission modes with single modefibers [79] or by a holed mask [78, 80] it is possible to generate path- (or linearmomentum-) entanglement. In the next sections I will investigate in detail thisentanglement source. The optical Fourier transform of the two-photon state allowsto measure their transverse spatial correlation [81]. By a different scheme, eachphoton is forced to pass through two slits corresponding to two orthogonal states[82]. Transverse spatial correlations are controlled by manipulating the pump laserbeam. By making the biphotons passing only through symmetrically opposite slits,entangled states are generated.

Energy-time entanglement

A further available photon DOF is given by the conjugate variables energy and time.Let’s define |ω⟩ = a†ω|0⟩ and consider only two definite spatial modes (as done byselecting the radiation with two single mode fibers) and a Gaussian pump profile.In the limit of perfect phase matching and assuming constant refractive indicesno(ω) ∼ no(ω0), the (normalized) SPDC two-photon state is expressed as (here weset the emission time t0 = 0):

|Ψ⟩ =

√τpT

π

∫dωsdωi e

−T 24 (ωs−ωi)2

e−τ2

p4 (ωs+ωi−ωp)2 |ωs⟩|ωi⟩ (1.8)

with T = no(ω0)wθc , w related to the beam waist of the pump and the selected photons

and θ corresponding to the emission angle. By defining |t⟩ = 1√2π

∫dωe−iωt|ω⟩ the

SPDC state may be expressed as

|Ψ⟩ =√

1πTτp

∫dt1dt2 e−

(t1−t2)2

4T 2 e− (t1+t2)2

4τ2p eiω0(t1+t2)|t1⟩|t2⟩ (1.9)

For T = τp the SPDC state is entangled in energy (or time). When τp ≫T , frequency anticorrelations and emission time correlations are observed: in thiscase, the photons are emitted at the same time and emission events occurring atdifferent times are coherent within τp. A scheme invented by Franson [69] enablesthe measurement of energy-time entanglement by post-selecting the state (see Figure1.3b)). For each photon, it consistes of an unbalanced Mach-Zehnder interferometerwith a short (S) and a long (L) arm (see Figure 1.3a)). If cτp ≫ L−S, interferenceis observed when the two photons are detected in coincidence since it is not possibleto determine whether they followed both the short or the long paths.

When used for nonlocality experiments, the Franson’s scheme suffers of post-selection loophole, that may be removed by using a modified version of the setup[71, 83].

OAM-angle entanglement

Besides spin, photons possess a further angular momentum, namely orbital angularmomentum (OAM). In the paraxial approximation, photons described by a modefunction expressed by a Laguerre-Gaussian mode |ℓ, p⟩ are eigenstates of the OAMoperator with eigenvalues ℓ~ (ℓ = 0,±1,±2, . . .). The integer quantum numberp is related to the radial profile of the beam and the integer ℓ, referred to as the


Figure 1.3. a) Single qubit encoded in the time-energy DOF. It consists of an unbalancedMach-Zehnder interferometer with a short (S) and a long (L) arm. The input photon, afterthe interferometer is in a coherent superposition of the two possible events |S⟩ and |L⟩. It ispossible to obtain a general state by unbalancing the transmittivity and reflectivity of theBSs. The relative phase ϕ between |S⟩ and |L⟩ can be varyed by using a thin glass plateplaced in one of the two arms. It is important to stress that in this case the interferometerhas to satisfy the condition: L−S > τph, where τph represents the single photon coherencetime. b) Experimental scheme proposed by Franson in 1989 [69]. The EPR source in thiscase was represented by an atom emitting two photons during the decay. Each photon wasintroduced in a Mach-Zehnder interferometer and, by using a post-selection process, it waspossible to obtain the entangled state 1/

√2(|SS⟩12 + eiϕ|LL⟩12) where ϕ = ϕ1 + ϕ2.

topological winding number, describes the helical structure of the wave front arounda wave front singularity. The two-photon SPDC state may be expressed as [84]

|Ψ⟩ =∑ℓs,ps

∑ℓs,ps

Cℓs,ℓips,pi|ℓs, ps⟩|ℓi, pi⟩, (1.10)

enlightening possible entanglement in the OAM degree of freedom. By using therelationship between OAM and its conjugate variable, angular position, it is possibleto observe entanglement in the angular domain.

Under collinear phase matching conditions, when the pump beam is a |ℓ0, p0⟩mode, the two-photon state at the output of the nonlinear crystal contains onlyterms such that ℓs + ℓi = ℓ0. In many OAM applications, one considers only modeswith pi = ps = 0. In this subspace, when the pump beam is a Gaussian TEM00


beam, the two-photon state is expressed as

|Ψ⟩ =+∞∑

ℓ=−∞

√Pℓ|ℓ⟩s| − ℓ⟩i (1.11)

where Pl represents the probability of creating a signal photon with orbital angularmomentum ℓ and an idler photon with −ℓ. The state of Eq. (1.11) representsan OAM entangled state. It is worth stressing that the OAM space is infinitedimensional. In typical QI experiments, qubits are encoded into subspaces of thewhole Hilbert space [85, 86].

1.3.2 Hyperentanglement in different degrees of freedom

The above described techniques may be combined to generate photon states that areentangled in more than one DOF, so-called hyperentangled states. The first proposalof energy-momentum-polarization HE state with a type-II phase matching was givenin 1997 [55]. The first experimental realizations of HE states were provided on 2005[48, 52, 78, 87]. In the following I will describe the different kinds of hyperentangledstates so far realized.

Polarization-momentum hyperentanglement

By using the source of polarization entanglement developed in Rome, the extensionto path entanglement is straightforward: the two polarization entangled photons(labeled as A and B) are emitted over two symmetric directions belonging to theexternal surface of the degenerate cone. By selecting with a holed mask two pairsof correlated modes, thanks to the spatial coherence property of the source, thephotons are also entangled in path. Precisely, if |r⟩ and |ℓ⟩ represent the two modesin which each photon can be emitted, the hyperentangled state may be written as:

1√2

(|H⟩A|V ⟩B + |V ⟩A|H⟩B)⊗ 1√2

(|r⟩A|ℓ⟩B + |ℓ⟩A|r⟩B) . (1.12)

This state encodes 4 qubits into 2 photons [48, 78]. In Sec. 1.5 I will report adetailed description of the engineered source.

An alternative approach to realize polarization-momentum hyperentanglementexploits a double passage of pump beam in a Type-II nonlinear crystal [87, 88].The Type-II phase matching allows to create polarization entanglement. The twopossible emissions (forward or backward) generate momentum entanglement. The|r⟩ and |ℓ⟩modes are identified with the two possible directions in which each photoncan be emitted.

Polarization-OAM-time hyperentanglement

Using the 2-crystal source of polarization entanglement and operating with a longpump coherence time, the following polarization-OAM-time hyperentangled statewas generated [52]:

(|HH⟩+ |V V ⟩)︸︷︷︸polarization

⊗ (| − 1,+1⟩+ α|0, 0⟩+ |+ 1,−1⟩)︸︷︷︸OAM

⊗ (|SS⟩+ |LL⟩)︸︷︷︸time

. (1.13)


In the previous equation |±1⟩ and |0⟩ represent the OAM eigenstates and α describesthe OAM spatial-mode balance prescribed by the source and selected via the mode-matching conditions.

1.4 Hyperentanglement for quantum information

1.4.1 Quantum nonlocality tests

Hyperentanglement allows to generalize the Greenberger-Horne-Zeilinger (GHZ)theorem [89, 90] with only two entangled particles. The GHZ theorem, sometimesreferred as “Bell’s theorem without inequalities” or “All-Versus-Nothing (AVN)”proof of Bell’s theorem, shows a contradiction between quantum mechanics (QM)and local realistic theories even for definite predictions. The quantum nonlocalitycan thus, in principle, be manifest in a single run of a certain measurement. Whilethe GHZ argument requires at least three particles and, consequently, three space-like separated observers, with hyperentanglement an AVN nonlocality proof maybe derived with only two photons [91]. The contradiction between QM and localrealistic theories arises from perfect correlations. On the other hand, in a real exper-iment, perfect correlations and ideal measurement devices are practically impossible.To face this difficulty, it is possible to introduce an operator O, whose expectationvalue on the hyperentangled state of Eq. (1.12) is ⟨O⟩ = 9. However, local hid-den variable (LHV) theories predict an upper bound on the observed values of O,⟨O⟩LHV ≤ 7, which is in contradiction with quantum mechanical predictions. Theexperimental realizations of [80, 87] show that, under the fair sampling assumption,the inequality is violated. Stronger AVN inequalities can be found with 4 [49, 92]and 6 qubits [53] encoded in two photons.

One of the main limitations of the nonlocality tests performed with photons isrepresented by the so-called “detection loophole”: if the particle detection efficiencyis lower than a certain threshold level, the undetected events can be exploited by alocal model to reproduce the quantum predictions.

1.4.2 Bell state analysis and dense coding

Bell state analysis, i.e. the complete and deterministic discrimination between thefour orthogonal and maximally entangled “Bell states” of Eq. (1.2) are central inmany quantum information applications and processing, such as quantum densecoding [25, 93, 94] teleportation [24, 30, 31, 32], entanglement swapping [95], cryp-tography [27] and direct characterization of quantum dynamics [96]. However, itis not possible to completely and deterministically discriminate between the fourstates by using only linear operations and classical communication. Moreover, theoptimal probability of success in these cases is only 50% [93, 97].

By working in a larger Hilbert space, i.e., by employing hyperentangled states,a complete analysis of Bell states with only linear optical elements can be achieved[98, 99, 100, 101]. The method adopting polarization-path hyperentanglement isexplained as follows. Let’s consider the four hyperentangled states of the form|Ξ⟩ = |Π⟩π⊗|ψ+⟩k, where |Π⟩π is one of the four polarization Bell states of Eq. (1.2)and |ψ+⟩ = 1√

2(|rℓ⟩ + |ℓr⟩). For a given momentum state |ψ+⟩, the discriminationof the four hyperentangled states is equivalent to distinguish among the four Bell

1.4 Hyperentanglement for quantum information 9

polarization states. The method is based on the following equations:

|Φ±⟩|ψ+⟩ = 12

[±|σ+⟩A|τ±⟩B ∓ |σ−⟩A|τ∓⟩B + |τ+⟩A|σ±⟩B − |τ−⟩A|σ∓⟩B],

|Ψ±⟩|ψ+⟩ = 12

[±|σ+⟩A|σ±⟩B ∓ |σ−⟩A|σ∓⟩B + |τ+⟩A|τ±⟩B − |τ−⟩A|τ∓⟩B] .(1.14)

which allows to express the four possible states |Π⟩π|ψ+⟩k in terms of the singlephoton Bell basis |σ±⟩ = 1√

2 [|H⟩|ℓ⟩ ± |V ⟩|r⟩] , |τ±⟩ = 1√2 [|V ⟩|ℓ⟩ ± |H⟩|r⟩]. Each

product state on the r.h.s. of Eq. (1.14) identifies unambiguously one of the states|Ξ⟩. To distinguish among the four Bell polarization states it is sufficient to mea-sure each particle into the single photon Bell states. Projection into |τ±⟩, |σ±⟩ isachieved by implementing a controlled-Not (CNOT) gate between the polarizationand the momentum of each particle which transforms them into separable states.By using a half waveplate (with the optical axis oriented at 45 with respect tovertical direction) on the |ℓ⟩ mode [100] were able to implement a CNOT gate withmomentum and polarization playing the role of control and target, respectively. Therole of polarization and momentum is exchanged by sing a PBS to implement theCNOT [99]. A similar scheme with polarization-time HE state was implemented by[101].

Bell state analysis is crucial for dense coding [25], one of the fundamental QIprotocols. It works as follows: two observers, Alice and Bob, share an entangled pairand each observer possesses a particle. One observer, say Bob, can encode a 2-bitmessage by applying one of four unitary operations on his particle, which he thentransmits to Alice. Alice decodes the message by performing a Bell state analysis.Since a deterministic discrimination of all the four Bell states with linear optics isnot possible the attainable channel capacity is reduced from 2 to log2 3 ≃ 1.585bits. As explained above, entanglement in an extra-degree of freedom enables thecomplete and deterministic discrimination of all Bell states. Using a polarization-OAM hyperentangled state a dense-coding experiment breaking the conventionallinear-optics threshold was reported [102].

1.4.3 Quantum computing

Hyperentanglement or, in general, the possibility of encoding more qubits in differ-ent DOFs of the same particle is a useful tool for quantum computation (QC). Therealization of multiqubit states can be achieved with relevant advantages in termsof generation rate and state fidelity, compared to multiphoton states. Indeed, byincreasing the number of qubits encoded in different DOFs of the same particle, theoverall detection efficiency and hence the repetition rate of detection is constant,since it scales as ηN (N being the number of photons and η the detector quantumefficiency). Furthermore, an entangled state built on a larger number of particlesis in principle more affected by decoherence because of the increased difficulty ofmaking photons indistinguishable. However, it is worth to remember that increas-ing the number n of involved DOFs implies an exponential requirement of resources.For instance, 2n modes must be exploited to encode n qubits into a photon. How-ever, according to the current optical technology, working with few DOFs (such asn = 2, 3, 4) offers still more advantages than working with a corresponding numberof photons, because of the higher repetition rate and state generation and detection


efficiency. On a medium-term time scale a hybrid approach to QC (i.e., multi-DOFand multiphoton states) may represent a convenient solution to overcome the struc-tural limitations in generation and detection of quantum photon states.

Several quantum algorithms have been realized by exploiting multi-DOF statesin the one-way framework of QC [26]. Cluster states are particular quantum statesassociated to a graph with N vertex and L links. A qubit in the state |+⟩ =

1√2(|0⟩ + |1⟩) is associated to each vertex and while a Controlled-Z (CZij) gate:|0⟩i⟨0|11j + |1⟩i⟨1|(σz)j is associated to each link. Cluster states represent the basicresource for the realization of a quantum computer operating in the one-way model.In the standard QC approach any quantum algorithm can be realized by a sequenceof single-qubit rotations and two-qubit gates on the physical qubits. Deterministicone-way QC is based on the initial preparation of the physical qubits in a clusterstate, followed by a temporally ordered pattern of single-qubit measurements andfeed-forward operations. In this way, nonunitary measurements on the physicalqubits correspond to unitary gates on the logical qubits.

A HE state can be transformed into a more general cluster state by applyingsuitable quantum gate operations between qubits belonging to the same particle.With two-photon four-qubit cluster states built from polarization-path HE states,the Grover algorithm and a CZ gate [103], a generic single qubit rotation withactive feed-forward [50], a CNOT gate and the Deutsch algorithm [36, 104, 105]were implemented.

More complex algorithms have been realized with 6-qubit cluster states. Forinstance, in [36] an all-optical implementation of the Deutsch-Josza (DJ) algorithmfor n = 2 bits is presented. The DJ algorithm allows to discriminate in one runif a boolean n-bit function f is constant or balanced (i.e. it takes the value 0 onhalf inputs and the value 1 on the remaining halves). Classically, 2n−1 + 1 runsof the algorithm are necessary to deterministically solve the problem. At variancewith the simple case n = 1, the DJ algorithm allows to take advantage of theexponential growing of the computational speed-up for increasing values of n. Thecorrect output is identified at a frequency of almost 1 kHz without feed-forward,a result which overcomes by several orders of magnitude what is usually achievedwith a six-photon cluster state created by SPDC.

1.5 Hyperentanglement source

The SPDC source used in this work [78] is based on the simultaneous entanglementof 2 photons in the polarization-longitudinal momentum DOFs. The scheme of thesource is shown in Fig. 1.4. Polarization entanglement is created by double exci-tation (back and forth, after reflection on a spherical mirror) of a 1 mm Type IBBO crystal by a UV laser beam. The backward emission determines the so calledV − cone, with SPDC photon polarization transformed from horizontal (H) to verti-cal (V) by double passage of the two photons through a quarter waveplate (QWP).The forward BBO emission corresponds to the H − cone. Temporal and spatialsuperposition guarantees indistinguishability of the two emission cones and allowsfor the creation of the polarization entangled state 1√

2(|H⟩A|H⟩B + eiγ |V ⟩A|V ⟩B),by assuming the following relations between physical and logical qubits: |H⟩ → |0⟩,|V ⟩ → |1⟩.

1.5 Hyperentanglement source 11

Figure 1.4. Source of hyperentangled photon states. The relative phase between the|HH⟩AB and |V V ⟩AB contributions can be adjusted by translation of the spherical mirror.A lens L located at a focal distance from the crystal transforms the conical emission into acylindrical one. The dimensionality of the state may be increased by selecting further pairsof correlated modes on the mask.

The two photons are emitted with equal probability over symmetrical directionson the overlapping cone surface then transformed into a cylinder by the lens L[See. Fig. 1.4]. By selecting different pairs of correlated emission modes withsingle mode fibers [79] or with a 4-hole screen [80] path- (longitudinal momentum-) entanglement is created. In the experiments presented in this thesis, the state

1√2(|r⟩A|ℓ⟩B + eiδ|ℓ⟩A|r⟩B) has been generated by selecting 2 pairs of correlated

modes. Here |r⟩ (|ℓ⟩) stands for the optical path followed by the photons in theright (left) direction, with the following relation between physical states and logicalqubits, |r⟩ → |0⟩, |ℓ⟩ → |1⟩. The obtained HE state is written as follows:

|HE4⟩ = 12

(|HH⟩AB + eiγ |V V ⟩AB)⊗ (|rℓ⟩AB + eiδ|ℓr⟩AB) (1.15)

By selecting only a single pair of correlated photons it is possible to obtain a two-qubit Bell state encoded in the polarization DOF of the particles. The single qubitof this Bell state can be manipulated by using suitably rotated waveplates. Thiscorresponds to implement single qubit local operations.

The above described scheme has also been used to explore a higher-dimensionalHilbert space [36, 53, 106]. With a larger number of optical paths, more qubits maybe added to the state.

Precisely, by selecting four pairs of modes a six-qubit two-photon hyperentangledstate can be generated [106]. In this experiment two qubits were encoded in thepolarization DOF while four qubits were encoded in the path DOF [See Fig. 1.5for more details]. A generalization of the above described scheme, including theenergy-time DOF, has been proposed for the preparation of six-qubit polarization-momentum-time entanglement [107]. In this proposal two qubits were encoded in


Figure 1.5. Four correlated pairs of SPDC modes selected by the holed mask. Photon A(B) can be collected with equal probability into each one of the four modes. This schemeallows to implement a four-qubit hyperentangled state encoded in two different DOFs ex-ploiting the optical path followed by the emitted particles. In fact we can relabel forconvenience the modes 1A, ..., 4A belonging to the A side as |E, ℓ⟩A, |I, ℓ⟩A, |I, r⟩A, |E, r⟩Awhere ℓ (r) and E (I) refer to the left (right) and external (internal) emission modes. Theyare respectively correlated to the B emission modes |E, r⟩B , |I, r⟩B , |I, ℓ⟩B , |E, ℓ⟩B , labeledas 1B , ..., 4B in the figure. The four qubit hyperentangled state is obtained by consider-ing the equally weighted superposition of these four events and can be written as follows:|Ψ⟩k = 1

2∑4j=1 e

iϕj |j⟩A|j⟩B . being |j⟩A (|j⟩B) the A (B) photon mode of the jth modepair and ϕj the corresponding phase.

each DOF. This conceived scheme was based on the exploitation of a Michelsoninterferometer which can be used to add two qubits encoded in the energy-timeDOF.

An ideal source of eight-qubit hyperentagled states could be obtained by con-sidering two qubits encoded in the polarization DOF, four qubits encoded in thepath DOF and two qubits encoded in the energy-time DOF. The main difficulty ofthis scheme would be represented by its non trivial alignment and optimization. Afurther increasing of the state dimensionality, even if possible in principle, would benot easily achievable.

The source described in this paragraph represents a fundamental part of theexperiments performed during my PhD.

Chapter 2

Multipartite photonic quantumstates

2.1 Introduction

The family of symmetric Dicke states [108] has been found to play a major rolein the context of distributed quantum information processing, embodying preciousresources for the realization of important protocols such as the remote cloning ofquantum states and the multi-destinatary quantum teleportation. The crucial rolerepresented by the Dicke states is due to the interesting entanglement structureshared among the qubits of the state. In this Chapter I will describe several experi-ments involving the family of the Dicke states. I start by describing the engineeredsource of Dicke and Phased Dicke (PD) states (Sec. 2.2.1) while the first experi-ment concerns the introduction of white noise in the PD state and the detection ofmultipartite entanglement by using a particular witness operator (Sec. 2.2.2). Therealization of the second experiment has allowed to provide a tomographic char-acterization of the Dicke state upon subjecting part of it to specific single-qubitprojections (Sec. 2.3.1). The last experiment concerns the implementation of twoquantum networking protocols realized by adding a further qubit to the engineeredsource of Dicke state (Sec. 2.4).

2.2 Hyperentangled Mixed Phased Dicke States

2.2.1 4-qubit hyperentangled Phased Dicke states

Let us consider the following state |ξ⟩ ≡ 1√6(|0010⟩ − |1000⟩ + 2|0111⟩). It is easy

to show that the phased Dicke state can be obtained by applying a unitary trans-formation1 U to the state |ξ⟩:

|D(ph)4 ⟩ = Z4CZ12CZ34CX12CX34H1H3|ξ⟩ ≡ U|ξ⟩ (2.1)

where Hj and Zj stands for the Hadamard and the Pauli σz transformations on qubitj, CXij = |0⟩i⟨0|11j + |1⟩i⟨1|Xj is the controlled-NOT gate and CZij = |1⟩i⟨1|11j +

1 We used the transformation given in (2.1) in order to compensate the optical delay introducedby the CX gates in the Sagnac loop of Fig. 2.1b).

13

14 2. Multipartite photonic quantum states

Figure 2.1. a) Engineered source of the state |ξ⟩. The polarization-longitudinal momen-tum hyperentanglement source has been properly modified to engineer the state reportedin Eq. (5.8). The quarter waveplate QWP1 rotates the polarization of the SPDC photonsemitted by the first excitation of the crystal while the quarter waveplate QWP2 allows tounbalance the relative weight between the |HH⟩ and the |V V ⟩ contributions. The ℓ andr modes on the V − cone are intercepted by two beam stops in order to cancel the term|V V ⟩AB|ℓr⟩AB in the HE state (1.15). b) Generation of Phased Dicke state and measure-ment setup. A thin glass plate, placed before the Sagnac interferometer, allows to set themomentum phase δ = π. The Phased Dicke state has been obtained by applying the Uni-tary transformation U , shown in Eq. (2.1), to the state |ξ⟩. The BS allows to implementthe Hadamard gates in the path DOF while the half waveplate (HWP) at 45 (0), inter-cepting both the photons, allows to implement the gates CXA

12CXB34 (CZA12CZ

B

34). ThePauli operators, in the path DOF, have been measured by exploiting the second passagethrough the BS and the thin glass plates ϕA and ϕB . The necessary measurements in thepolarization DOF have been realized by using an analysis setup, the PA box, before thetwo detectors. This is composed by HWP, QWP and polarizing beam splitter (PBS).

2.2 Hyperentangled Mixed Phased Dicke States 15

|0⟩i⟨0|Zj the controlled-Z. I realized the Dicke state by using 4-qubits encoded intopolarization and path of two parametric photons [A and B in Fig. 2.1a)]. The |0⟩and |1⟩ states are encoded into horizontal |H⟩ and vertical |V ⟩ polarization or intoright |r⟩ and left |ℓ⟩ path. Explicitly, the following correspondence between physicalstates and logical qubits has been used: |0⟩1, |1⟩1 → |r⟩A, |ℓ⟩A, |0⟩2, |1⟩2 →|H⟩A, |V ⟩A, |0⟩3, |1⟩3 → |r⟩B, |ℓ⟩B and |0⟩4, |1⟩4 → |H⟩B, |V ⟩B. Accord-ing to these relations the state |ξ⟩ reads:

|ξ⟩ = 1√6

[|HH⟩(|rℓ⟩ − |ℓr⟩) + 2|V V ⟩|rℓ⟩] (2.2)

and may be obtained by suitably modifying the source used to realize polarization-momantum hyperentangled states [48, 53]. In each “ket” of (5.8) the first (second)term refers to particle A (B). A vertically polarized UV laser beam (P = 40mW )impinges on a Type I β-barium borate (BBO) nonlinear crystal in two oppositedirections, back and forth, and determines the generation of the polarization entan-gled state corresponding to the superposition of the spontaneous parametric downconversion (SPDC) emission at degenerate wavelength [See Fig. 2.1a)]. A 4-holemask selects four optical modes (two for each photon), namely |r⟩A, |ℓ⟩A, |r⟩Band |ℓ⟩B, within the emission cone of the crystal. The SPDC contribution, due tothe pump beam incoming after reflection on mirror M , corresponds to the term|HH⟩(|rℓ⟩ − |ℓr⟩), whose weight is determined by a half waveplate intercepting theUV beam (see [109] for more details on the generation of the non-maximally polar-ization entangled state). The other SPDC contribution 2|V V ⟩|rℓ⟩ is determined bythe first excitation of the pump beam: here the |ℓr⟩ modes are intercepted by twobeam stops and a quarter waveplate QWP transforms the |HH⟩ SPDC emissioninto |V V ⟩ after reflection on mirror M . The relative phase between the |V V ⟩ and|HH⟩ is varied by translation of the spherical mirror.

The transformation (2.1) |ξ⟩ → |D(ph)4 ⟩ is realized by using waveplates and one

beam splitter (BS): the two Hadamards H1 and H3 in (2.1), acting on both pathqubits, are implemented by a single BS for both A and B modes. For each controlled-NOT (or controlled-Z) gate appearing in (2.1) the control and target qubit arerespectively represented by the path and the polarization of a single photon: ahalf waveplave (HWP) with axis oriented at 45 (0) with respect to the verticaldirection and located into the left |ℓ⟩ (right |r⟩) mode implements a CX (CZ) gate.

After these transformations, the optical modes are spatially matched for a secondtime on the BS, closing in this way a “displaced Sagnac loop” interferometer thatallows high stability in the path Pauli operator measurements [See Fig. 2.1b)].Polarization Pauli operators are measured by standard polarization analysis setupin front of detectors DA and DB (not shown in the figure). The overall detection rateis ∼ 500Hz. Note that, the |0⟩ (|1⟩) states are identified by the counterclockwise(clockwise) modes in the Sagnac loop. It is worth stressing the high stability allowedin path analysis by the Sagnac interferometric scheme. This particular configuration,operating on both the up- and down-photon of the state, has made possible toperform a detailed investigation of the robustness of a multipartite entangled state.


2.2.2 Structural Entanglement Witness

The generation and detection of multipartite entangled states is a remarkable chal-lenge that needs to be accomplished in order to fully explore and exploit the genuinequantum features of quantum information and many-body physics. So far only alimited number of families of pure multipartite entangled states has been experimen-tally produced. In view of future applications, it is particularly important to testthe robustness of the generated states in the presence of unavoidable noise comingfrom the environment. In this section I will present how we engineered a new familyof multipartite entangled states, how we experimentally introduced certain types ofnoise in a controlled way and tested the robustness properties of the states.

The experimental generation of multipartite entangled states that I present inthis section is based on the hyperentanglement source [48, 52] described in Sec.1.5, which allows to produce symmetric and Phased Dicke states. Dicke stateshave recently attracted much interest, due to their resistance against photon lossand projection measurements [110], and have been produced in experiments withphotons [110, 111, 112]. Phased Dicke states are achieved by introducing phasechanges starting from ordinary Dicke states [113]. They are defined as follows:

|D(ph)4 ⟩ = 1√

6(|0011⟩+ |1100⟩+ |1001⟩+ |0110⟩ − |1010⟩ − |0101⟩), (2.3)

and they can be obtained by the symmetric state |D(2)4 ⟩ by applying a simple

Unitary:

|Dph4 ⟩1234 = Z1Z3|D(2)

4 ⟩1234 (2.4)

They do not belong to the symmetric subspace and offer new possibilities formultipartite communication protocols, in particular because of their degree of en-tanglement which is higher or equal with respect to the symmetric ones [113].

In order to test the presence of multipartite entanglement we may adopt differentkinds of entanglement witnesses [110, 111, 112, 114]. In the performed experiment,the presence of entanglement in the generated Phased Dicke states was tested byadopting a recently proposed class of entanglement witnesses, so-called structuralwitnesses [113]. This class of operator has been extended in order to achieve higherefficiency in entanglement detection. Moreover, we test the robustness of the phasedDicke states by introducing dephasing noise in a controlled fashion and provide ameasurement of the lower bound on the robustness of entanglement. In this waywe provide a new experimental tool to investigate the entanglement properties ofmultipartite mixed states.

The method adopted to create 4-qubit phased Dicke states is based on 2-photonhyperentanglement. This technique makes possible the realization of such multipar-tite states, with relevant advantages in terms of generation rate and state fidelitycompared to 4-photon states. The measurements were performed by a closed-loopSagnac scheme that allows high stability. Moreover, we were able to control thenoise in a photonic 4-qubit experiment.

An entanglement witness is defined as a Hermitian operator W that detectsthe entanglement of a state ρ if it has a negative expectation value for this state,


⟨W ⟩ρ = Tr(ρW ) < 0 while at the same time Tr(σW ) ≥ 0 for all separable states σ[115, 116].

For a composite system of N particles, the structural witnesses [113] have theform

W (k) := 11N − Σ(k) (2.5)

where k is a real parameter (the three dimensional wave-vector transferred in ascattering scenario), 11N is the identity operator and

Σ(kx, ky, kz) = 1B(N, 2)

[cxSxx(kx) + cyS

yy(ky) + czSzz(kz)], (2.6)

with ci ∈ R, |ci| ≤ 1. Here B(N, 2) is the binomial coefficient and the structurefactor operators Sαβ(k) are defined as

Sαβ(k) :=∑i<j

eik(ri−rj)Sαi S

βj , (2.7)

where i, j denote the i-th and j-th spins, ri, rj their positions in a one-dimensionalscenario, and Sα

i are the spin operators with α, β = x, y, z. A suitable structuralwitnessW for the Phased Dicke state can be constructed by considering kx = ky = πand kz = 0:

W = 11N −16

[Sxx(π) + Syy(π)− Szz(0)] . (2.8)

This operator represents a generalization of the one proposed in [113], the expecta-tion value of this witness for the Phased Dicke state is given by Tr(|Dph

4 ⟩⟨Dph4 |W) =

−23 , thus leading to a robust entanglement detection in the presence of noise.

I report in Table 2.1 the experimental values for each operator appearing in theWitness (2.8).

The witness W measured for the Phased Dicke state [117], is

⟨W⟩exp = −0.382± 0.012 (2.9)

Following the approach of quantitative entanglement witnesses [118], we can alsouse the experimental result on the expectation value of the witness to provide a lowerbound on the random robustness of entanglement Er. This is the maximum amountof white noise that one can add to a given state ρ before it becomes separable. When⟨W⟩ is negative, a lower bound on Er(ρ) is given by

Er(ρ) ≥ D|Tr(ρW)|Tr(W)

, (2.10)

where D is the dimension of the Hilbert space on which ρ acts. In the experimentthe witness from Eq. (2.8) and its expectation value given in Eq. (2.14) lead to

Er(ρ) ≥ |⟨W⟩exp| = 0.382± 0.012 (2.11)

Other bounds for different values of q2 are shown in figure 2.3. Equation (2.10)thus relates the value of W in the presence of the collective noise (2.13) with theresilience of entanglement under the presence of a general white noise.


Table 2.1. Experimental values of the operators needed to estimate the structural witnessin Eq. (2.8). The uncertainties are determined by associating Poissonian fluctuations tothe coincidence counts. Here k refers to the longitudinal momentum DOF while π refers tothe polarization DOF. The second passage through the BS allows high stability in the pathPauli operator measurements [See Fig. 2.1b) and Sec. 2.2.1]. Polarization Pauli operatorsare measured by standard polarization analysis setup in front of the detectors.

Operators Involved Local ResultsQubits Settings

1234 (13)k(24)π

Sxx14 X11X (X1)k(1X)π −0.458± 0.013Sxx

24 1X1X (11)k(XX)π 0.531± 0.012Sxx

34 11XX (1X)k(1X)π −0.384± 0.013Sxx

12 XX11 (X1)k(X1)π −0.545± 0.012Sxx

13 X1X1 (XX)k(11)π 0.597± 0.011Sxx

23 1XX1 (1X)k(X1)π −0.620± 0.011Syy

14 Y11Y (Y1)k(1Y)π −0.617± 0.009Syy

24 1Y1Y (11)k(YY)π 0.590± 0.009Syy

34 11YY (1Y)k(1Y)π −0.528± 0.009Syy

12 YY11 (Y1)k(Y1)π −0.550± 0.009Syy

13 Y1Y1 (YY)k(11)π 0.523± 0.010Syy

23 1YY1 (1Y)k(Y1)π −0.425± 0.010Szz

14 Z11Z (Z1)k(1Z)π −0.327± 0.024Szz

24 1Z1Z (11)k(ZZ)π −0.304± 0.024Szz

34 11ZZ (1Z)k(1Z)π −0.314± 0.024Szz

12 ZZ11 (Z1)k(Z1)π −0.354± 0.024Szz

13 Z1Z1 (ZZ)k(11)π −0.308± 0.024Szz

23 1ZZ1 (1Z)k(Z1)π −0.315± 0.024

2.2.3 Decoherence

I will now describe how it is possible to introduce a controlled decoherence into thesystem. Consider a single photon in a Mach-Zehnder interferometer with two arms(left and right). Varying the relative delay ∆x = ℓ−r between the right and left armcorresponds to a single qubit path decoherence channel given by ρ→ (1−p)ρ+pZρZ.The parameter p is related to ∆x: when ∆x > cτ , where τ represents the photoncoherence time, then p = 1

2 , while when ∆x = 0 we have p = 0. This can beunderstood by observing that two time bins exist (one for each path). By varyingthe optical delay, we entangle the path with the time bin DOF. Hence, by tracingover time we obtain decoherence in the path DOF depending on the overlap betweenthe two time bins. In our setup, this can be obtained by changing the relativedelay ∆x = ℓ − r between the right and the left modes of the photons in thefirst interferometer shown in Fig. 2.1b). Since the translation stage moving themirror acts simultaneously on both photons, this operation corresponds to two pathdecoherence channels:

ρ→ (1− q2)2ρ+ q2(1− q2) [Z1ρZ1 + Z3ρZ3] + q22Z1Z3ρZ1Z3 (2.12)


Figure 2.2. Values of q2 corresponding to different values of the path delay ∆x.

where the parameter q2 is related to ∆x in the following way. Let us consider thepath terms in the |HH⟩ contribution in |ξ⟩, namely |ψ−⟩ = 1√

2(|rℓ⟩ − |ℓr⟩). Thedecoherence acts by (partially) spoiling the coherence between the |rℓ⟩ and |ℓr⟩term giving the state 1

2(|ℓr⟩⟨ℓr| + |rℓ⟩⟨rℓ|) − 12(1 − 2q2)2(|ℓr⟩⟨rℓ| + |rℓ⟩⟨ℓr|). By

assuming that for |ψ−⟩ the decoherence (2.12) is the main source of imperfections,the measured visibility2 Vexp(∆x) of first interference on BS may be compared withthe calculated value V = (1−2q2)2: then, the relation between ∆x and q2, shown inFig. 2.2, is obtained. It is worth noting that at ∆x = 0 we have q2 = 0.0175±0.0001which corresponds to a maximum visibility V0 = 0.9313± 0.0005 at ∆x = 0.

The decoherence channel (2.12) acts on the state |ξ⟩. However, it can be in-terpreted as a decoherence acting on the phased Dicke state. Using equation (2.1)and the relations UZ1U† = −Y1Y2 and UZ3U† = Y3Y4, the channel (2.12) may beinterpreted as a collective decoherence channel on |D(ph)

4 ⟩:

|D(ph)4 ⟩⟨D(ph)

4 | →4∑

j=1Bj |D(ph)

4 ⟩⟨D(ph)4 |B†j (2.13)

with B1 = (1 − q2)11, B2 =√q2(1− q2)Y1Y2, B3 =

√q2(1− q2)Y3Y4 and B4 =

q2Y1Y2Y3Y4 (see the Appendix A for a detailed discussion). A collective decoherenceis a decoherence process that cannot be seen as the action of several channels actingseparately on two (or more) qubits.

Two other main sources of imperfections must be considered in our setup (seethe Appendix A for a detailed discussion): the first one is due to a non perfect su-perposition between forward and backward SPDC emission, i.e. between the |HH⟩and |V V ⟩ contributions. This imperfection can be modeled as a phase polarizationdecoherence channel acting on qubit 2: ρ → (1 − q1)ρ + q1Z2ρZ2. By selecting in|ξ⟩ the correlated modes |rl⟩ and by suitably setting the HWP on the pump beamwe obtain the following state: 1√

2(|HH⟩AB + eiγ |V V ⟩AB)|rl⟩. Even in this casethe value of the measured polarization visibility (Vπ ≃ 0.90) can be related to thepolarization decoherence channel as q1 = 1−Vπ

2 ≃ 0.05. The second interference onthe BS (i.e. after the Sagnac loop) has been also investigated. In the measurement

2The measured visibility is defined as Vexp(∆x) = B−CB

where B are the coincidences measuredout of interference (i.e. measured for ∆x much longer than the single photon coherence lenght)and C the coincidences measured in a given position of ∆x.


Figure 2.3. Experimental values of the witness W and the bound on Er as a function ofq2. The curves correspond to the theoretical behaviour obtained by setting q1 = 0.05 andq3 = 0.05.

condition we obtained an average visibility of Vk2 ≃ 0.80 corresponding to a deco-herence channel ρ → (1− q3)2ρ+ q3(1− q3) [Z1ρZ1 + Z3ρZ3] + q2

3Z1Z3ρZ1Z3 withq3 = 0.05.

2.2.4 Measurements

The witness operator (2.8) was measured for different values of q2. The resultsare shown in figure 2.3. The dark curve corresponds to the theoretical expectationobtained by considering all the three decoherence channels described above andsetting q1 = 0.05 and q3 = 0.05 [See Appeandix A for details on the theoreticalcurve]. Notice that the noise parameter for which the witness expectation valuevanishes gives a lower bound on the robustness of the entanglement of the producedstate with respect to the implemented noise. The witness W measured for thephased Dicke state is

⟨W⟩exp = −0.382± 0.012 (2.14)

I have also measured a witness Wmult, introduced in [119], to demonstrate thegenuine multipartite nature of the generated state. This operator is defined asfollows:

Wmult = 2 · 11 + 16

(J2x + J2

y − J4x − J4

y ) + 3112J2

z −712J4

z (2.15)

where J2i = 1+ 1

2 Sii(ki) and J4

i = 1+ Sii(ki)+ 14(Sii)2(ki), i=x,y,z and kx = ky = π,

kz = 0. It comes out that this equation, in terms of the operators Sii(ki) definedin Eq. (2.7), reads:

Wmult = 18

(2 · 11− 2Sxx(π)− 2Syy(π) + Szz(0)

− 7Szzzz − 2Sxxxx − 2Syyyy) (2.16)

2.3 Characterization of the engineered multiDOF Dicke states 21

Table 2.2. Experimentally measured expectation values of collective spin operators for thePhased Dicke state. The uncertainties are determined by associating Poissonian fluctuationsto the coincidence counts.

Operators Local Settings Results

1234 (13)k(24)π

X1X2X3X4 (XX)k(XX)π 0.673± 0.011Y1Y2Y3Y4 (YY)k(YY)π 0.635± 0.009Z1Z2Z3Z4 (ZZ)k(ZZ)π 0.922± 0.010

with Szzzz = Z1Z2Z3Z4, Sxxxx = X1X2X3X4, Syyyy = Y1Y2Y3Y4, here the sub-scripts indicate the qubits involved in the measurement. The measured values ofthe operators Sii(ki) are reported in Table 2.1. By taking into account also theresults reported in Table 2.2, we obtained

⟨Wmult⟩ = −0.341± 0.015 (2.17)

Varying the noise parameter q2, we obtained a negative expectation value of Wmult

for q2 ≤ 0.114, thus proving the existence of genuine multipartite entanglement upto this noise level. This witness allows us to obtain also a lower bound on thefidelity of the generated state (See Appendix A for the details), here I report onlythe obtained result:

⟨Wmult⟩ = −0.341± 0.015 → F ≥ 0.780± 0.005 (2.18)

2.3 Characterization of the engineered multiDOF Dickestates

2.3.1 Tomographic characterization

In this section, I report the indirect characterization of the entanglement-sharingstructure within a four-qubit photonic symmetric Dicke state [120]. The sourcedescribed in the previous section allows to engineer both the Phased and the sym-metric Dicke state. In fact they are related by the single qubit unitary opoerationreported in Eq. (2.4). The approach followed in this section is to characterize thefaithfulness of the Dicke-state generator by projecting out a growing number of ele-ments of the computational register and testing the faithfulness of the reduced statesthus generated to the expected ideal states. In particular, I report the experimentalQST of three-qubit Dicke states with one and two excitations, as well as two-qubit(Bell) states and single-qubit ones. The values off the state fidelity associated witheach family of projected states witnesses the closeness of the original quadripartiteresource to the ideal Dicke one.

The second task of this section moves from the high quality of the three-qubitstates generated as described above to address the monogamy of correlations withina tripartite state. In particular, I provide the first experimental evaluation of therelation formulated by Koashi and Winter in Ref. [121]. In particular, a state-symmetry simplified expression is given for each entry of the Koashi-Winter (KW)


Figure 2.4. Experimental setup for the generation and analysis of three-qubit |W (1,2)⟩states obtained from the projection of the momentum qubit d of the quadripartite Dickestate |D(2)

4 ⟩ onto |r⟩ (left panel) and |ℓ⟩ (right panel), respectively. The red (green) modesrepresent the optical path followed by photon A (B).

relation and it is demonstrated that, for the class of three-qubit Dicke states (withone excitation), the latter can be experimentally tested by making use of only fivemeasurements settings, implemented in our experiment. The reasons behind thedeviations of the experimental value of the KW relation from the expected valuehave been also investigated by providing a plausible analysis.

The Experiment

The four qubits sharing the |D(2)4 ⟩ have been encoded in two DOFs of two photons,

as said. The following relation between the logical and physical qubits has beenused:

a → πA

b → πA

c → kA

d → kB

where πA (kB) represents the polarization (momentum) of photon A (B).In the basis of the physical information carriers, the engineered state reads:

|D(2)4 ⟩=[|HHℓℓ⟩+ |V V rr⟩+ (|V H⟩+ |HV ⟩)(|rℓ⟩+ |ℓr⟩)]/

√6. (2.19)

Key information on the properties of this state can be indirectly gathered by address-ing the family of entangled states generated from |D(2)

4 ⟩ by performing projectivemeasurements on part of the qubit register. Let’s address this point more in detail.

It is straightforward to recognize that

|D(2)4 ⟩ = 1√

2

(|0⟩j |W (2)⟩j + |1⟩j |W (1)⟩j

), (2.20)


Figure 2.5. Tomographically reconstructed density matrix (real part) of |W (2)⟩abc (left)and |W (1)⟩abc (right) obtained by the procedure described in the text. The contributionsof the imaginary part are negligible. I have a state fidelity with the target states F =⟨W (p)|ϱp|W (p)⟩ ≃ 0.87, consistently for p = 1, 2.

with j labeling any of the four qubits in the set a, b, c, d and j standing for thereduced three-qubit set a, b, c, d/j. Here, |W (2)⟩ = (|011⟩ + |101⟩ + |110⟩)/

√3

and |W (1)⟩ = (|100⟩+ |010⟩+ |001⟩)/√

3 are three-qubit Dicke states with two andone excitation, respectively. Hence, the strategy will be to test how close are theexperimentally generated states by projecting out one of the qubits of the computa-tional register to the expected states |W (1)⟩ and |W (2)⟩. On the experimental sidewe projected out the momentum qubit d by suitably selecting the modes emergingfrom the BS, as explained in Fig. 2.4.

I then performed the tomographic reconstruction of the density matrices ϱ1,2describing the states of the remaining three (polarization and momentum) qubitsby projecting each of them onto the elements of a (statistically complete) subsetof 64 states extracted from the 216 ones obtained by taking the tensor products of|H⟩, |V ⟩, |±x⟩, |±y⟩ with |+k⟩ (|−k⟩) the eigenstates of the k-Pauli matrix (k =x, y) with eigenvalue +1 (-1). The two qubits encoded in the polarization DOF wereprojected onto the necessary elements by using a usual analysis setup before eachdetector, composed by a quarter-waveplate, a half-waveplate and a polarizing beamsplitter. The momentum encoded qubit was measured by exploiting the secondpassage through the BS which allows to project onto the eigenstates of the necessaryPauli operators. The results of the tomographic analysis are shown in Fig. 2.5, wherethe value of the state fidelity F = ⟨W (p)|ϱp|W (p)⟩ = (0.87 ± 0.01), consistently forthe two projected states, is also reported.

The study of the inherent entanglement sharing structure within |D(2)4 ⟩ continues

with a further projection. Indeed, by following the same procedure of Eq. (2.20),one finds

|D(2)4 ⟩ = 1√

6(|0011⟩abcd + |1100⟩abcd) +

√23|ψ+⟩ab|ψ+⟩cd (2.21)


Figure 2.6. Experimental setup used for generation and analysis of the two-qubit |ψ+1,2⟩

states obtained from the projection of the momentum qubits c, d of the quadripartite Dickestate |D(2)

4 ⟩ onto |ℓr⟩AB (left) and |rℓ⟩AB (right), respectively. The red (green) modesrepresent the optical path followed by photon A(B).

where we have introduced the Bell state |ψ+⟩=(|01⟩+|10⟩)/√

2. Therefore, by pro-jecting the quadripartite Dicke state onto |01⟩cd or |10⟩cd, we are able to obtain thereduced two-qubit register prepared in |ψ+⟩ab. We have experimentally verified thatthis is indeed the case by projecting the Dicke state created by our state-generationprocess onto |10⟩cd and |01⟩cd and performing a two-qubit QST on the rest of theregister. Here the necessary projections were performed by suitably selecting themodes emerging from the BS. In the previous case we needed to select only onemode since we projected only qubit d. In order to project also qubit c we had toselect an optical mode emerging from the BS also for photon A.

The results of such analysis are given in Fig. 2.7, where I have distinguished thereduced states achieved by projecting onto |10⟩cd or |01⟩cd by labeling them as |ψ+

1 ⟩or |ψ+

2 ⟩, respectively (needless to say, ideally there should be no difference betweenthem). The associated state fidelity ≥ 90%, independently of the projection, thusdemonstrating the quality of the reduced two-qubit states.

I conclude this part by describing the results of a further reduction performedon the initial four-qubit Dicke resource. I thus implement the projection onto threequbits of the register. I already discussed how the projections onto |01⟩cd or |10⟩cd

were implemented. The last reduction was implemented by selecting the state |1⟩a(b)and performing the QST on the qubit b(a). It is worth reminding that qubits a andb were encoded in the polarization DOF, so the projection of these qubits could beperformed by selecting the physical state |V ⟩ before the detector. In Fig. 2.8 I showthe results of the projections onto the states |101⟩acd, |110⟩acd, |101⟩bcd, |110⟩bcd.Needless to say, the increasing quality of the reduced states is due to the decreasingsize of the computational register, which makes state characterisation much moreagile, and the filtering effects of the projective measurements [111, 112].


Figure 2.7. Tomographically reconstructed density matrices of |ψ+1,2⟩ab achieved by pro-

jecting |D(2)4 ⟩ onto |10⟩cd (|ℓr⟩AB, left) and |01⟩cd (|rℓ⟩AB , right), respectively. We show

the real part of the elements of each density matrix, the imaginary parts being negligible.The fidelities, obtained with respect to the theoretical state, are F|ψ+

1 ⟩= 0.92 ± 0.07 and

F|ψ+2 ⟩

= 0.92± 0.06.

2.3.2 Quantum and classical correlations in a tripartite system

The good quality of the tripartite state produced through the projection abovedescribed allows us to go beyond the reconstruction of the density matrix of thesystem. In particular, in what follows I concentrate on an analysis of the trade-offbetween quantum and classical correlations in a multi-qubit state, a topic that iscurrently enjoying a strong and extensive attention by the community interested inquantum information processing [122]. In the remainder of the section I thus con-sider the KW relation [121] and apply it to the experimental |W (1)⟩ state. In orderto provide a self-contained account of our goal, in what follows I briefly introduce

Figure 2.8. Tomographically reconstructed density matrices by projecting three qubits ofthe Dicke states onto the shown states. The fidelities have been calculated with respect tothe expected theoretical states. I show the real part of the elements of each density matrix,the imaginary parts being negligible.


the KW relation and identify its context of relevance.

The KW relation

Quantum correlations beyond entanglement have been the centre of an extensivequest for the understanding of their significance, relevance and use in the panoramaof quantum-empowered information processing and for the study of non-classicality [122].Such investigation passes through the ab initio definition of quantum correlations[123, 124, 125, 126], their evaluation for different classes of systems [126], and theiroperational interpretation given in terms of the role that they have in quantum com-munication and computation protocols [127, 128, 129]. In this context, in Ref. [121]the KW relation was introduced to capture the trade-off between entanglement andclassical correlations in a pure tripartite system. This has been recently used as thestepping-stone for various results, such as conservation laws [130].

The KW relation is based, in some sense, on the monogamous nature of entan-glement: if a system α is correlated with both β and γ, it cannot be maximallycorrelated with any of them [131]. The KW relation shows that the degree of cor-relations that, say, β can share with the other two is limited by its von Neumannentropy. In more quantitative terms and considering only pure tripartite states forthe moment, the KW relation reads

J(β|α) + E(β, γ) = S(β), (2.22)

where any permutation of the systems’ indices will lead to similar equalities. Here,E(β, γ) is the entanglement of formation shared by β and γ [132], S(B) = −Tr[ρβ log2(ρβ)]is the von Neumann entropy of β only and J(β|α) ≡ maxEα J(β|Eα) measures theamount of classical correlations within the state of systems α and β upon the perfor-mance of a measurement (described by the positive-operator-valued-measurementEα) over α. Here, J(β|Eα) = S(β) − S(β|Eα), with S(β|Eα) the quantum con-ditional entropy of the state of system β [133]. The maximization inherent in thedefinition of J(β|α) is necessary to remove any dependence on the specific choice ofEα. Moreover, it was shown in [134] that orthogonal projective measurements areoptimal for rank-2 states and provide a very tight bound for rank 3 and 4, thereforethe maximisation can be performed within this class of measurements.

As mentioned above, by permuting the systems’ labels, we find five more anal-ogous KW relations. Notice that the relations are six because classical correlationsare asymmetrical by definition (as one needs to specify the part the measurementis carried on). If the tripartite state is mixed, the left-hand side of Eq. (2.22) isalways upper-bounded by the right-hand side. In the last part of the section I usethe quantity defined as the difference between the right and left-hand side of Eq.(2.22), that is

KW = S(β)− J(β|α)− E(β, γ). (2.23)

Needless to say, the value achieved by a pure state is zero.

Experimental investigation of the KW relation

The maximization necessary for the evaluation of the amount of classical correlationsmakes the explicit evaluation of the KW relation difficult, in general. However, for


the particular case of our experimental investigation, state symmetries come toour aids. Indeed, the three-qubit reduced states ϱ1,2 that I have achieved startingfrom |D(2)

4 ⟩ has a large overlap with the ideal |W (1,2)⟩. We can thus exploit thesymmetries inherent in the latter to derive a manageable expression for the KWrelation evaluated over the experiment state. Needless to say, we will have thecaveat of dealing with the KW inequality rather than Eq. (2.22). In order to fixthe ideas, we will consider the subspace of the three-qubit Hilbert space spannedby |001⟩, |010⟩, |100⟩, so that our results will be valid for |W (1)⟩ (although similarconclusions can be drawn for |W (2)⟩). We start evaluating the quantities enteringEq. (2.23) for a state of systems α, β and γ of the following form

σ=

0 0 0 0 0 0 0 00 p c 0 c 0 0 00 c p 0 c 0 0 00 0 0 0 0 0 0 00 c c 0 p 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

, (2.24)

where we deliberately keep all the quantities symbolic so as to have an expressionthat depends explicitly on the populations of the state |001⟩, |010⟩, and |100⟩ andon the respective coherences. We get

S(β) = −p(2 + 3 log2 p),

E(β, γ) = −12

(1 +√

1− 4p2) log2

[12

(1 +√

1− 4p2)]

− 12

(1−√

1− 4p2) log2

[12

(1−√

1− 4p2)],

J(B|A) = −p log2 p− 2p log2 2p

+ 12p

(3p2 −

√4c2p2 + p4) log2

[12

(1−

√4c2p2 + p4

3p2

)]

+ (3p2 +√

4c2p2 + p4) log2

[12

(1 +

√4c2p2 + p4

3p2

)]. (2.25)

The amount of classical correlations were calculated by maximisation over the com-plete set of orthogonal projectors given by Πj = |θj⟩⟨θj | with j = 1, 2, |θ1⟩ =cos θ|0⟩ + eıϕ sin θ|1⟩ and |θ2⟩ = e−ıϕ sin θ|0⟩ − cos θ|1⟩. For the specific state cho-sen it turns out that the maximisation does not depend on the ϕ parameter andis achieved for θ = π/4. Taking an experimental viewpoint, we notice that thepopulations p and coherences c can be written explicitly as function of the followingcorrelators (calculated over the state of the three-particle system)

p = [−⟨ZαZβZγ⟩+ ⟨IαIβIγ⟩ − P (⟨ZαZβIγ⟩)/3 + P (⟨ZαIβIγ⟩)]/8,c = [P (⟨XαXβZγ⟩) + P (⟨YαYβZγ⟩) + P (⟨XαXβIγ⟩) + P (⟨YαYβIγ⟩)]/24,

where P (·) performs the sum of the permutation over α, β and γ of its argument.With this at hand, it is now clear that the KW relation can be evaluated by means


of only eight correlators. In turn, the latter could be experimentally reconstructedstarting from only five measurement settings [135]. Interestingly, due to the statesymmetries, such correlators are exactly those needed to evaluate the fidelity-basedentanglement witness for |W (1)⟩, when optimally decomposed into local measure-ment settings [135]. As such, a single set of local measurements would allow for thedetermination of the genuine tripartite entangled nature of our projected state [136]and the analysis of the sharing of correlations among its constituents. In the specificcase of our experimental implementation, though, a decomposition into the seveninequivalent settings that can be identified in Eq. (2.26) was more convenient andhas thus been used.

When applied to the experimental state ϱ1, whose experimentally reconstructedcorrelators are given in Table 2.3, with the associations α = a, β = b and γ = c, thecombination of the expressions in Eqs. (2.25) gives us the value KW = 0.04± 0.02,which is very close to the value expected for |W (1)⟩. Needless to say, this resultshould be considered as a lower bound to the actual value that Eq. (2.23) wouldtake over an experimentally generated state, in particular due to the lack of perfectsymmetries which played an important role in the determination of the expressionsin Eqs. (2.25). This is, again, fully in line with similar issues affecting the evaluationof entanglement witnesses over experimental states that do not fulfill the expecteddegree of symmetry that characterise ideal states. Indeed, the evaluation of Eq.(2.23) over the tomographically reconstructed state returns a value (averaged overthe various permutations of systems’ indices) of 0.36 ± 0.04, which is significantlydifferent from the one estimated above based on only 8 correlators and the under-lying symmetry assumptions. The argument above can be made more quantitativeby noticing that, while the numerical functions J(a|Πb) and J(b|Πa) evaluatedover ϱ1 are almost independent of ϕ and reach their maximum for θ close to π/4, thesame does not apply for classical correlation involving the c qubit, thus witnessingthe asymmetry of the experimental state. This can be ascribed to the fact thatqubit c is physically encoded in the momentum DOF of a light mode, differentlyfrom a and b, which are both encoded in polarization.

While we are currently working on a formulation that could encompass possibleasymmetries in the tested state yet retaining the handiness of an expression thatcould be experimentally determined by only a handful of measurement settings, we

Table 2.3. Table of the correlators needed in order to evaluate the KW relation forsymmetric three-qubit states leaving in the single-excitation subspace. The experimentalvalues are reported with their associated uncertainties determined by associating Poissonianfluctuations to the coincidence counts needed to reconstruct each correlator.

Correlator ValueZaZbZc 0.87± 0.02ZaZbIc 0.35± 0.04ZaIbIc 0.26± 0.04XaXbZc 0.55± 0.04YaYbZc 0.70± 0.03XaXbIc 0.66± 0.03YaYbIc 0.52± 0.04


mention that a model that is in better agreement with the experimental values ofthe KW would involve a mixture of |W (1)⟩ with a fully mixed state of three qubits.This model is motivated by considering that, experimentally, such projected three-qubit state was created by performing a measurement over a quadripartite havinga non-ideal estimated fidelity greater than 78% with the ideal Dicke resource andthat, for all practical purposes, can be rightly written as ρDicke = p(|D(2)

4 ⟩⟨D(2)4 |) +

1−p16 I16 [117, 136], where the statistical mixture parameter most compatible with

the above mentioned fidelity is p = 0.765±0.005. It is straightforward to check thatsuch a model provides a value of Eq. (2.23) equal to 0.25±0.01, that is significantlycloser to the actual experimental value, still using only eight correlators overall. Bygeneralising this model to anisotropic noise added to an ideal |W (1)⟩ state, we willbe able to come up with a broadly applicable expression for the handy experimentalestimate of the expectation value of Eq. (2.23).

2.3.3 Entanglement witnesses

I performed further tests in order to determine the closeness of the state experimen-tally produced to the ideal Dicke state and characterize its entanglement-sharingstructure.

Testing the Multipartite Entanglement

First, I have ascertained the genuine multipartite entangled nature of the state athand by using tools designed to assess the properties of symmetric Dicke states [113,119, 137]. I have considered the multipartite entanglement witness

Wm = [2411 + J2x Sx + J2

y Sy + J2z (3111− 7J2

z )]/12, (2.26)

which is specific of |D(2)4 ⟩ [119] and requires only three measurement settings. Here,

Sx,y,z=(J2x,y,z−11)/2 with Jx,y,z=

∑i∈Q σ

x,y,zi /2 collective spin operators, σj (j=x, y, z)

the j-Pauli matrix and Q = a, b, c, d. The expectation value of Wm is positiveon any bi-separable four-qubit state, thus negativity implies multipartite entangle-ment. Its experimental implementation allows to provide a lower bound to thestate fidelity with the ideal Dicke state as F

D(2)4≥ (2− ⟨Wm⟩)/3. When calculated

over the resource that I have created in the lab, I achieve Wm= − 0.341 ± 0.015,which leads to F

D(2)4≥ (78 ± 0.5)%. The genuine multipartite entangled nature of

our state is corroborated by another significant test: I consider the witness test-ing bi-separability on multipartite symmetric, permutation invariant states like our|D(2)

4 ⟩ [111, 112, 137]

Wcs(γ) = b4(γ)11− (J2x + J2

y + γJ2z ) (γ∈R). (2.27)

Here b4(γ) is the maximum expectation value of the collective spin operator J2x+J2

y +γJ2z

over the class of bi-separable states of four qubits and can be calculated for any valueof the parameter γ. [137]. Finding ⟨Wcs(γ)⟩<0 for some γ implies genuine multi-partite entanglement. The witness requires only three measurement settings andis thus experimentally very convenient. The bi-separability bound bn(γ) is now afunction of parameter γ and can be calculated numerically using the procedure de-scribed in Ref. [137]. In general, bn(γ)<bn(0) for γ < 0. Consequently, we restrict


-2.5 -1.5 -0.5

-1.0

-0.6

-0.2

uncertainty

Figure 2.9. Functional form of ⟨Wcs(γ) =⟩ against γ, as determined by the measuredexpectation values of collective spin operators (cf. Table tavola). A negative value of⟨Wcs(γ)⟩ signals genuine multipartite entanglement of the state experimental state underscrutiny. The associated experimental uncertainty [See Eq. (2.28)] increases only veryslowly as |γ| grows.

ourselves to the case of negative γ. In Table 2.4 I provide the experimental valuesof ⟨J2

x,y,z⟩ through which we have evaluated Eq. (2.27), which is plotted against γin Fig. 2.9. While ⟨Wcs(γ)⟩ soon becomes negative as γ<− 0.1 is taken, the uncer-tainty associated with such expectation value, calculated by propagating errors inquadrature as

δ⟨Wexpcs (γ)⟩ =

√ ∑j=x,y

(δ⟨J2j ⟩)2 + γ2(δ⟨J2

z ⟩)2, (2.28)

grows only very slowly with γ, therefore signaling an increasingly significant viola-tion of bi-separability.

Testing the Entanglement in the projected states

In order to provide an informed and experimentally not-demanding analysis onthe state being generated, we have decided to resort to indirect yet highly signif-icant evidence on its properties. In particular, we have exploited the interestingentanglement structure that arises from |D(2)

4 ⟩ upon subjecting part of the qubitregister to specific single-qubit projections. In fact, by projecting one of the qubitsonto the logical |0⟩ and |1⟩ states, we maintain or lower the number of excita-tions in the resulting state without leaving the Dicke space, respectively. Indeed,

Table 2.4. Experimentally measured expectation values of collective spin operators forthe symmetric four-qubit Dicke state prepared in our experiment. The uncertainties aredetermined by associating Poissonian fluctuations to the coincidence counts.

Expectation value (with uncertainty) Value⟨J2

x⟩ ± δ⟨J2x⟩ 2.568±0.015

⟨J2y ⟩ ± δ⟨J2

y ⟩ 2.617±0.011⟨J2

z ⟩ ± δ⟨J2z ⟩ 0.039±0.028

2.4 Quantum Networking via Dicke states 31

we achieve |D(2)3 ⟩ = (|011⟩ + |101⟩ + |110⟩)/

√3 when projecting onto |0⟩, while

|D13⟩=(|100⟩ + |010⟩ + |001⟩)/

√3 is obtained when the projected qubit is found in

|1⟩. Needless to say, these are genuinely tripartite entangled states, as it can beascertained by using the entanglement witness formalism. For this task we haveused the fidelity-based witness [138]

WD

(k)3

= (2/3)11−|D(k)3 ⟩⟨D

(k)3 |, (k = 1, 2) (2.29)

which is positive for any separable and biseparable three-qubit state, gives −1/3when evaluated over |Dk

3⟩ and whose optimal decomposition requires five local mea-surement settings [135, 138].

Without affecting the generality of our discussion, here we concentrate on thecase of a qubit-projection on qubit d giving outcome |1⟩d, thus leaving us with state|D(1)

3 ⟩abc. The fidelity-based witness that I have implemented is given in Eq. (2.29)and is decomposed in five measurement settings as [135]

WD

(1)3

= 124

1711+7σz

aσzb σ

zc +3Π[σz

a11bc]+5Π[σaσb11c]

−∑

l=x,y

∑k=±

(11a+σza+kσl

a)(11b+σzb +kσl

b)(11c+σzc +kσl

c) (2.30)

where Π[·] performs the permutation of the indices of its argument. The decompo-sition is optimal in the sense that W

D(1)3

cannot be decomposed with lesser mea-surement settings. Experimentally, I have used the following rearrangement of theprevious expression

WD

(1)3

= 124

1311abc+3σz

aσzb σ

zc−Π[σz

a11bc]+Π[σzaσ

zb 11c]

−2Π[σxa σ

xb 11c]−2Π[σy

aσyb 11c]−2Π[σx

a σxb σ

zc ]−2Π[σy

aσyb σ

zc ],

(2.31)

which was easier to implement with our setup.I have implemented the witness for states obtained projecting qubit d (i.e. mo-

mentum of photon B), achieving ⟨Wexp

D(1)3⟩=− 0.21±0.01 and ⟨Wexp

D(2)3⟩=− 0.24±0.01

(the apex indicates their experimental nature) corresponding to lower bounds forthe fidelity with the desired state of 0.876±0.003 and 0.908±0.003, respectively.

I can thus confidently claim to have a very good Dicke resource, which puts usin the position to experimentally implement the mentioned quantum protocols.

2.4 Quantum Networking via Dicke statesNetworking offers the proven benefits of enhanced connectivity and sharing, oftenallowing for tasks that single parties would not be able to accomplish on their own.This is clearly true for computing, where grids of connected processors outperformthe computational power of single machines or allow the storage and sharing ofmuch larger database. It is thus not at all surprising that similar advantages aretransferred, de facto, to the realm of quantum information processing. Quantumnetworking, where a computational/communication task is pursued by a lattice oflocal quantum nodes sharing (possibly entangled) quantum channels, is emerging as


a realistic scenario for the implementation of quantum protocols requiring registersof medium-to-large size.

In this scenario, photonics is playing a very important role: the high re-configurabilityof photonic setups, together with the technical improvements, have been responsiblefor the birth of a whole new generation of linear-optics experiments (performed bothin bulk optics and, more recently, in the promising context of on-chip integratedphotonic circuits [139, 140]) that have demonstrated multi-photon quantum controltowards the implementation of high-fidelity computing with registers of a size fullyinaccessible until only recently [36, 46, 50, 51, 54]. The Dicke class has emergedas an interesting arena for the characterization of multipartite entanglement inthe proximity of fully symmetric states and its robustness to decoherence [117],as well as a potentially useful resource for the implementation of important pro-tocols for distributed quantum communication, from quantum secret sharing [141]to quantum telecloning (QTC) [142] and (probabilistic) open destination telepor-tation (ODT) [37, 143]. Such opportunities have only been theoretically examinedand indirectly confirmed so far [110, 111, 112].

In this Section, I report the experimental demonstration of 1→ 3 QTC and ODTof logical states using a four-qubit symmetric Dicke state.

The entanglement-sharing structure of the state has been characterized as ex-plained in the previous sections. All those criteria have confirmed the theoreticalexpectations with a high degree of significance. As for the protocols themselves,the qubit state to teleclone/teleport is encoded in an extra degree of freedom ofone of the physical information carriers entering such multipartite resource. Thishas been made possible by the use of a displaced Sagnac loop [56] [cf. Fig. 2.10b)],which introduced unprecedented flexibility in the setting, allowing for the realiza-tion of high-quality entangling two-qubit gates on heterogeneous degrees of freedomof a photon within the Sagnac loop itself. The high fidelities achieved betweenthe experiments and theory (as large as 96%, on average, for ODT) demonstratethe usefulness of Dicke states as resources for distributed quantum communicationbeyond the limitations of a “proof of principle”.

Let us remember the relation between logical and physical qubits:

a → πA

b → πA

c → kA

d → kB

where πA (kB) represents the polarization (momentum) of photon A (B). In thebasis of the physical information carriers, the Dicke state reads

|D(2)4 ⟩=[|HHℓℓ⟩+ |V V rr⟩+ (|V H⟩+ |HV ⟩)(|rℓ⟩+ |ℓr⟩)]/

√6. (2.32)

2.4.1 1→3 QTC and ODT

Telecloning [142] is a communication primitive that merges teleportation and cloningto deliver approximate copies of a quantum state to remote nodes of a network.Differently, ODT [143] enables the teleportation of a quantum state to an arbitrarylocation of the network. Both require, as the key element, shared multipartite


Figure 2.10. Experimental setup. The panel a) shows the displaced Sagnac loop neededto add the qubit to clone/teleport in the QTC/ODT protocol. Panel a) [b)] shows the pathfollowed by the upper [lower] photon A [B]. Here ϕA,B,X are thin glass plates that allow usto vary the relative phase between the different paths within the Sagnac interferometer.

entanglement. A deterministic version of ODT can be formulated using GHZ-typeentanglement [37], while the optimal resource for QTC is embodied by symmetricstates having the form of superpositions of Dicke states with k excitations [142,144, 145, 146]. High-fidelity continuous-variable QTC was demonstrated in [147].Although a symmetric Dicke state is known to be suitable for the implementationof such protocols (ODT being reformulated probabilistically) [110], no experimentaldemonstration has been reported, to the best of our knowledge: in Ref. [110], onlyan estimate of the fidelity of generation of a two-qubit Bell state between sender andreceiver was given, based on data for |D(2)

4 ⟩. As I will explain later, the flexibilityof the setup allows to perform both QTC and probabilistic ODT.

2.4.2 Description of the protocols.

I start by discussing the 1→3 QTC scheme realized using |D(2)4 ⟩, which is a variation

of the protocol put forward in Ref. [142]. I consider the arbitrary qubit state toclone |α⟩X = α|0⟩X + β|1⟩X (|α|2 + |β|2=1), held by a client X. The agents of aserver composed of qubits a, b, c, d and sharing the Dicke resource agree on theidentification of a port qubit p.The state of pair (X, p) undergoes a Bell measurement(BM) performed by subjecting them to a controlled-NOT gate CXXp followed by aprojection of X (b) on the eigenstates of σx (σz). They publicly announce the results


Figure 2.11. Quantum circuit for the realization of 1→3 QTC and ODT. Qubits a, b, c, dare prepared in |D(2)

4 ⟩ while X is the one to clone/teleport. I show the symbol for a CXXb,which is common to both the protocols. For QTC, such gate should be complemented by theprojection of the qubit X(b) on the eigenstates of σx (σz), so as to complete a BM. Finally,operation O depends on the protocol to be implemented: for QTC (ODT), a set of localPauli gates P (single-qubit projections) are required. P is determined by the outcome of theBM to perform in the telecloning scheme, according to the given table. I have introducedthe Bell basis |ψ±⟩, |ϕ±⟩. For ODT (with receiver, say, qubit c), the operations within thedashed boxes should be removed.

of their measurement, which leaves us with⊗j∈Stc

Pj(α|D(1)3 ⟩+ β|D(2)

3 ⟩)Stc ⊗ |ψ+⟩Xp, (2.33)

where Stc=a, b, c, d/p is the set of server’s qubits minus the port p, |D(k)3 ⟩ is a three-

qubit Dicke state with k=1, 2 excitations and the gates Pj (which are identical forall the qubits in Stc) are determined by the outcome of the BM as illustrated in thetable given in Fig. 2.11. The protocol is now completed and a copy of the client’squbit has been cloned into the state of the elements of Stc. In order to see this moreclearly, I trace out two of the elements of such set and evaluate the state fidelitybetween the density matrix ρr of the remaining qubit r and the client’s state, whichreads F(θ)=[9− cos(2θ)]/12, where α= cos(θ/2). Clearly, the fidelity depends onthe input state to clone, achieving a maximum (minimum) of 5/6 (2/3) at θ = π/2(θ = 0, π). This value exceeds the fidelity achieved by a universal symmetric 1→ 3cloner (equal to 7/9) due to the state-dependent nature of the protocol at hand [SeeAppendix B for more details].

I now introduce the Dicke state-based ODT protocol. As for QTC, ODT canbe formulated in terms of a simple quantum game with a client and a multi-partiteserver. As before, the client holds qubit X, into which he encodes the qubit state


a) b)

0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.5 1.5 2.5

0.70

0.80

0.90

Figure 2.12. a) Experimental QTC: for an input state |1⟩X , after the BM on (X, b)giving outcome |ϕ+⟩Xb, the ideal output state reads (2|0⟩j⟨0|+ |1⟩j⟨1|)/3, regardless of thequbit label j=a, c, d [left column of the panel]. I have verified that the state of qubit j,after the application of the experimental QTC protocol, has an almost ideal overlap withthe theoretical state. The right column of the panel shows the experimental single-qubitdensity matrices. b) Theoretical QTC fidelity and experimental density matrices of theclone (qubit a), evaluated for various input states. The measured fidelities, calculatedbetween the experimental input states and the experimental clones (cf. Sec. 2.4.3), areindicated with the associated statistical uncertainties determined by attaching Poissonianfluctuations of the coincidence counts. The dashed line shows the theoretical fidelity forpure input states of the client’s qubit, while the dashed area encloses the values of thefidelity achieved for a mixed input state of X and the use of an imperfect Dicke resourcecompatible with the states generated in our experiment (cf. Sec. 2.4.3)

|α⟩X that he would like to teleport, while the elements of the server share theresource embodied by |D(2)

4 ⟩. The client has the privilege to decide which is theserver’s party r that should receive the qubit to teleport.Both r and p can be any ofthe qubits a, b, c, d and, most importantly, r can be decided at the very last stepof the scheme. The client performs a CXXp (but not a complete Bell measurement,at variance with the previous scheme). At this stage, the information on the qubitto teleport is spread across the server state and this is when the client declares whois the one destined to receive it. Depending on his choice, the server’s members inSodt=a, b, c, d/r, p project their qubits onto |01⟩Sodt

, which leaves us with

[α(|001⟩+ |010⟩)Xpr + β(|111⟩+ |100⟩)Xpr]⊗ |01⟩Sodt. (2.34)

The scheme is completed by projecting the client and port qubits onto | + 1⟩Xp

with |+⟩=(|0⟩+ |1⟩)/√

2. Analogous results are found for server’s projections onto|10⟩Sodt

[See Appendix B for more details].

2.4.3 Experimental implementations of 1→3 QTC.

The setup in Fig. 2.10 a) and b) allows for the implementation of the protocols dis-cussed here. It represents a significant improvement over the scheme used in [117]and gives us the freedom necessary for the faithful realization of our tasks. Theshown displaced Sagnac loop and the use of the lower photon B allow us to add the


client’s qubit to the computational register. This is encoded in the sense of circula-tion of the loop by such field: modes |r⟩ and |ℓ⟩ of photon B impinge on differentpoints of beam splitter BS2, so that the photon entering the Sagnac loop can followthe clockwise path, thus being in the |⟩≡|0⟩ logical state, or the counterclockwiseone, thus being in |⟩≡|1⟩. Note that the photon A does not pass through the BS2.The probability |α|2 of being in the former (latter) state is related to the trans-mittivity (reflectivity) of BS2. This probability can be varyed by using intensityattenuators intercepting the output modes of the BS2. At this stage, the state ofthe register is |D(2)

4 ⟩abcd ⊗ (α|⟩ + eiϕx√

1− |α|2|⟩)X where ϕx is varied by tilt-ing the glass plate placed in the loop. The CXXp gate has been implemented withqubit X as the control, qubit b (i.e. the polarization of photon B) as the port pand taking a HWP rotated at 45 with respect to the optical axes, placed only onthe counterclockwise circulating modes of the displaced Sagnac loop. The secondpassage of the lower photon in the BS2 allows to project qubit X on the eigenstatesof σx

X . To complete the Bell measurement on qubits (X, p) I have placed a HWPand a PBS before the detector in order to project qubit p on the eigenstates of σz

p.The remaining qubits (a, c and d), which are now in state (2.33), embody threecopies of the qubit X. Their quality has been tested by performing QST over thereduced states obtained by tracing over any two qubits. Pauli operators in the pathDOF have been measured by exploiting the second passage of both the photonsthrough BS1. The glass plates ϕA,B allow projections onto 1√

2(|r⟩ + eiϕA(B) |ℓ⟩)c(d).To perform QST on the polarization DOF we used an analyzer composed of HWP,QWP and PBS before the photo-detector. The trace over polarization has beenimplemented by removing the analyzer. To trace over the path, a delayer has beenplaced on either |r⟩ or |ℓ⟩ coming back to the BS1, thus making them distinguishableand spoiling their interference.

In Fig. 2.12a) I show the experimental results obtained for the input states |1⟩X ,when p=b. QST on qubit j=a, c, d shows an almost ideal fidelity with the theoreticalstate, uniform with respect to the label j, thus proving the symmetric nature of QTC.The flexibility of the setup allows us to teleclone arbitrary input states. In orderto illustrate the working principles and efficiency of the telecloning machine, I haveconsidered the logical X states |0⟩X and |+⟩X and |1⟩X (i.e. states determined bytaking θ≃0, π/2 and π) and measured the corresponding copies in qubit a (i.e. thepolarization DOF of photon A). States |0⟩X and |1⟩X were generated by suitablyselecting the modes in the displaced Sagnac. In the first (second) case I consideredonly the modes |⟩ (|⟩), while state |+⟩X was generated by considering both themodes and setting the relative phase with the glass plate ϕX . It is worth notingthat by varying this phase ϕX I could explore the whole phase-covariant case.

In Fig. 2.13 I give the single-qubit density matrix obtained through quantumstate tomography [148] of the receiver’s state in the QTC protocol. The teleclonedstates reported in Fig. 2.13 have been shown in Fig. 2.12b). The three differentinput client’s states are reported. For each of them, I have measured the teleclonedstate on the qubit a.

Although the experimental results are very close to the theoretical expectationsfor F(θ) [cf. Fig. 2.12b)], some discrepancies are found close to θ = π/2. Inparticular, the theoretical expectations seem to underestimate (overestimate) theexperimental fidelity of telecloning, close to θ = π/2 (θ = 0, π). These effects


Figure 2.13. I report the reconstructed density matrices of the telecloned states measuredon qubit a, for three different input client’s states (qubit X).

can be ascribed to the mixed nature of the state of qubit X entering the displacedSagnac loop as well as the non-unit fidelity between the experimental Dicke resourceand the ideal one. In fact, the experimental input state corresponding to takingθ ≃ π/2 has fidelity 0.91± 0.02 with the desired |+⟩X due to depleted off-diagonalelements of the corresponding density matrix. We have thus formulated a morefaithful model for quantum telecloning of dephased input client states based on theuse of a mixed Dicke channel of sub-unit fidelity with the ideal |D(2)

4 ⟩. The detailsof such model are presented in the following paragraphs. Here it is sufficient tomention that, by including the uncertainty in the value of the estimated F

D(2)4

, a θ-dependent region of telecloning fidelities can be identified where the fidelity betweenthe experimental state of the clones and the input client state should be expectedto fall. As illustrated in Fig. 2.12b), this provides a much better agreement betweentheory and experimental data.

Experimental measurement of the client’s qubit

A few remarks are reported in order to explain the experimental measurement of theclient’s qubit X in the actual implementation of the quantum telecloning protocol.

Due to slight unbalance at BS2 of Fig. 2.10b), the blue and yellow paths in theSagnac loop used to encode qubit X are unbalanced. I have thus corrected for suchan asymmetry by first measuring the state of qubit X generated entering the looponly with |r⟩ modes [i.e. the blue path in Fig. 2.10 b)]. I have then done the samewith the |ℓ⟩ modes (yellow paths). Finally, I have traced out the path degree offreedom embodied by |r⟩, |ℓ⟩ by summing up the corresponding counts measuredfor every single projection that is needed for the implementation of single-qubitquantum state tomography, therefore reinstating symmetry.


Fidelity of quantum telecloning for mixed states of the client

Here I provide a model for the solid (red) line of Fig. 2.12b) accounting for thefidelity of quantum telecloning of a client’s mixed state. The evaluation of thetheoretical fidelity of telecloning given in the previous paragraphs does not takeinto account the mixed nature of the client’s state, as well as the non-ideality of theexperimental Dicke channel used for the scheme. As already announced, these twoimperfections are the main sources of discrepancy between the experimental resultsand the theoretical predictions. The model includes these imperfections and allowsfor a more faithful comparison between theoretical predictions and experimentaldata.

The starting point is the observation that mixed input states of the client cancorrespond to telecloning fidelities larger than the theoretical values predicted byF(θ) = [9−cos(2θ)]/12. This can be straightforwardly seen by running the quantumtelecloning protocol with a decohered state resulting from the application of a de-phasing channel to a pure client’s state of the form α|0⟩X +β|1⟩X with α = cos(θ/2)(this is illustrated in Fig. 2.12). Quite intuitively, as the input client’s state loses tiscoherences, the fidelity of telecloning improves. The second observation we makeis that the entangled channel used in the experiment, although being of very goodquality, has a non-unit overlap with an ideal Dicke resource. Taking into accountthe major sources of experimental imperfections, along the lines of the investiga-tion in [117], a reasonable description of the four-qubit resource produced in ourexperiment is the Werner-like state

ρD = p|D(2)4 ⟩⟨D

(2)4 |+ (1− p)11/16 (2.35)

with 0 ≤ p ≤ 1. The entangled Dicke component in such state is evaluated con-sidering that our experimental estimate for the lower bound on the state fidelity isF

D(2)4

= (0.78 ± 0.5). Moreover, I have checked that slight experimental imperfec-tions in the determination of the populations of the input client’s states (within therange observed experimentally) do not affect the overall picture significantly. Wehave thus incorporated the effects of a coherence-depleted input states of qubit Xinto the protocol for 1 → 3 quantum telecloning performed using a mixed Dickeresource as in Eq. (2.35). The dephasing parameter used in the model for mixedclient’s state has been adjusted so that, at θ = π/2, we get the real part of the ex-perimentally reconstructed off-diagonal elements of the density matrix of qubit X(fixed relative phases between |0⟩X and |1⟩X do not modify our conclusions). Theresulting state fidelity, shown in Fig. 2.12b), shows a very good agreement with theexperimental data.

2.4.4 Experimental implementations of ODT

In ODT the client holds qubit X, which is added to the computational registerusing the same displaced Sagnac loop discussed so far. The client’s qubit has beenteleported to elements a and b of the server (i.e. the polarization DOF of bothphotons A and B). The necessary CXXp gate has been implemented, as explainedin the previous paragraph, by taking X as the control and p=b as the target qubit.The server’s elements c, d have been projected onto |01⟩cd and |10⟩cd. On theother hand, here the receiver r can be either a or b. Depending on our choice, the


Figure 2.14. I report the reconstructed density matrices of the various receiver states forfour different input client’s states (qubit X) and projections of the server’s qubits onto both|01⟩Sodt

and |10⟩Sodt.

scheme is experimentally implemented by the projections onto | + 1⟩Xa(b) and theimplementation of QST of the teleported qubit b(a). While the projection onto |+⟩Xhas been realized by exploiting the second passage of the lower photon through BS2,a projection onto |1⟩a(b) is achieved by projecting the physical qubit onto |V ⟩a(b). InTable 2.5 the experimental results obtained for several measurement configurationsand different teleportation channels are shown.

In Fig. 2.14 I give the single-qubit density matrix obtained through quantumstate tomography of the receiver’s state in the ODT protocol. The values of statefidelity included in this figure are those reported in Table 2.5.

Table 2.5. Experimental fidelities of the teleported qubit (a or b) with respect to theexperimental state of qubit X (determined by the angle θ). The uncertainties are found byassociating Poissonian fluctuations to the coincidence counts.

Projection θ Fidelity Projection θ Fidelitycd⟨10| 0 Fa=0.93±0.01 cd⟨01| π Fa=0.98±0.01cd⟨10| 0 Fb=0.95±0.01 cd⟨01| π Fb=0.97±0.01cd⟨01| 0 Fa=0.97±0.01 cd⟨10| 1.46 Fa=0.92±0.02cd⟨01| 0 Fb=0.97±0.01 cd⟨10| 1.46 Fb=0.98±0.01cd⟨10| π Fa=0.96±0.01 cd⟨01| 1.37 Fa=0.97±0.02cd⟨10| π Fb=0.98±0.01 cd⟨01| 1.37 Fb=0.96±0.02


2.5 ConclusionsIn this Chapter I have described several experiments based on the engineering ofmultipartite mlutiDOFs photonic states. This states, based on the hyperentangle-ment of two photons in two different DOFs, have been fully characterized by usingseveral entanglement witnesses. The experimental results have shown the high qual-ity of the engineered state.

The KW relation has been experimentally evaluated by means of only eightcorrelators over a tripartite multiDOFs photonic state obtained by subjecting onequbit to a Dicke state. This result has demonstrated also the high flexibilty ofthe experimental setup allowing not only to engineer both Dicke and Phased Dickestates but also to explore the subspaces obtained by subjecting one, two or threequbits to the initial register, i.e. a four-qubit Dicke state.

The high capabilities of the multiDOFs quantum states have been exploited forthe experimental implementation of the QTC and ODT protocols. The engineeredinterferometric setup adopted for these experiments has the necessary flexibility toadd a fifth qubit to the Dicke state and to manipulate each qubit as requested bythe protocols. Our results go significantly beyond the state-of-the-art in the analysisand manipulation of experimental multi-qubit Dicke states and the realization ofinteresting schemes for quantum networking.

Chapter 3

Non-Markovianity

3.1 Introduction: Open quantum system

3.1.1 Decoherence

Decoherence represents a fundamental aspect in any quantum information process[149]. Indeed the unavoidable coupling of a quantum system with the surrondingenvironment brings to quantum noise phenomena. Therefore, it is of crucial impor-tance the study of the decoherence which is the main responsible of the collapsingof many quantum processes into classical ones. In fact the interaction system-environment spoils the coherent superposition of quantum states and thus elimi-nates one of the basic structural elements of quantum theory, i.e. the superpositionprinciple. Hence the decoherence represents a limitation for the implementation ofthe quantum protocols/algorithm.

Let us consider a model where system and environment are represented respec-tively by two single qubit states. In this model the interaction is given by a CNOTgate. The initial density matrix of the system is ρS = |ψ⟩⟨ψ| with |ψ⟩S = α|0⟩+β|1⟩while the environment is in the state |0⟩E . In the considered case, the CNOT hasthe system as control and the environment as target and the process can be de-scribed as follows:

|ψ⟩S ⊗ |0⟩ECNOT−−−−→ α|0⟩S |0⟩E + β|1⟩S |1⟩E . (3.1)

By tracing the environment, the state of the system has become:

ρS =(|α|2 0

0 |β|2

)(3.2)

The simple model allows to explain how the interaction with the environment canspoil the quantum properties of a system. In fact in the considered case the outputstate of the system has no coherence terms. This process can be interpreted asa flow of information between the system and the environment which acquires acertain amount of information about the state of the system after the interaction.Indeed, depending on the state of the system, the environment changes or doesnot change its state. The CNOT gate can be thought as a “measure” performedby the environment and a flow of information takes place from the system to theenvironment.

41

42 3. Non-Markovianity

3.1.2 Evolution of Open Quantum Systems

In the the case of closed quantum systems, there is no flow of informations or energywith another system or with the environment. The evolution of these systems is welldecribed by the Schrödinger equation. In the description of the nature it is realisticto consider the closed quantum systems only as a theoretical assumption. Indeed,the interaction between the system and anohter system, usually embodied by theenvironment, is unavoidable. This interaction may introduce quantum noise anddecoherence as described in the previous section. Hence it is necessary to introducea new formalism in order to describe the open quantum systems [150]. In quantuminformation, the instruments used to study the open quantum system have beennamed quantum operation [151]. They are employed to describe every evolution ofa quantum system and can be introduced by three equivalent approaches reviewedin the following paragraphs.

Assiomatic Definition

A map E , from the density operators of the input space Q1, to the density operatorsof the output space Q2, is a quantum operation if it satisfyes the following properies:

1. the map preserve the Hermitianity.

Tr[E(ρ)] represents the probability that the process described by E happenswhen the initial state is ρ. It follows that 0 < Tr[E(ρ)] < 1, if Tr[E(ρ)] = 1the map is named trace-preserving and the process is deterministic1 while ifTr[E(ρ)] < 1 the map is named non trace-preserving and the process is proba-bilistic.

2. the map has to be linear and convex, hence E(∑

i piρi) =∑

i piE(ρi) withE(0) = 0.

3. the map is completly positive because it maps physical states into physicalstates.

Physical Interpretation

In this case an interaction is introduced between system and environment. Letus consider a system in the state ρS ∈ HS and the environment embodied by anancillary state |e0⟩⟨e0| ∈ HE . If the interaction is described by the unitary transfor-mation U , the output state of the system, obtained after a projective measure onthe ancilla, can be written as:

ρk = TrE [|ek⟩E⟨ek|U(ρS ⊗ |e0⟩E⟨e0|)U†|ek⟩E⟨ek|]Tr[U(ρS ⊗ |e0⟩E⟨e0|)U†|ek⟩E⟨ek|]

=EkρSE

†k

Tr[EkρSE†k]

(3.3)

where EK = |ek⟩U|e0⟩ and Tr[EkρSE†k] represents the probability of measuring the

event k, pk. Thus, the final state is a mixture of the states ρk with associatedprobabilities pk. In this approach, the interaction is equivalent to an aleatory trans-formation of the initial state of the system ρS into final states ρk pk.

1It is worth to remember that a trace-preserving map is not necessary a unitary transformation

3.1 Introduction: Open quantum system 43

Kraus representation

By this approach, the open system S and the enviroment E are considered as asingle closed bipartite system. The interaction is described by a unitary evolutionU acting on the whole system SE:

ρSEU−→ ρ′SE = UρSEU† (3.4)

with ρSE = ρS ⊗ρE (i.e. we consider no correlations in the initial state betweenS and E). Since we are interested only in the evolution of S, it is necessary to traceE:

ρ′S = TrE [U(ρS ⊗ ρE)U†]. (3.5)

The environment is thought to be in a pure state, precisely in the state |0⟩Ehence it is always possible to obtain this situation by using the purification method[149]. By this assumption, the state in Eq. (3.5) becomes:

ρ′S = TrE [U(ρS ⊗ |0⟩E⟨0|)U†] =∑

k

E⟨k|U|0⟩E ρS E⟨0|U†|k⟩E . (3.6)

where |k⟩E is a basis for HE . The terms ⟨k|U|0⟩ are the Kraus operators2

acting on the subspace HS and the evolution of the system can be written as alinear map:

E : ρS → ρ′S =∑

k

EkρSE†k, (3.7)

The map E is a quantum operation and ρ′S =∑

k EkρSE†k is the Kraus representation

of E .The map reported in Eq. (3.7) may be extended to time-dependent evolution

as follows:E(t) : ρS(0)→ ρS(t) =

∑k

Ek(t)ρS(0)E†k(t), (3.8)

and this is called dynamic map.Let us now remember the properties of the map in Eq. (3.7):

• ρ′S is Hermitian if ρS is Hermitian

• ρ′S has unitary trace if ρS has unitary trace

• ρ′S is a non-negative operator if ρS is a non-negative operator.

Master Equation

This approach represents a complementary point of view with respect to the for-malism of the quantum operation which has been presented in the three previousparagraphs. In this case, differential equation are employed in order to describe theevolution of open quantum systems, the main aim of this method is the descriptionof non-unitary dynamics. We consider an open quantum system S coupled with an

2Since U is unitary, it follows that∑

kEkE†

k = 11S


environment E, the whole system belongs to the Hilbert space H = HS⊗HE whichcan be thought as a closed system obeying to the following Hamiltonian:

H(t) = HS ⊗ 11E + 11S ⊗HE +HSE(t). (3.9)

Here HS (HE) represents the free evolution of the system (environment) and HSE(t)describes the interaction.

S changes as a consequence of the internal dynamics HS and because of theinteraction HSE(t). Furthermore, the dynamics is no longer unitary because of thecorrelations between S and E and this represents a difference with respect to theother three approaches.

The master equation is generally expressed in the Lindblad form:

dρ

dt= − i

~[H, ρ] +

∑j

[LjρL†i − L

†jLj , ρ] (3.10)

where , is the anticommutator, H is the hamiltonian of the system, and Lj

are the Lindblad operators representing the coupling between the system and theenvironment. The initial state is represented by a product state of S and E, it isworth noting that in this approach Tr[ρ(t)] = 1 ∀t.

The solution of a master equation is not always easy, and it allows to determinethe temporal evolution of a density matrix, thus the result can be written as adynamic map with Krauss representation: ρ(t) =

∑k Ek(t)ρ(0)E†k(t) where the

Ek(t) are the time-dependent Kraus operators, found by the solution of the masterequation.

The quantum operation can describe the non-Markovian dynamics presented inthe following sections, since they deal with non-continous change of state in thetemporal domain while it is extremely difficult to find master equation for thatkind of dynamics. Obviously, even the quantum operation cannot be applied to thedescription of every evolution of a open quantum system.

3.1.3 Markovian Dynamics

A stochastic process X(t) is considered Markovian when it shows a short memoryduring the evolution. Since these kind of processes forget fastly the past, markovianis often a synonimus of memory-less.

The evolution of the stochastic systems is described by a probabilistic approach.This means that the system, at each step of the evolution, is described by an en-semble of states with a given probabilty ditribution.

Let us give a mathematical description of a Markovian process. X(t) is analeatory variable in the space of the states S and P (X(t) = i) represents the prob-ability of finding this variable in the state i at the time t. In order to describe itsevolution, we can suppose that there is a correlation between the intial state andthe following one. It follows that the probability of finding X in the state j at thestep t+ 1, depends on the n previous states as follows:

P (Xt+1 = j|(Xt = i) ∩ (Xt−1 = i1) ∩ ... ∩ ∩(Xt−n+1 = in−1)) (3.11)

which represents a conditonal probability.


If we introduce in this model the memory loss at each step, we obtain that theevolution depends only on the state at time t and does not depend on the previuoshystory of the system. By this assumption, the evolution reads:

P (Xt+1 = j|(Xt = i)∩(Xt−1 = i1)∩...∩∩(Xt−n+1 = in−1)) = P (Xt+1 = j|(Xt = i)).(3.12)

Eq. (3.12) represents the Markov condition which embodyes the fact that aMarkovian process is short memory or memoryless.

More in general, a Markov process with a finite memory s, is described by atransition function with this form: Wij(s) = P (Xt+s = j|(Xt = i)), ∀t > 0 whereWij(s) is the probability of being in the state i at t and in the state j at t+ s3. Itis worth noting that it depends only on the distance between t and t+ s.

The Markovian processes satisfy the Chapman-Kolmogorov condition. Let usconsider three different moments t1 < t2 < t3 in the evolution of the variable X,associated to the state i1, i2, i3. The probability of going from the state i1 att1 to the state i3 at t3 is obtained by the product of the intermediate transitionsi1, t1 → i2, t2 and i2, t2 → i3, t3, and by summing up the possible interme-diate positions. Thus, the Chapman-Kolmogorov equation can be written also interm of probility function as follows:

Wt1+t3(i1, i3) =∑i2

Wt1(i1, i2)Wt3(i2, i3) (3.13)

where Wij(s) = Ws(i, j).The properties of the Markovian dynamics have been described in the previ-

ous paragraph by considering classical systems. Nevertheless, even the quantumMarkovian processes are characterized by memoryless evolution and have the sameproperties of the classical case but transferred in the quantum framework. By usingthe formalism of the quantum operation, introduced in the Sec. 3.1.2, a Markoviandynamics of an open quantum system S is described by a dynamic map

Φ(t) : ρS(0)→ ρS(t) =∑

k

Ek(t)ρS(0)E†k(t), (3.14)

where ρS(0) is the initial state (t = 0) of the open system S, ρS(t) is the state at thetime t after the interaction with the environment, and Ek(t) is the Kraus operatorrepresenting the interaction.

The action of the dynamic map is usually summarized in the following way:

ρSE(0) = ρS(0)⊗ ρEUnitary evolution−−−−−−−−−−−→ ρSE(t) = U(t, 0)ρSE(0)U†(t, 0)

T rE

y yT rE

ρS(0) −−−−−−−−→dynamic map

ρS(t) = Φ(t)ρS(0)

(3.15)

The map Φ(t) describes the evolution for a given time t ≥ 0 but it is necessaryto vary the parameter t in order to describe a complete evolution of the open systemin an interval [0, T ].

3∑j∈S

Wij(s) = 1


Figure 3.1. a) Markovian Dynamics. In this case there is a flow of energy and coherenceonly from S towards E. b) Non-Markovian Dynamics. In this case there is a flow of energyand coherence S → E , but also S ← E (i.e. there are memory effects).

This can be done by creating a family of dynamic maps Φt|0 ≤ t ≤ T whereΦ(0) is the identity map. The dynamic map Φ, concerning the Markovian processes,must satisfy the assiomatic properties [See Sec. 3.1.2] and the semi-group one:

Φ(t1)Φ(t2) = Φ(t1 + t2), for t1, t2 ≥ 0 (3.16)

which can be written also as:

Φ(t3, t1) = Φ(t3, t2)Φ(t2, t1) =, for t3 ≥ t2 ≥ t1 ≥ 0 (3.17)

In this form, it is equivalent to the Chapman-Kolmogorov equation presented forthe classical case.

It is possible to avoid the memory effect in the dynamic of the system S whenthe decay of the correlation function between S and the environment is faster thanthe time characterizing the evolution of S.

The Markovian dynamic of an open quantum system can be completly describedeven by the master equation formalism [See Eq. (3.10)], but two approximationsare necessary and have to be satisfyed: the interaction is not strong (i.e. Bornapproximation), the property reported in Eq. (3.12) has to be valid (i.e. Markovapproximation). The latter allows to neglect the short-range correlations betweenS and the enviroment.

3.1.4 Non-Markovian Dynamics

The Markovian open quantum systems have been widely studied in the last years,however many physical systems do not satisfy the markovianity conditions and theybehave in a different way. These systems interact strongly with the environmentwith which show correlations. The Non-Markovian picture is used to describe thiskind of systems.

The master equation formalism is not suitable in this case because of the diffi-culty of finding the appropriate differential equation.

Indeed the non-Markovian quantum systems represent a fruitfull field of researchdue to the lack of a complete comprehension of these processes (e.g. as far as Iknown there is not a definition generally accepted). In the first approach, every


master equation which could not be written in the Lindblad form was considereda signature of non-markovianity. In the following studies other features of themarkovian dynamics were found [See Sec. 3.1.3] and other approaches to the non-markovianity were developed. For instance a dynamic map was considered non-Markovian if it did not satisfy the semi-group property [Eq. (3.16) and Eq. (3.17)].

Nowadays a significant difference with respect to the markovianity is considereda singnature of non-markovianity [152, 153].The fundamental idea is that a givensystem is non-markovian if its evolution shows memory effects.

Many physical processes in quantum optics, solid state, quantum chemistry,show correlations with the surrounding environment for long time. Because of thisreason they have to be modelled as non-markovian.

In quantum information, the information encoded in a quatum system is affectedby the decoherence and hence by the interaction with the environment. A modelused to describe these systems is based on the fact that the quantum informationcan flow from the system to the environment [152]. In a Markovian process thesystem losses the information due to the interaction, as if the information wasdissipated in the surrounding enviroment with a unidirectional flow [See Fig. 3.1a)].In a non-Markovian process there can be memory effect and part of the dissipatedinformation can come back to the system [See Fig. 3.1b)].

According to this idea, some proposal have been done in order to define thedegree of non-Markovianity and to provide also a signature of non-Markovianity.The methods that I will present here do not depend on the particular consideredsystem and they do not require the existence of a master equation. The techniqueis based on the measurement of the states of a physical system for several time andon the evaluation of the trace-distance. Given two quantum states of the system ρ1and ρ2, the trace-distance is defined as:

D(ρ1, ρ2) = 12

(Tr|ρ1 − ρ2|) (3.18)

where |A| =√A†A for a generic operatorA. The trace distance satisfies the property

0 ≤ D ≤ 1. Other features are:

• it does not change under unitary transformation, D(Uρ1U†,Uρ2U†) = D(ρ1, ρ2);

• in the case of the completly positive and trace-preserving maps Φ it happensthat D(Φρ1,Φρ2) ≤ D(ρ1, ρ2)

The trace-distance can be considered as a measure of the distinguishability of thestates. The Markovian processes dissipate information and destroy the differencesbetween the physical states. The distinguishability can increase for a given intervalonly in presence of non-Markovian dynamics which implies that there is a feedbackof information from the environment in the evolution of the open quantum system.

The presented method is not considered the last step in the study of the non-Markovian systems because the trace-distance is not the only measure of distancebetween quantum states.

3.1.5 Signatures of Non-Markovianity

The method used in the experiment described in the following section, exploits agenuine trait of the quantum world: the entanglement. Precisely it can be used


!!

Figure 3.2. a)Local interaction of an open quantum system with the surrounding environ-ment. The system is initially maximally entangled with an ancillary system which is alwaysisolated. b)Non-Markovian behaviour of the entanglement during the evolution.

as a witness of the presence of correlations between the open system and the envi-ronment [153] and it can be exploited in this framework also because of its strictrelation with the decoherence. It has been demonstrated in other experiment thatthe entanglement depends on the effect of the environment [56, 154, 155, 156]. Ina Markovian process, due to the leakage of information, the entanglment vanisheswhile in the non-Markovian case the memory effect can cause a revival of it. Thestrategy used to quantify the non-Markovianity of dynamics in presence of entran-glement, is based on the monitoring of the quantum correlations behaviour in acomposite system. Precisely, only a part of it is subjected to local interactions withthe environment. This allows to measure the non-markovianity of a open quantumsystem even when the form of the dynamics is completely unknown. It is necessaryto prepare an intial maximally entangled state of the open quantum system withan ancillary one which has to remain isolated. The source of decoherence is rep-resented by the environment as sketched in Fig. 3.2a). Due to the fact that thetrace-preserving dynamic maps do not increase the degree of entanglement, fromthe Eq. (3.17) it comes out that the decay of the entanglement between the systemand the ancilla has monotonic behaviour for markovian dynamics.

In the case of non-Markovian dynamics a monotonic behaviour is not expectedsince the correlations with the environment can enhance the bipartite entanglementduring the evolution [See Fig. 3.2b)].

The idea presented in this section is that, by measuring the degree of entangle-ment between the open system and the ancilla for different times within an interval[t0, tmax], it is possible to verify the presence of a markovian/non-markovian dynam-ics if the decay is strictly monotonic/if there is at least a revival.

3.2 Experimental quantum simulation of (non-) Markovian dynamics 49

3.2 Experimental quantum simulation of (non-) Marko-vian dynamics

The simulation of quantum processes is a key goal for the grand programme aimingat grounding quantum technologies as the way to explore complex phenomena thatare inaccessible through standard, classical calculators [3, 4, 5, 6, 38, 42, 57, 60, 61].

The simulation of quantum processes has recently been extended to open quan-tum evolutions [157], marking the possibility to investigate important features of theway a quantum system interacts with its environment, including the so-called sud-den death of entanglement induced by a memoryless environment [56]. As said in theprevious section, this interaction can destroy the most genuine quantum propertiesof the system, or involve exchange of coherence between system and environment,giving rise to memory effects and thus making the dynamics “non-Markovian" [150].Characterizing non-Markovian evolutions is currently at the centre of extensive the-oretical and experimental efforts [158, 159]. In this section I report the experimentaldemonstration of the (non-)Markovianity of a process where system and environ-ment are coupled through a simulated transverse Ising model [See Appendix C].By engineering the evolution in a fully controlled photonic quantum simulator, weassess and demonstrate the role that system-environment correlations have in theemergence of memory effects.

To describe the performed experiment I use the paradigmatic description ofopen quantum dynamics which involves a physical system S evolving in a free way,according to the Hamiltonian HS and embedded in an environment E (whose freedynamics is ruled by HE). System and environment interact via the HamiltonianHSE [150], which we assume to be time-independent for easiness of description.While the dynamics of the joint state ρSE is closed and governed by the unitaryoperator Ut = e−ı(HS+HE+HSE)t, the state of S, which is typically the only object tobe directly accessible, is given by the reduced density matrix ρS(t) = TrEρSE(t).For factorized initial states ρSE(0) = ρS(0) ⊗ ρE(0) and moving to an interactionframe defined by the free Hamiltonian of the total system, such reduced evolutioncan be recast in the operator-sum picture ρS(t) =

∑µKµ(t)ρS(0)K†µ, where Kµ

is the set of trace-preseving, non-unitary Kraus operators of S that are responsi-ble for effects such as the loss of populations and coherence from the state of thesystem [151].

The roots of such decoherence mechanism have long been studied, together withtheir intimate connection to the so-called measurement problem and the implica-tions of the collapse of the wave function [160]. Numerous experimentally orientedtechniques have been proposed to counteract decoherence [151, 161, 162]. Yet, acomplementary viewpoint can be taken, where the possibility to engineering struc-tured environments and tailored S-E couplings is seen as a resource to achievelonger coherence times of the system [163], prepare entangled states, perform quan-tum computation, and realise quantum memories. This calls for the exploitationof memory-effects typical of a non-Markovian dynamics as a useful tool for theprocessing of a quantum state. Unfortunately, a satisfactory understanding of non-Markovianity is yet to be reached, which motivates the recent and intense effortsperformed towards the rigorous characterisation of non-Markovian evolutions, theformulation of criteria for the emergence of non-Markovian features, and the propo-


sition of factual measures for the quantification of the degree of non-Markovianity ofa process [152, 153, 164]. Some of them have been recently used in order to charac-terize, both theoretically and experimentally, the character of system-environmentinteractions for few- and many-body quantum systems [158, 159].

3.2.1 The Experiment: non-Markovian dynamics

The realized experiment is based on a photonic setup to simulate a system-environmentcoupling ruled by a transverse Ising spin model [See Appendix C]. Such interactiongives rise to the non-Markovian evolution of S, as witnessed by the non-monotonicbehavior of its entanglement with an ancilla A that is shielded from the environmen-tal effects. Our goal has been the experimental investigation of the fundamentalconnection between system-environment correlations and non-Markovianity.

The simulator that we propose allows for the implementation of various freeevolutions of S and E, as well as the adjustment of their mutual coupling, thusmaking possible the transition from deeply non-Markovian dynamics, all the waydown to a fully forgetful regime. It is realised using different DOFs of the photonicdevice shown in Fig. 3.3, which consists of the concatenation of two Mach-Zehnderinterferometers and a single Sagnac loop. The information carriers are two pho-tons (referred hereafter as “high" and “low"): the system S is embodied by thepolarisation of the low photon (which can be either horizontal |H⟩ ≡ |0⟩ or vertical|V ⟩ ≡ |1⟩), while the environment E is encoded in the longitudinal momentum de-gree of freedom (the path) of the same photon (which will be right |r⟩ ≡ |0⟩ or left|l⟩ ≡ |0⟩). The ancilla A is embodied by the polarisation of the high photon. Highand low photons are emitted by the source of polarization-entanglement describedin Sec. 1.5.

The ancilla is the key tool for our goals. Indeed, to investigate the emergenceof non-Markovianity in the evolution of S due to its interaction with E, we usethe method proposed in Ref. [153]: we focus on the modifications induced by theS-E coupling on a prepared entangled state of S and A. If the S-A entanglementdecays monotonically in time, the dynamics of S is fully Markovian. Differently, iffor certain time-windows there is a kick-back from E that makes such entanglementincrease, the dynamics is necessarily non-Markovian. In fact, if the local actionof the environment is no longer represented by a continuous family of completelypositive maps, the S-A entanglement is no longer constrained to decrease monoton-ically. This is evidence of the flow-back of coherence on the system and results inan increment of the S-A entanglement.

As in other digital quantum simulators, the dynamics is approximated by astroboscopic sequence of quantum gates. Conceptually, the simulation consists ofthe forward evolution of S over discrete time slices [165] according to a Trotter-Suzuki decomposition [166] of the total time propagator. This approach is knownto be effective for quantum simulation [6] and is implemented here by the sequence ofoperations shown in Fig. 3.4a). The free evolution of the environment is accountedfor by the Hadamard gate HE = (σE

z +σEx )/√

2 (σjm is the m = x, y, z Pauli matrix

of qubit j = S,E and Hj is the Hadamard gate of qubit j = S,E), while HS ∝σS

z . The S-E interaction is engineered by implementing the controlled-rotationGES = |0⟩⟨0|E ⊗ 11S + |1⟩⟨1|E ⊗RS(φ), which rotates the system according to thegeneral single-qubit operationRS(φ) = cosφ11S−ı sinφσS

y depending on the state of


Figure 3.3. Experimental setup. The black box shown in the figure represents the sourceused for the generation of the hyperentangled state encoded in the polarization and pathdegrees of freedom of the photons produced by spontaneous parametric down-conversion[See Sec. 1.5]. For our pourposes we need to select only a pair of correlated directions,belonging to the emission cone surface, along which two photons travel, obtaining in thisway the high and low photons shown in red and yellow. The red line represents the highphoton, whose polarisation is immediately detected. The low photon, depicted by the yellowand blue paths, goes through the set of gates acting on its polarisation and momentum,implemented by BS, HWP, and glasses plates ϕi. The optical axis of HWP∗ is kept free soas to implement the last step of the simulation. The thin glass plates ϕ1, ϕ2 and ϕ3 placedin the interferometer allow to set the phase of the environment evolution at each step. TheHadamard gates are implemented by setting ϕ1 = ϕ2 = ϕ3 = 0 [cf. Eq. (3.22)].

the environment. This class of conditional operations is obtained from Hamiltoniangenerators of the two-qubit transverse Ising form [See Appendix C], which motivatesour choice and is thus the class of system-environment interactions that is simulatedin this work. Experimentally, we fix φ = π/4 and the rotation, in this case, readsRS(π/4) = 1/

√2(11S − ıσS

y ) which is equivalent to a σSxHS gate as shown in the

following:

RS(π/4) = 1√2

(1 00 1

)− ı√

2

(0 −ıı 0

)= 1√

2

(1 −11 1

)(3.19)

σSxHS =

(0 11 0

)⊗ 1√

2

(1 11 −1

)= 1√

2

(1 −11 1

)= RS(π/4) (3.20)

By using the identity σSz = HSσS

xHS , the following equivalence can be straight-forwardly proved:

GES = |0⟩⟨0|E ⊗ 11S + |1⟩⟨1|E ⊗ σSxHS = (|0⟩⟨0|E ⊗ σS

z + |1⟩⟨1|E ⊗HS)σSz (3.21)

This result demonstrates that this controlled gate is locally equivalent to the com-position of a controlled Hadamard and a controlled anti-Z (i.e. a gate applying σS

z

only when E is in |0⟩), namely GES ≡ CHkπCZkπ = |0⟩⟨0|E ⊗ σSz + |1⟩⟨1|E ⊗HS .

The extra σSz needed to make the two conditional gates equivalent can be absorbed

in HS (i.e. the evolution of S), which thus becomes 11S .


Figure 3.4. Scheme of principle of the simulation. a) A maximally entangled state of Aand S is prepared. S and E then evolve through the application of free single-qubit gates(HE and HS) and joint two-qubits (CHkπCZkπ) ones. Time increases from bottom to topof the figure. Here πh(ℓ) stands for the polarization of the high (low) photon, while kℓ forthe momentum of the low photon. b) Each block of the scheme is technically realised bythe concatenation of a BS and two HWPs oriented at different angles.

This set of gates is experimentally realised in the photonic simulator as sketchedin Fig. 3.4b). The HE gate is implemented by means of a beam splitter (BS) inconditions of temporal and spatial indistinguishability of the optical modes. Thisscheme allows to evolve the input modes |r⟩ and |ℓ⟩ at each step as

|r⟩ → 1√2

(|r⟩+ eıϕi |ℓ⟩), |ℓ⟩ → 1√2

(|r⟩ − eıϕi |ℓ⟩), (3.22)

where the phases ϕi are varied by rotating thin glass plates intercepting one of theoptical modes entering the BS. In order to perform a more general rotation of Eit will be necessary to unbalance the output modes (using intensity attenuators)making the probability of occurrence of |r⟩ and |ℓ⟩ unequal.

The controlled gate CHkπ is realised by placing a half-wave plate (HWP) onone of the two output modes with optical axis at 22.5 with respect to the verticaldirection. The temporal delay introduced by this waveplate is compensated by afurther HWP on the opposite output mode with optical axis set at 0 with respectto the vertical direction. This implements the CZ gate, as shown in Fig. 3.4b).In order to assess non-Markovianity, ancilla and system are prepared in the maxi-mally entangled state |ϕ+⟩SA

π = (|HH⟩ + |V V ⟩)/√

2. This two-qubit Bell state isengineered using the polarization entanglement source described in Chapter 1. Theenvironment is initialised in |χ⟩Ek = (α|r⟩+

√1− α2|ℓ⟩)E

k (α ∈ R is set by the trans-mittivity of BS1 [cf. Fig. 3.3] and can be varied by placing an intensity attenuatoron one of the two output modes), a state endowed with quantum coherence as it iskey due to the conditional nature of the dynamics that we simulate. The stability,


modular structure and long coherence time of our interferometric setting allows forthe repeated iteration of each block of gates GES(HE ⊗ 11S).

AsA does not evolve with the environment, the polarisation state of the high pho-ton (depicted in red in Fig. 3.3) is immediately detected and only the low photon (inyellow in Fig. 3.3) goes through each sequence of gates at the various steps. A stan-dard optical setup for the performance of quantum state tomography (QST) [148] isused to reconstruct the state of the S-A system after the evolution. The radiationis collected by using an integrated system composed of GRaded INdex (GRIN) lensand single-mode fibre [79, 167], and is then detected by single-photon counters. Foreach step of the simulated dynamics, we measure the state of the environment byprojecting the polarisation qubits on the states |HH⟩, |HV ⟩, |V H⟩, |V V ⟩. Finally,we trace out the degrees of freedom of the S-A system by summing up the corre-sponding counts measured for every single projection needed for the implementationof single-qubit QST. The Pauli operators for E, generated after the first passagethrough the BS1, are measured using BS2. The same procedure is followed toperform the QST of the environmental state at each step. The state of the S-Asystem is reconstructed in a similar way, by summing up the counts collected afterprojecting E onto |r⟩ and |ℓ⟩.

3.2.2 Experimental results: non-Markovian dynamics

The figures 3.5, 3.6 and 3.7 report the results obtained by running through theevolution of the overall system. In order to quantify entanglement we use the en-tanglement of formation EOF(SA) [132, 168] between S and A, which is operativelylinked to the cost of engineering a given state by means of Bell-state resources. Weare also interested in the correlations shared by the environment with the rest of thesystem, hence we evaluate the von Neumann entropy of the environment, definedas S(E) = −Tr[ρE log2(ρE)], which under the assumption of pure total ASE statequantifies the entanglement in the partition AS against E.

The relation formulated by Koashi and Winter in [121] provides a direct con-nection between the entanglement of formation of the system-ancilla state and thevon Neumann entropy of the environment, thus establishing a relation between thecorrelation in the tripartite system ASE. Even for this reason we have chosenthe entanglement of formation and the von Neumann entropy to characterize thesimulated dynamics.

Figure 3.5 shows the experimental entanglement of formation at each time step(black-square points) against the results of a theoretical model (red line) that, in-cluding all the most relevant sources of imperfections, deviates from the ideal picturesketched above. First, the BSs are not entirely polarisation-insensitive: for BS1 andH (V) polarisation the reflectivity over transmittivity ratio R/T = 42/58 (45/55),while for BS2 is R/T = 45/55 (55/45). Second, although the desired input state|ϕ+⟩SA

π is created with high fidelity (≃ 93%, see Appendix C), the entangled-statesource generates spurious |HV ⟩ and |V H⟩ components, accounting for about 5% ofthe total state, which reduce the initial system-environment entanglement to about0.8. The inclusion of such imperfections makes the agreement between theory andexperimental data very good up to the fourth step of our simulation, showing at leastone revival of the S-A entanglement and thus witnessing the non-Markovian natureof the evolution [153]. The fifth experimental point is significantly far from the


1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

EO

F (

SA

)

STEP

Figure 3.5. Results of the simulation of the (non-)Markovian regimes. Theoretical predic-tion (red line) against experimentally inferred data (black squares) for EOF(SA) as againstthe steps of the non-Markovian simulation. Two revivals of EOF(SA) are clearly visible.

theoretical behaviour because of the not-ideal setting of the phases ϕi (i = 1, 2, 3).In fact, we have verified that their values are not completely polarization inde-pendent. This determines a slight difference for the four contributions |HH⟩SA,|HV ⟩SA,|V V ⟩SA,|V H⟩SA entering the state. This imperfection affects the perfor-mance of each Hadamard gate and becomes significant expecially for the last step,where the cumulative effect of three HE gates should be considered. Nonetheless,the last point reveals successfully the occurrence of a second entanglement revival,thus strengthening our conclusions.

Figure 3.6 compares the experimentally inferred EOF(SA) (black squares) tothe von Neumann entropy of the environment SE (green circles) quantifying thecorrelations shared between E and S-A. These figures of merit appear to be per-fectly anti-correlated, thus giving evidence of a trade-off between the amount ofentanglement that S and A can share at the expenses of S-E correlations. Thisstrengthens the idea that correlations with the environment play a fundamentalrole in this process, a point that has been addressed in Ref. [169] where it is shownthat the establishment of system-environment correlations is a necessary conditionsfor the emergence of non-Markovianity. This result can be bridged with our analysisconsidering that the mixed system-environment state of Ref. [169] can be purifiedby enlarging the Hilbert space of the system including an appropriate ancilla.


1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0 EOF (SA)

S (E)

STEP

Figure 3.6. Results of the simulation of the (non-)Markovian regimes. Comparison be-tween the experimental evolution of EOF(SA) (black squares) and S(E) (green circles) rep-resenting between E and S-A. The two evolutions are clearly anti-correlated. In order tomeasure the Pauli operators needed to reconstruct the state of E at the fifth step, a furtherBS will be necessary.

3.2.3 The Experiment: Markovian dynamics

The Markovian counterpart of the system-enviroment dynamics was engineered bysuitably adjusting the setup used for the non-Markovian case. For this experimentwe considered the scheme reported in Fig. 3.4a) but in this case the unitary evolu-tion of E was replaced with an incoherent map that resets the environment into thevery same state at each step of the evolution. As the key role in the S-E interac-tion is played by quantum coherence, the environment was re-set into a completelymixed state. This is realised by spoiling the temporal indistinguishability of theoptical modes entering the BS by mutually delaying the |r⟩ and |ℓ⟩ components. In-tuitively, by making the state of the environment rigid, we wash out any possibilityfor system-environment correlations, thus pushing the dynamics towards Marko-vianity. This intuition is fully confirmed by the experimental evidences collectedvia QST: the black squares in Figure 3.7 show a monotonic (quasi-exponential)decay of EOF(SA) (matching our theoretical predictions, red line), which is in per-fect agreement with the absence of S-E quantum correlations as signalled by the


1 2 3 4 50.0

0.2

0.4

0.6

0.8

1.0

STEP

EO

F (

SA

)

Figure 3.7. Theoretical behaviour (red line) and experimental data (black squares) forEOF(SA) in the Markovian simulation.

positivity of the partially transposed S-E state.

3.3 ConclusionsBy simulating a non-trivial two-qubit coupling model, I have demonstrated the non-Markovianity of the evolution induced on S by a dynamical environment and asystem-environment interaction allowing for kick-back of coherence. By adopting awitness that makes use of the effects that non-Markovianity has on entanglement, Ihave explored experimentally the link between the emergence of non-Markovianityand system-environment correlations. The next step in this endeavour will be theexperimental proof that non-Markovianity can be used as a resource for the ad-vantageous processing of information, such as the preparation of interesting states,along the lines of previous studies on state engineering and information manipula-tion through Markovian processes [170, 171]. Our setup will be particularly wellsuited for this task, in light of the effective control over both system and environ-ment that can be engineered both in space (thanks to the modular nature of thesetting) and in time.

Chapter 4

Noisy Channels

4.1 Introduction

Quantum noise is unavoidably present in any realistic implementation of quantumtasks, ranging from quantum communication protocols [172] to quantum informa-tion processing devices and quantum metrology [173, 174]. The performance andthe optimisation of quantum tasks quite often depend on the level of noise which ispresent in the physical realisation considered.

In this chapter I will describe two experiments dealing with quantum noise.Precisely in the first case a protocol allowing the optimal estimation of the levelof noise in a Pauli channel [151] (Sec. 4.3) has been experimentally implemented,while in the second case it was possible to achieve an optimal flow of informationbetween a sender and a receiver in presence of a depolarizing channel [151] (Sec.4.4).

The noisy Pauli channels (PCs) act on the density operator ρ of a qubit asfollows:

Γp[ρ] =3∑

i=0piσiρσi (4.1)

where σ0 is the identity operator, σi (i = 1, 2, 3) are the three Pauli operatorsσx, σy, σz respectively, and pi represent the corresponding probabilities (

∑3i=0 pi =

1).The family of Pauli channels regards a wide class of noise processes, that includes

several physically relevant cases such as the the dephasing, the bit-flip channels andthe depolarising channel which is defined as reported in Eq. (4.1). Note that, inthe case presented here p0 = 1− p (with p ∈ [0, 1]) and pi = p/3 for i = 1, 2, 3.

4.2 Experimental implementation of a PC and a DC

Different techniques have been exploited to experimentally implement a PC actingon a single qubit state [175, 176, 177, 178, 179].

In the performed experiments, the general Pauli channel (PC) consists of asequence of liquid crystal retarders (LC1 and LC2) [See Fig. 4.1]. The LCs actas phase retarders, with the relative phase between the ordinary and extraordinaryradiation components depending on the applied voltage V . Precisely, Vπ and V11[See Fig. 4.1] correspond to the case of LCs operating as half-waveplate (HWP) and

57

58 4. Noisy Channels

Figure 4.1. Scheme of the implemented Pauli channel. t1, t2,t3 represent the time intervalsof σx, σy or σz activation. Both t1, t2, t3 and the repetition time T can be varied by aremote control.

as the identity operator, respectively. The LC1 and LC2 optical axes are set at 0and 45 with respect to the V-polarization. Then, when the voltage Vπ is applied,the LC1 (LC2) acts as a σz (σx) on the single qubit. It was possible to switchbetween V11 and Vπ in a controlled way and independently for both LC1 and LC2.The simultaneous application of Vπ on both LC1 and LC2 corresponds to the σy

operation. It was also possible to adjust the temporal delay between the intervals inwhich the Vπ voltage is applied to the two retarders. I define t1, t2, t3 respectivelyas the activation time of the operators σx, σy or σz and T as the period of the LCsactivation cycle, as shown in Fig. 4.1.

4.3 Optimal estimation of quantum noise in Pauli chan-nels

It is of great interest to develop experimental methods allowing to estimate thelevel of noise in the system under examination as precisely as possible. QuantumProcess Tomography (QPT) [151], which has already been implemented in variousexperimental realizations [180, 181], represents a well-known method to identify anunknown noise, but it lacks the notion of efficiency. In many realistic scenarios,however, some a priori information on the kind of noise is available and thereforethe problem of measuring it is equivalent to estimate few noise parameters in themost efficient way.

The aim of this experiment is to provide the first genuine application of quantumchannel estimation theory. The experimental realization presented here is based on

4.3 Optimal estimation of quantum noise in Pauli channels 59

a quantum optical setup, but it opens new perspectives of applications to a greatvariety of physical scenarios and quantum technologies, from atomic to solid statesystems.

I have experimentally tested an optimal protocol to estimate the 4 probabilitiespi characterizing the single-qubit PC [182].

The optimal channel estimation scheme is achieved as follows [183]. The inputstate is represented by a Bell state for two qubits, for example the singlet state|ψ−⟩ = 1√

2(|01⟩−|10⟩), where only one of the qubits is affected by the noisy channelwhile the other one is left untouched. In the experiment the 2-qubit Bell state wasencoded in the polarization DOF of two photons, obtained by tracing out the pathDOF in the 4-qubit HE state 1.5.

The optimal measurement consists of a Bell measurement on the two qubits atthe channel output, this corresponds to the projective measurement:

M =|ψ−⟩⟨ψ−|, |ψ+⟩⟨ψ+|, |ϕ−⟩⟨ϕ−|, |ϕ+⟩⟨ϕ+|

. (4.2)

The outcome probabilities then provide an optimal estimation of the channel pa-rameters pi.

This scheme is optimal for any number of input qubits. Actually, no additionalentanglement among the input qubits and no collective measurements at the outputcan increase the efficiency of the present scheme [183]. Moreover, it can be alsosimply generalised to estimate any general noise process of the form (4.1), where theσ operators are replaced by any set of unitary operators Vi such that Tr[ViV

†j ] = 2δij .

The same scheme can also extended to estimate any generalised Pauli channel forquantum systems in arbitrary finite dimension [183].

The method has been first applied to estimate a general Pauli channel, with in-dependent values of the probabilities pi. Then it has been applied to a depolarizingchannel (DC), corresponding to the case of isotropic noise, with p1 = p2 = p3 = p

3 ,where the parameter p completely specifies the channel itself. In this case the proce-dure simplifies and, in the following, I show that only two projective measurements,M ′ = |ψ−⟩⟨ψ−|, 1− |ψ−⟩⟨ψ−|, are needed.

4.3.1 The Experiment: Anisotropic Noise

A general PC was generated by varying the four time intervals t1, t2, t3 and T . Theintervals ti are related to the probabilities pi (i = 1, 2, 3), introduced in (4.1), by thefollowing expression: pi = ti

T . The probability p0 of the identity operator is givenby p0 = 1− δ

T (with δ = t1 + t2 + t3).The optimal noise estimation protocol was implemented by the interferometric

scheme shown in Fig. 4.2. Precisely, a two-photon entangled state source [77]generates the two-qubit singlet state |ψ−⟩ = 1√

2(|HV ⟩AB − |V H⟩AB), where twoqubits are encoded in the polarization degree of freedom, with H (V ) referring tothe horizontal (vertical) polarization of photons A and B. In our setup, the singlequbit noisy channel is operating only on one of the two entangled particles (i.e.photon A).

In order to obtain the probabilities associated to the four projection operatorsM ,I measured the coincidence counts between the two outputs of the BS. In fact, theseprobabilities are related to the interference visibility measured by the interferometer


!

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%

Figure 4.2. Experimental setup. Photons A and B are spatially and temporally super-imposed on a symmetric beam splitter (BS). The optical path delay ∆x allows to varythe arrival time of the photons on the BS. Photons are collected by using an integratedsystem, composed by a GRIN lens (GL) and a single mode fiber, and then detected bysingle photon counters. The same setup allows to perform the Ancillary Assisted QuantumProcess Tomography (AAQPT) after removing the BS. Quantum State Tomography (QST)[148] on the output state is performed by using quarter-waveplates (QWPs), half-waveplates(HWPs) and polarizing beam splitters (PBSs).

in Fig. 4.2. The half-waveplate (HWP∗) and quarter-waveplate (QWP∗) of Fig. 4.2were used to project the noisy state onto the four different Bell states.

Different configurations of the noisy channel were investigated by implementingthe optimal noise protocol estimation for each configuration. A summary of fourrelevant experimental results, corresponding to different probabilities associated tothe Bell states, are given in Fig. 4.3.

In the measurements shown in Fig. 4.3 the case a) correspond to a noiselesschannel (identity transformation) while the cases b), c) and d), correspond to dif-ferent complete noisy channels with p0 = 0 (i.e. I set T = δ). For each process, thefirst column shows the relative weights between the Pauli operators acting in thechannel. Starting from these values it is possible to calculate the theoretical ones.For instance, let us consider the process d) where the σz, σy and σx act respectivelyfor T

8 , 4T8 and 3T

8 . The expected values of pi are, for this process, p0 = 0, p1 = 38 ,

p2 = 48 and p3 = 1

8 . The slight disagreement between the expected theoretical val-ues and the experimental measured ones are mainly due to the finite rise and decaytimes of the electrical signal driving the LC devices.

I have implemented the protocol by using always the same input state andprojecting it on the Bell basis. It is worth noting that this is totally equivalent toenter the PC with the four Bell states and to project them into the |ψ−⟩ state.


Figure 4.3. Experimental probabilities of measuring the four Bell states obtained for fourdifferent cases of anisotropic noise. The black boxes report the corresponding theoreticalvalues. a) Identity: noiseless channel. b) σy: only one Pauli matrix, σy is acting on thestate |ψ−⟩. c) Partially anisotropic DC : σx and σz operate for the same time interval, infact the probabilities of measuring the states |ψ+⟩ and |ϕ+⟩ are equal. d) Totally anisotropicDC : each Pauli operator operates for a different time interval.

4.3.2 The Experiment: Isotropic Noise

The condition t1 = t2 = t3 corresponds to the depolarizing channel, with the threePauli operators acting on the single qubit with the same probability p = δ

T =t1+t2+t3

T . This parameter was changed by fixing the times ti and varying the periodT . The optimal protocol to estimate the value of p was realized by using the Bellstate |ψ−⟩, as mentioned above.

The DC was activated on photon A. In this case the projective measurementM ′ = |ψ−⟩⟨ψ−|, 11 − |ψ−⟩⟨ψ−|, consisting of just two projectors, is sufficient tooptimally estimate p and was performed for several noise degrees. For each level ofnoise, it was possibile to estimate the channel parameter pexp as pexp = Nss

Nss+Cint

where Cint are the coincidences between the two outputs of the BS in interferencecondition and Nss is the number of events in which the two photons are detected onthe same BS output side. Nss was estimated by knowing the amount of coincidencesout of interference. The typical peak interference measured for the state |ψ−⟩ as afunction of the path delay ∆x is shown in Fig. 4.4.

In Fig. 4.5a) I report the experimental values pexp corresponding to the differentvalues of T . It is in good agreement with the expected theoretical behaviour. Inthe corresponding inset I show the pexp errors evaluated by propagating the Cint

and Nss Poissonian errors. In this case the standard deviations follow the behavior


Figure 4.4. Dip (peak) of the measured coincidence counts as a function of the opticalpath delay for a state |ϕ−⟩(|ψ−⟩) entering the BS in absence of noise.

√p(1− p)/N where N is the dimension of the sample used to evaluate p. In the

next paragraph I will present the experimental realization of the Ancillary AssistedQuantum Process Tomography (AAQPT) which has been performed in order toestimate the noise parameter p. I will show that the optimal protocol allows tominimize the standard deviations with respect to the AAQPT.

4.3.3 Ancillary Assisted Quantum Process Tomography

The experimental results, just discussed for the optimal estimation of the depolar-izing channel, have been compared with the probability values of p which can beobtained by exploiting the AAQPT [177, 184, 185].

The action of a generic channel operating on a single qubit can be written asE [ρ] =

∑3i,j=0 χijσiρσj , where the matrix χij characterizes completely the process.

AAQPT is based on the following procedure: i) prepare a two-qubit maximallyentangled state and reconstruct it by Quantum State Tomography (QST) [148]; ii)send one of the two entangled qubits through the channel E ; iii) reconstruct theoutput two-qubit state by QST and obtain, in this way, the matrix χij from thetwo-qubit output density matrix. For a DC, the matrix χij is expressed as [151]:

χT heop =

(1− p) 0 0 0

0 p3 0 0

0 0 p3 0

0 0 0 p3

(4.3)

I implemented the AAQPT algorithm by injecting the state |ψ−⟩ into the DCand I reconstructed by QST the density matrices of the input and output states forseveral noise degrees [see Fig. 4.2]. The experimental matrix χexp was obtained fordifferent values of T and, for each value of T , it was possible to find the parameterp which maximizes the fidelity between the experimental χexp and the theoreticalχT heo

p process matrices. The experimental results are shown in Fig. 4.5b).


Figure 4.5. Noise parameter estimation for the DC case. a) Measured values of pexpvs. δ

T by implementing the M ′ projective measurements. Continuous red line correspondsto the theoretical behavior. Inset: experimental values of the standard deviations for theoptimal protocol implemented by the M ′ projective measurements. They are obtained bypropagating the poissonian uncertainties. The solid line represents the expected theoreticalbehaviour. b) Experimental probabilities associated to the experimental matrix χ vs δ

T .Values of p are obtained by maximizing the fidelity F between theoretical and experimentalmatrix χ. Error bars are calculated by considering the poissonian uncertainty associatedto the coincidence counts, and simulating different matrices of the process, obtaining, inthis way, different values of p. Inset: experimental values of the standard deviations for theAAQPT. These have been calculated by a simulation based on the poissonian uncertaintyassociated to the coincidence counts. The solid line represents the optimal bound.

The standard deviations, shown in the inset in Fig. 4.5b), have been calculatedby a simulation realized by MATHEMATICA 5.0. It was based on the followingprocedure:

• a poissonian uncertainty was associated to the coincidence counts

• hundred different matrices of the process were simulated

• the fidelity between these matrices and the theoretical ones, reported in themain text, was calculated

• the routine NMAXIMIZE allowed to find numerically the value of p maximiz-ing the fidelity for each simulated process

• this sample, composed of hundred values of p, was used to calculate bothaverage and standard deviation.

The experimental values of the inset in Fig. 4.5b) correspond to a sample of∼1600 coincidences per second.

Even in this case the theoretical behaviour is fully satisfied. However, comparingthese results with those obtained by the optimal protocol, I observe that the latterleads to the same results, but with a much lower number of measurements. Infact, in this case, only the two projections M ′ are needed while, to implement theAAQPT algorithm, 16 measurements are necessary. Moreover, by adopting ourexperimental setup I was able to demonstrate that the value of p and the DC actiondo not depend on the input state. In fact the AAQPT was realized with all the


Figure 4.6. Experimental (left side) and theoretical (right side) matrices χ for T = δ andT ≈ 3δ.

four Bell states entering the DC, obtaining the same results of those shown in Fig.4.5b).

It is worth noting that, even if the AAQPT gives a more complete informationon the process compared to the implemented optimal protocol, the latter allowsto achieve a more accurate value of p. The inset in Fig. 4.5a) shows that, forthe optimal protocol, the measured standard deviation reaches the curve given by√p(1− p)/N , where N is the dimension of the sample used to evaluate p. I show

in the inset in Fig. 4.5b) the standard deviations, well above the optimal bound,obtained with the AAQPT. The lower optimal bound represented by the black curveis below the experimental data, demonstrating that AAQPT is far away from theoptimal estimation protocol presented in this section.

4.4 Entanglement assisted capacity for the depolarizingchannel

It is of great importance to design strategies that allow to optimise the flow ofinformation transmitted in presence of noise. In the most general scenario informa-tion can be transmitted by quantum states and several notions of efficiency can be

4.4 Entanglement assisted capacity for the depolarizing channel 65

defined according to the considered task and the resource available along the trans-mission channel. For example, a quantum communication channel can be designedto transmit classical, private classical or quantum information [186], and it can beemployed on its own or with the addition of other resources, such as entanglement.The case of an entanglement assisted quantum communication channel occurs whenclassical information is transmitted and entanglement is a priori available betweensender and receiver [187]. The capacity is then given by the maximum amountof information that can be transmitted over the channel by optimising the inputsignals and the output decoding procedure. In this section I consider the latter sce-nario and I present an experimental demonstration of the performance obtained byan entanglement assisted depolarizing channel for qubits, which allows to achievethe information capacity. The experimental realization presented here relies on aquantum optical implementation, but it lays the ground for new perspectives ofapplications to a great variety of communication scenarios. This is the first exper-imental demonstration of the information capacity for a controlled noisy quantumcommunication channel, since previous experimental realizations consider just thecase of a noiseless channel [102] or classical noise [188].

An entanglement assisted communication scenario is given by a (generally noisy)quantum channel along which quantum states can be transmitted by assuming thatan unlimited amount of noiseless entanglement is a priori available between thesender and the receiver [187]. The entanglement assisted classical capacity (EACC)is then given by the maximum amount of classical mutual information that can betransmitted over such a channel.

In this case the entanglement assisted classical capacity takes the simple form[187]

C = 2 + (1− p) log2(1− p) + p log2(p/3) . (4.4)

The scheme considered in this work is composed of two qubits, initially preparedin the singlet state |ψ−⟩ = 1√

2(|01⟩ − |10⟩), where the states |0⟩ and |1⟩ are abasis for each qubit. The singlet state then represents the noiseless entangled statethat is a priori shared by the sender and the receiver. Classical information isthen encoded by the sender by performing, with equal probabilities, either theidentity or one of the three Pauli operators on his qubit, which is then transmittedalong the depolarizing channel to the receiver. The receiver finally performs a Bellmeasurement in order to retrieve the information encoded in the two-qubit system.This scenario allows to achieve the capacity (4.4) [189].

In this work an experimental scheme corresponding to the above scenario isimplemented, moreover it is demonstrated that the capacity (4.4) can actually beachieved, allowing the optimal transmission of classical information through thechannel and therefore the use of the channel at the best of its possible performances.Then the classical information transmitted through the channel in terms of theclassical mutual information is measured, it can be expressed as

I =∑

x

p1(x)∑

y

p(y|x) log2p(y|x)p2(y)

(4.5)

where x and y are the input/output variables, with corresponding probabilitydistributions p1(x) and p2(y), while p(y|x) represents the conditional probability of


!

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#

!

"

!

"

$

Figure 4.7. I) Spontaneous parametric down conversion (SPDC) source of polarizationentanglement. It has been obtained by selecting only a pair of correlated modes in thehyperentanglement source described in Sec. 1.5. II) Alice encodes two bits of classicalinformation by applying single qubit Pauli transformations on one entangled particle. Thiscan be realized by applying a suitable voltage to the liquid crystal modulator, LC1 andLC2 acting as waveplates and implementing the transformations σi, pi. III) Photon Ais transmitted through a depolarizing channel (DC), here implemented by liquid crystalmodulators LC3 and LC4. The LCs’s activation time corresponding to the three Paulioperators σx, σy, σz is the same, i.e. t1 = t2 = t3.

receiving y given transmission of x. In our experiment the variables x and y corre-spond to the four Bell states, p1(x) = 1/4 and p(y|x) is the conditional probabilityof detecting the Bell state y given that the Bell state x is transmitted.

4.4.1 General scheme

I describe here the general experimental scheme to implementing the scenario de-scribed above. The two-photon Bell states are engineered by exploiting the po-larization entanglement source [77, 48] [See Fig. 4.7I)] which allows to generateaccurately the states |ϕ±⟩ = 1√

2(|00⟩ ± |11⟩), with qubit |0⟩ (|1⟩) corresponding tothe horizontal H (vertical V ) polarization of the photon. The other Bell states|ψ±⟩ = 1√

2(|01⟩± |10⟩) can be obtained by applying simple single-qubit local opera-tions. Let us now assume that the initial state, shared by the sender Alice and thereceiver Bob, is represented by the singlet |ψ−⟩AB = 1√

2(|01⟩ − |10⟩). In this caseAlice encodes classical information by performing local transformations on photonA, which is then transmitted to Bob through the depolarising channel, while photonB belongs to Bob.

In the experiment the information encoding was performed by employing twoliquid crystals (LC1 and LC2) acting simultaneously on photon A. This correspondsto apply the single qubit local operation σi as described in Sec. 4.2. As described inFig. 4.7II), LCs were suitably activated by a remote control in order to perform thefour Pauli operators with equal probabilities. In this way, depending on the trans-

4.4 Entanglement assisted capacity for the depolarizing channel 67

formation implemented by the LCs, Alice encodes two bits of classical informationin the shared Bell state:

11 → 00σz → 01σx → 10σy → 11

The manipulated qubit was then sent through a depolarizing channel (DC).In the performes experiment, the DC was implemented according to the schemeproposed in [182] [See Fig. 4.7III)] and described in Sec. 4.2. The necessary logicaloperations were implemented by careful adjustment of the voltage applied to twoLCs and of the activation interval of each LC system.

Figure 4.8. Experimental setup. Alice and Bob share a two-qubit singlet state. LC1and LC2 implement the four Pauli operators σi with probability pi. The LC3 and LC4systems perform the DC operation. By properly setting both temporal delay and spatialmode superposition, the two photons arrive in the BS in condition of complete undistin-guishability. The noisy state is projected onto the four Bell states by exploiting the BS andthe transformation U . Each polarization analysis (PA) setup consists of a half-waveplate, aquarter-waveplate and a polarizing BS placed before the detectors.

In order to evaluate the joint probabilities, the four possible outputs y need tobe measured for each input state x. This final Bell measurement was realised with aset of projections onto the Bell states |ψ−⟩⟨ψ−|, |ψ+⟩⟨ψ+|, |ϕ−⟩⟨ϕ−|, |ϕ+⟩⟨ϕ+| byusing a suitable interferometric setup and by performing those local transformationswhich allow to select the desired Bell state. The present measurement schemedoes not allow to perform the complete Bell state discrimination realized in otherexperiments based on the simultaneous entanglement of the photons in more degreesof freedom [100, 101, 102].


4.4.2 The Experiment

Let us consider the expression of the classical mutual information given in Eq. (4.6).As mentioned above, after encoding, photon A is transmitted through the depolar-izing channel. The experiment was performed by starting from the shared singletstate |ψ−⟩ and for each one of the other Bell states obtained after the encodingoperation. The DC was performed by two liquid crystal retarders (LC3 and LC4)inserted within the path of one of the two photons. As said, I was able to switchbetween V11 and Vπ in a controlled way and independently for both LC3 and LC4.I could also adjust the temporal delay between the intervals corresponding to a Vπ

voltage applied to the two retarders. The condition t1 = t2 = t3 (i.e the activationtime of the operators σx, σy or σz respectively) corresponds to the case of a DC,here I call pexp = δ

T = t1+t2+t3T . In the experiment, the parameter pexp was varyed

by changing the interval δ for a fixed period T .Fig. 4.8 reports the actual interferometric setup adopted to perform projec-

tive measurements |ψ−⟩⟨ψ−|, |ψ+⟩⟨ψ+|, |ϕ−⟩⟨ϕ−|, |ϕ+⟩⟨ϕ+|. The transformationU , sketched in the same figure, represents the different local unitaries necessary totransform each Bell state |ϕ+⟩, |ϕ−⟩, |ψ+⟩ into the singlet state |ψ−⟩ , i.e. the onlyone that allows to measure coincidences between the two output modes of the BS incondition of complete undistinguishability. They are implemented, as explained inSec. 4.3 by proper setting of the optical axes of a quarter-wave plate and a half-waveplate. In this way I could perform a complete Bell measurement by performing thefour projections at different times.

For each value of the noise degree, the probabilities associated to the measure-ment of each Bell state, corresponding to the conditional probabilities p(y|x) of Eq.(4.6), were obtained by considering the coincidence counts measured by the inter-ferometer for each projection normalized over the results of the four measurements.After the transmission through the beam splitter (BS), photons were coupled intosingle-mode fibers by using GRaded INdex (GRIN) lenses [79, 167] and detected bysingle photons detectors.

4.4.3 Experimental results

Let us consider Eq. (4.6), with x,y=1,..,4 representing the four Bell states. In ourcase 1 → |ψ−⟩, 2 → |ψ+⟩, 3 → |ϕ−⟩, 4 → |ϕ+⟩. Since the probabilities of the fourinput Bell states are equal I have p1(x) = 1

4 . Therefore Eq. (4.6) reads

Imeas = 14∑x,y

p(y|x) log2p(y|x)p2(y)

. (4.6)

The probabilities p(y|x) in the above expression were measured by consideringseparately each Bell state x and by projecting it onto the four Bell states after theaction of the DC. The experiment was carried out for each state obtained after theencoding operation and for several values of the noise degrees, i.e. of the parameterpexp. The values of Imeas were then obtained from the measured probabilities as inEq. (4.6) by summing the contributions of the four Bell states for each value of p.

The experimental data were analyzed by taking into account the non perfectpurity of the actual input singlet state. This was evaluated for the singlet state |ψ−⟩from the visibility V ≈ 94% of the coincidence count peak measured in condition of

4.5 Conclusions 69

temporal and spatial undistinguishability [see the inset of Fig. 4.9]. According tothis value, I could express the actual input state entering the DC as

ρV = 13

(1 + 2V)|ψ−⟩⟨ψ−|+ 2/3(1− V)114

(4.7)

When the visibility V = 1 (V = −0.5) only the contribution of the state |ψ−⟩⟨ψ−|(11/4) is present and a peak (dip) in the coincidence counts is measured. A −50%visibility is expected by the complete polarization mixed state included in the secondterm of Eq. (4.7). In fact, it can be decomposed in the computational basis as(|HH⟩⟨HH| + |HV ⟩⟨HV | + |V H⟩⟨V H| + |V V ⟩⟨V V |)/4, with terms |HH⟩⟨HH|,|V V ⟩⟨V V | giving a −100% visibility, while the terms |HV ⟩⟨HV |, |V H⟩⟨V H| givea 0% contribution to visibility.

The above state can be interpreted as the result of the action of a preliminarDC channel on the pure state |ψ−⟩⟨ψ−|, characterized by the noise parameter p′ =3(1 − V)/4. The action of the sequence of two DCs may be then expressed as aglobal DC with the following global noise parameter

p = Vpexp + p′ . (4.8)

I report in Fig. 4.9 the experimental results for the transmitted informationwith the theoretical value of the capacity (4.4) with p given by Eq. (4.8). The errorbars have been obtained by propagating the poissonian uncertainties associated tothe coincidence counts. The agreement between the experimental data analyzed asexplained above and the theoretical behaviour is very high.

The theoretical curve in Fig. 4.9 allows to single out three working regimescorresponding to three different values of the noise parameter p:

• p = 0 (absence of noise). The maximum allowed transferred informationconsists of 2 bits.

• p = 0.75. The noisy state is given by the two-qubit maximally mixed state 114 .

In these conditions there is no correlation between the output and the inputstate, hence no information can be transmitted.

• p > 0.75. In spite of the large amount of noise a partial revival of the mutualinformation is observable due to the non uniform distribution of the condi-tional probabilities p(y|x).

The capability of exploring the three regimes strongly depends on the purity ofthe entangled input state. It is remarkable that the purity, attainable from thevisibility V in absence of any active noise applied by LC3 and LC4 (correspondingto an effective noise parameter p = 0.03), easily allows to achieve a quite high valueof Imeas. Precisely, in this case we obtained Imeas = 1.655± 0.014 demonstrating ahigh channel capacity of our system.

As a further consideration, in these experimental conditions, the growing ofmutual information occurring for large values of p (> 0.75) may be clearly detected.

4.5 ConclusionsIn this section I have shown how to manipulate a single qubit encoded in the polar-ization DOF of a single photon. The presented tecnique has allowed to optimally


Figure 4.9. Measured values (black dots) of the mutual information and theoretical curve(blue line) for the entanglement assisted capacity for several values of the parameter p whichfully characterize the total amount of depolarizing noise introduced in the experiment. Theerror bars have been obtained by propagating the poissonian uncertainties associated to thecoincidence counts measured in 10 seconds. Inset: peak of the coincidence counts measuredin 2 seconds as a function of the optical path delay ∆ for a state |ψ−⟩⟨ψ−| entering the BSin the absence of controlled noise.

implement a general Pauli channel and a particual case of it, that is the Depolarisingchannel.

An optimal protocol allowing the most efficient estimation of a noisy Pauli chan-nel has been experimentally implemented. The efficiency of this method has beencompared to the one achieved by quantum process tomography, demonstrating thatthe optimal protocol allows to achieve a lower level for the errors and to perform theestimate of the noisy channel with a lower number of measurements. This methodcan be profitably applied when some knowledge on the noise process is availableand can be successfully implemented in quantum-enhanced technologies involvingthe management of decoherence.

With the second experiment I have provided a proof-of-principle experimen-tal demonstration of the entanglement assisted capacity for classical informationtransmission over a depolarizing quantum communication channel, where classicalinformation is encoded locally on a preshared maximally entangled state of twoqubits and a controlled noise is then introduced on the transmitted qubit. Theexperimental implementation of the protocol demonstrates the achievement of theclassical information capacity theoretically predicted for the depolarising channel,

4.5 Conclusions 71

therefore showing the optimal way in which the depolarising channel can be usedwhen classical information is to be transmitted and a priori entanglement is avail-able.

Chapter 5

Other Experiments: briefreview

5.1 Extremal quantum correlations of 2-qubit states

5.1.1 Introduction

Entanglement does not embody the unique way in which quantum correlations(QCs) can be set among the elements of a composite system. When generic mixedstates are considered, QCs are no longer synonymous of entanglement: other formsof stronger-than-classical correlations exist and can be enforced in the mixed stateof a system. Among the quantifiers of QCs proposed so far, quantum discord [190](D) occupies a prominent position.

The study of the connections between entanglement and non-classicality hasseen an increasing interest in the last few years. Several forms of entanglementwitnesses have been proposed to reveal and quantify the entanglement of a par-ticular state or a particular class of states. Non-classicality indicators have beenproposed so as to quantify quantum correlations in two-qubit states. It has beendemonstrated that there are interesting situations in which non-entangled states (i.e.separable) can behave in a non-classical way [126]. Investigating on the interplaybetween quantum correlations and global state mixedness is an interesting topic[191]. Such study is dressed with even deeper fundamental significance when ex-tended to non-classical/not-entangled states. However, the more pragmatic aspectsof such endeavors should be stressed as well, given that non-entangled states canbe useful for quantum computing. In a recent paper [192] a new non-classicalityindicator, named AMID, has been proposed as an “ameliorated” version of mea-surement induced disturbance (or MID), which was originally proposed by Luo in[126]. AMID is a measure based on the concept of perturbation on a bipartitequantum state induced by joint local measurements. It represents an improvementas it embodies a faithful estimator of non-classicality that, at contrast with MID,vanishes exactly for fully classical states. Building on the framework provided bythe theoretical studies in Refs. [192], it has been possible to experimentally navigatethe space of two-qubit discorded states focusing our attention, in particular, on theclass of two-qubit maximally non-classical mixed states (MNCMS), i.e. those statesmaximizing the degree of quantum discord at assigned values of their global von

73

74 5. Other Experiments: brief review

Neumann entropy (VNE).Such a relation has been experimentally verified by adapting the polarization-

momentum hyperentanglement source [193] [See Sec. 1.5]. In particular, I engineermixedness in the joint polarization state of two photonic qubits by tracing outthe path degree of freedom (DOF). The properties of such residual states werethen analyzed by means of the quantum state tomography (QST) toolbox [148]and a quantitative comparison between their quantum-correlation contents and thepredictions on MNCMS was performed.

5.1.2 Quantum Discord and AMID

The discord is associated to the discrepancy between two classically equivalent ver-sions of mutual information [190]. For a bipartite state ρAB the latter is definedas I(ρAB)=S(ρA)+S(ρB)−S(ρAB). Here, S(ρ)=−Tr[ρ log2 ρ] is the VNE of thearbitrary two-qubit state ρ and ρj is the reduced density matrix of party j=A,B.One can also consider the expression J←(ρAB)=S(ρA)−HΠi(A|B) (the one-wayclassical correlation [190]) withHΠi(A|B)≡

∑i piS(ρi

A|B) the quantum conditionalentropy associated with the post-measurement density matrix ρi

A|B=TrB[ΠiρAB]/pi

obtained upon performing the complete projective measurement Πi on system B(pi=Tr[ΠiρAB]). The discord is defined as follows:

D←= infΠi

[I(ρAB)−J←(ρAB)], (5.1)

where the infimum is calculated over the set of projectors Πi. Discord is in generalasymmetric (D← =D→) with D→ obtained by swapping the roles of A and B. Thisoriginates the possibility to distinguish between quantum-quantum states having(D←,D→)=0, quantum-classical and classical-quantum ones, which are states hav-ing one of the two values of discord strictly null, and finally classical-classical statesfor which D←,D→=0, which are bipartite states that simply embed a classical prob-ability distribution in a two-qubit state. The asymmetry inherent in discord wouldlead us to mistake a quantum-classical state as a classical state. In order to bypasssuch an ambiguity we will consider the symmetrized discord D↔= max[D←,D→],which is zero only for classical-classical states.

AMID has been introduced in Ref. [192] as an alternative indicator of QCsdefined as follows:

A=I(ϱAB)−Ic(ϱAB), (5.2)

where Ic(ϱAB)≡supΩI(ϱΩAB) and ϱΩ

AB is the state resulting from the applicationof the arbitrary complete projective measurements Ωkl = ΠA,k ⊗ ΠB,l. Ic is theclassical mutual information (optimized over projective measurements), and canbe seen as a symmetric measure of bipartite classical correlations. A is thus thedifference between total and classical mutual information and has the prerequisitesto be an appropriate measure of QCs.

As shown in Ref. [192], when D↔ and S are taken as quantitative figures ofmerit for QCs and global mixedness, the class of MNCMS consists of four families

5.1 Extremal quantum correlations of 2-qubit states 75

of states, all of the form

ρXAB =

ρ11 0 0 ρ140 ρ22 ρ23 00 ρ∗23 ρ33 0ρ∗14 0 0 ρ44

, with∑

j

ρjj=1. (5.3)

The low-entropy region S∈[0, 0.9231) pertains to the rank-3 states ρRAB embodying

maximally entangled mixed states for the relative entropy of entanglement [77, 194]

ρRAB = 1− a+ r

2|Φ+⟩⟨Φ+|+ 1− a− r

2|Φ−⟩⟨Φ−|+ a|01⟩⟨01| (5.4)

with 0≤a≤1/3 and r a proper function of a [192]. In Eq. (5.4) we have used theBell state |Φ±⟩ ≡ |ϕ±(1/2)⟩AB. States ρR

AB span the leftmost trait in Fig. 5.1a).Next comes the family of Werner states

ρWAB(ϵ)=(1−ϵ)|Φ+⟩AB⟨Φ+|+ϵ114/4, (5.5)

which occupy the entropic sector S∈[0.9231, 1.410) for ϵ∈[0.225, 0.426) and the high-entropy region of S∈[1.585, 2] for ϵ∈[0.519, 1]. Such boundaries are clearly shown inFig. 5.1a). Evidently, two more families belong to the MNCMS boundary [cf. thetwo traits corresponding to S∈[1.410, 1.585)]. Such states are currently out of ourexperimental possibilities due to the small entropy-window they belong to, whichchallenges the tunability of the VNE achievable by our method.

It is worth noticing that quantum discord and AMID share the very same struc-ture of MNCMS, which can thus be rightfully regarded as the two-qubit states whoseQCs are maximally robust against state mixedness. This class of states are thusset to play a key role in realistic implementations of quantum information schemesbased on non-classicality of correlations as a resource [128, 129].

The relationship between A and D↔ is shown in Fig. 5.1b), where the solid linesshow that AMID embodies an upper bound to D↔ and is in agreement with thelatter in identifying classical-classical states with no QCs. Any physical two-qubitstate lives between the lower bound with A=D↔ and the upper one.

The lower bound in the AMID-discord plane is spanned by pure states of variableentanglement (i.e. for pure states A=D↔). However, it also accommodates boththe Werner states and the family

ρ↓AB(q) = (1− q)|Φ+⟩AB⟨Φ+|+ q|Φ−⟩AB⟨Φ−|, (5.6)

where q∈[0, 0.5], while the upper bound is spanned by

ρ↑(ϵ, p)AB=(1−ϵ)|ϕ+(p)⟩AB⟨ϕ+(p)|+ϵ|01⟩AB⟨01| (5.7)

for particular values of (ϵ, p) [192].

5.1.3 Resource-state generation

In this paragraph I will introduce the experimental techniques used in order toachieve the ample variety of states necessary for the performed investigation. Thekey element for the state engineering in our setup is embodied by state

|ξ⟩AB=√

1−ϵ|rℓ⟩AB|ϕ+(p)⟩AB +√ϵ|ℓr⟩AB|HV ⟩AB (5.8)


0.2 0.4 0.6 0.8 1.00.5 1.0 1.5 2.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0a) b)

Figure 5.1. a) Exploration of the two-way Discord D↔ ≡ max[D←,D→] vs. S plane. Thesolid line shows the MNCMS boundary. the color of each circles matches the trait of theboundary to which an experimental points belongs. b) Experimental comparison betweenAMID and D↔. The solid lines embody the bounds to A at set values of D↔. Both panelsshow experimental states and associated uncertainties.

Table 5.1. I report the experimental values of the parameters entering ρ↑(ϵ, p) and theiruncertainties.

Value and uncertaintyϵ 0.00±0.01 0.05±0.01 0.10±0.01 0.15±0.01 0.18±0.01 0.20±0.01p 0.50±0.02 0.70±0.01 0.80±0.01 0.90±0.01 0.95±0.02 0.99±0.02

with |ϕ±(p)⟩AB=√p|HH⟩AB±√

1− p|V V ⟩AB. In Eq. (5.8), four qubits are encodedin the polarization and path DOFs of optical modes A and B.

State |ξ⟩AB is produced by suitably adapting the polarization-momentum sourceof hyperentangled states presented in Sec. 1.5.

To generate |ϕ+(p)⟩, a UV laser impinges back and forth on a nonlinear crystal[cfr. Sec. 1.3.1]. The weight √p in the unbalanced Bell state |ϕ+(p)⟩AB, can bevaried by rotating a quarter-waveplate QWP placed between the crystal and themirror, which intercepts twice the UV pump beam [109]. The state thus producedreads |HE(p)⟩=(|rℓ⟩AB +eiγ |ℓr⟩AB)⊗|ϕ+(p)⟩AB/

√2. and by suitably manipulating

|HE(p)⟩ the state |ξ⟩AB has been obtained.The trace over the path DOF, which has been used to engineer several families

of boundary states, was performed by matching the left and right side of the modescoming from the four-hole mask in Fig. 1.4 on a beam splitter. When the differencebetween left and right paths is larger than the photon coherence time, an incoherentsuperposition of the state encoded in the polarization DOF is achieved on the outputmodes of the BS. Precisely, this tecnique allows to incoherently summing up the2-qubit polarization state encoded in the correlated modes |rℓ⟩ and the one encodedin the modes |ℓr⟩.Generation of ρ↑AB.- By tracing out the path DOF in |ξ⟩AB and using the corre-spondence between physical states and logical qubits |H⟩→|0⟩, |V ⟩→|1⟩, the densitymatrix for state ρ↑(ϵ, p) is achieved.

The values of the pairs (ϵ, p) determining the experimental states [shown as bluedots in Fig. 5.1b)] are given in Table 5.1, together with their uncertainties. Thevalues (ϵ, p)=(0,0.5) and (ϵ, p)=(0.2,1) correspond to the case of a pure state (having

5.2 Fully nonlocal quantum correlations 77

A=D↔=1) and a completely mixed state (with A=D↔=0) respectively.Generation of ρ↓AB.- The family embodied by ρ↓AB(q) can also be generated startingfrom the resource state |ξ⟩AB. By selecting only the correlated modes |rℓ⟩AB andsetting p = 1/2, I have generated the Bell state |Φ+⟩AB⟨Φ+|≡ρ↓(q=0). By insert-ing a birefringent quartz plate of proper thickness on the path of one of the twocorrelated modes, I controllably affect the coherence between the |HH⟩ and |V V ⟩states of polarization. Several quartz plates of different thickness ℓq have been usedto transform |Φ+⟩ into ρ↓(q). The value of q is related to the parameter ℓq.Generation of ρR,W .- The source of ρR and Werner state uses the setup previouslydescribed for the states ρ↑AB(ϵ, p). By setting p = 1/2 and by adding decoherencebetween |HH⟩ and |V V ⟩ (related to the parameter r) as previously explained, weare able to obtain ρR

AB from ρ↑AB. Regarding ρWAB, I have already addressed the

method to generate the |Φ+⟩AB⟨Φ+| component of the state on modes |rℓ⟩AB. Itis worth mentioning how to get the 114 contribution. This has been obtained byinserting two HWPs on the |ℓr⟩AB modes and suitably rotating both HWPs inorder to generate |ℓr⟩AB| + +⟩AB. By using two quartz plates introducing a delaylarger than τcoh between H and V , it is possible to obtain a fully mixed stateencoded in the polarization DOF of the modes |ℓr⟩AB. ρW

AB is finally achieved byincoherently summing up the contribution |Φ+⟩AB⟨Φ+| and the mixed state 114.

The properties of the states discussed above, are characterized by QST [148]allowing to obtain the physical density matrices and quantify D↔, S and A. ThePauli operators needed to implement the QST have been measured by using stan-dard polarization analyzers and two single photon detectors.

5.2 Fully nonlocal quantum correlations

5.2.1 Introduction

Quantum mechanics is a nonlocal theory [195, 196, 197], but not as nonlocal as thenon-signaling principle1 allows. The direct consequence is that quantum correlationshave a local content which is not always null. From a quantitative point of view, it ispossible to detect the local fraction of the correlations. This quantity measures thefraction of events that can be described by a local model. Consider all the possibledecompositions of the joint probability P (a, b|x, y) of a bipartite system (i.e. asystem composed by two sybsystems, namely Alice and Bob), where Alice(Bob)measures the observable x(y) obtaining the outcome a(b):

P (a, b|x, y) = qLPL(a, b|x, y) + (1− qL)PNL(a, b|x, y) (5.9)

Here PL(a, b|x, y)(PNL(a, b|x, y)) represents the content of the correlations that can(can’t) be described by a LHVT, with respective weights qL and 1 − qL, where0 ≤ qL ≤ 1. The local fraction of P (a, b|x, y) is defined as the maximum local

1Non-signaling principle can be explained in terms of joint probabilities as follows. We consider asystem composed by two subsystems, named Alice and Bob. P (a, b|x, y) is the probability to obtainthe outcomes a and b when Alice (Bob) measures the observable x (y). The non-signaling principlestates that the marginal probabilities P (a|x) and P (b|y) are independent of y and x, respectively.That is

∑b

P (ab|xy) =∑

bP (ab|xy′) ≡ P (a|x) ∀a, x, y, y′ or

∑a

P (ab|xy) =∑

aP (ab|x′y) ≡

P (b|y) ∀b, x, x′, y


weight over all possible decompositions: pL.= maxPL,PNL

qL. It can be understood as

a measure of the nonlocality of the correlations2.However, there exist quantum correlations that exhibit maximal nonlocality:

they are as nonlocal as any non-signaling correlations. Previous examples of maxi-mal quantum nonlocality between two parties required an infinite number of mea-surements, and the corresponding Bell violation is not robust against noise.

It has been demonstrated that, starting from the Peres-Mermin (PM) proof[198, 199] of the Kochen-Specker (KS) theorem [200], it is possible to obtain aBell inequality which is extremally violated [201]. This means that it involvesmaximally nonlocal quantum correlations, as they are able to attain the maximalBell violation compatible with the non-signaling principle. Moreover, this approachinvolves only few measurements and a finite-dimensional system (i.e. 4-qubit HEstate). Previous examples of maximal quantum nonlocality between two partiesrequire an infinite number of measurements, and the corresponding Bell violationwas not robust against noise.

In this section I present the experimental measurement of this inequality byexploiting the 4-qubit HE state encoded in the polarization and path DOFs 1.5.The lower value of pL ever reported has been obtained: pL = 0.218± 0.014 [201].

5.2.2 The inequality

Since the seminal work by Bell [195], we know that there exist quantum correlationsthat cannot be thought of classically. This impossibility is known as nonlocalityand follows from the fact that the correlations obtained when performing localmeasurements on entangled quantum states may violate a Bell inequality, whichsets conditions satisfied by all classically correlated systems.

The standard nonlocality scenario consists of two distant systems on which twoobservers, Alice and Bob, perform respectively ma and mb different measurements ofda and db possible outcomes. The outcomes of Alice and Bob are respectively labeleda and b, while their measurement choices x and y, with a = 1, . . . , da, b = 1, . . . , db,x = 1, . . . ,ma and y = 1, . . . ,mb. The correlations between the two systems areencapsulated in the joint conditional probability distribution P (a, b|x, y).

This probability distribution should satisfy the no-signalling principle. Quantumcorrelations in turn are those that can be written as P (a, b|x, y) = Tr(ρABM

xa ⊗M

yb ),

where ρAB is a bipartite quantum state and Mxa and My

b define local measurementsby the observers. Finally, classical correlations are defined as those that can bewritten as P (a, b|x, y) =

∑λ p(λ)PA(a|x, λ)PB(b|y, λ). These correlations are also

called local, as outcome a (b) is locally generated from input x (y) and the pre-established classical correlations λ.

The violation of Bell inequalities by entangled states implies that the set ofquantum correlations is strictly larger than the classical one. A similar gap appearswhen considering quantum versus general non-signalling correlations: there existcorrelations that, despite compatible with the no-signalling principle, cannot beobtained by performing local measurements on any quantum system. In particular,there exist non-signalling correlations which exhibit stronger nonlocality –in the

2Maximally nonlocal correlations feature pL = 0


!"#$

#$%&

'

!"#

!$

!%

$()*+

!$

!%&!"#

'

$()*+

%&

*, -,

Figure 5.2. Non-signalling, quantum and classical correlations. The set of nonsignalingcorrelations defines a polytope. The set of quantum correlations is contained in the setof non-signalling correlations. The set of classical correlations is again a polytope andis contained inside the quantum set. a) In general, the set of quantum correlations is nottangent to the set of non-signalling correlations. This means that the maximal value βQ of aBell inequality achievable by quantum correlations is above the local bound βL, but strictlysmaller than the maximal value βNS for non-signalling correlations. b) In the present Belltests, in contrast, quantum correlations are tangent to the set of non-signalling correlationsand, thus, attain the non-signalling value of a Bell inequality. The corresponding upperbound on the local fraction is zero, which discloses the full nonlocal nature of quantummechanics.

sense that they give larger Bell violations– than any quantum correlations [see Fig.5.2a)].

As said in the introduction, there are situations in which this second gap disap-pears: quantum correlations are then maximally nonlocal, as they are able to attainthe maximal Bell violation compatible with the no-signalling principle. Geometri-cally, in these extremal situations quantum correlations reach the border of the setof non-signalling correlations (see Fig. 5.2b). From a quantitative point of view, itis possible to detect this effect by computing the local fraction of the correlations.The quantity reported in Eq. (5.9) measures the fraction of events that can bedescribed by a local model.

Any Bell violation provides an upper bound on the local fraction of the correla-tions that cause it. In fact, a Bell inequality is defined as

∑Ta,b,x,yP (a, b|x, y) ≤ βL,

where Ta,b,x,y is a tensor of real coefficients. The maximal value of the left-hand sideof this inequality over classical correlations defines the local bound βL, whereas itsmaximum over quantum and non-signalling correlations give the maximal quantumand non-signalling values βQ and βNS , respectively. It follows that

pL ≤βNS − βQ

βNS − βL

.= pLmax. (5.10)

Thus, quantum correlations feature pL = 0 if they violate a Bell inequality as muchas any non-signalling correlations.

In this section, I report the experimental measurement of maximally nonlocalquantum correlations derived from the proof of the Kochen-Specker (KS) theorem[200] [See Appendix D].

Starting from the Peres-Mermin (PM) KS proof [198, 199] described in the


Appendix D, it is possible to arrive [201] at the following Bell inequality 3

β = ⟨a1 · b1|1, 1⟩+ ⟨a2 · b1|1, 2⟩+ ⟨a1 · b2|2, 1⟩+ ⟨a2 · b2|2, 2⟩+ ⟨a1 · a2 · b1|1, 3⟩+ ⟨a1 · a2 · b2|2, 3⟩+ ⟨a1 · b1 · b2|3, 1⟩⟨a2 · b1 · b2|3, 2⟩− ⟨a1 · a2 · b1 · b2|3, 3⟩ ≤ 7. (5.11)

Here ⟨f(a1, a2, b1, b2)|r, c⟩ denotes the expectation value of a function f of the outputbits for the measurements r and c. Remarkably, this inequality has already appearedin Ref.[202] in the context of “all-versus-nothing” nonlocality tests. It is defined ina scenario in which Alice and Bob perform 3 measurements of 4 outcomes. Alice’s(Bob’s) outcomes are decomposed into two bits a = (a1, a2) [b = (b1, b2)], which cantake values ±1. The terms ⟨f(ai, bj)|x, y⟩ denote the expectation value of f(ai, bj)for the measurements x and y. Measurements on maximally entangled states leadto quantum correlations saturating the algebraic violation of this inequality, i.e.βQ = 9.

5.2.3 Experimental realization

Consider next the following quantum realization: Alice and Bob share two two-qubitmaximally entangled states |ψ⟩ = 1√

2(|00⟩ + |11⟩)12 ⊗ 1√2(|00⟩ + |11⟩)34, which is

equivalent to a maximally entangled state of two four-dimensional systems. Alicepossesses systems 1 and 3 and Bob systems 2 and 4. Notice that state in Eq.(1.15) is recovered from the last one through the usual relations: |H⟩A,B ≡ |0⟩1,2,|V ⟩A,B ≡ |1⟩1,2, |r⟩A ≡ |0⟩3, |l⟩A ≡ |1⟩3, |l⟩B ≡ |0⟩4, and |r⟩B ≡ |1⟩4. Alice canchoose among three different measurements which correspond to the three rowsappearing in Table D.1 of the Appendix D. If Alice chooses input x, the quantummeasurement defined by observables placed in row r = x is performed. Note thatthe measurement acts on a four dimensional quantum state, thus there exist fourpossible outcomes (one for each eigenvector common to all three observables), inour scenario decomposed into two dichotomic outputs. We define ai to be the valueof the observable placed in c = i for i = 1, 2. The value of the third observable isredundant as it can be obtained as a function of the other two. Equivalently, Bob canchoose among three measurements which correspond to the three columns appearingin the table. If Bob chooses input y, outputs bj are the values of observables placedin c = y and r = j for y = 1, 2, 3 and j = 1, 2. This realization attains thealgebraic maximum βQ = βNS = 9 of the linear combination of probabilities givenby expression (5.11).

I performed a test of inequality (5.11) with two hyperentangled photons, A andB, generated by the source described in Sec. 1.5. The state (1.15) is also maximallyentangled between A and B, and therefore also allows for the maximal violation of(5.11).

Moving to the classical domain, the maximum value attainable by a local modelsis βL = 7 and, thus, Eq. (5.11) constitutes a valid Bell inequality maximally violatedby quantum mechanics.

3The details about the derivation of this inequality are not reported in the present thesis.


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Figure 5.3. Measurement setups used by Alice and Bob. See text for a detailed explana-tion of the measurements. BS: beam splitter. PBS: polarizing beam splitter. HWP: halfwaveplate.

Alicea1=−1, a2=−1 a1=−1, a2=1 a1=1, a2=−1 a1=1, a2=1

x = 1 |−⟩|l⟩ |+⟩|l⟩ |−⟩|r⟩ |+⟩|r⟩x = 2 |V ⟩|−⟩ |V ⟩|+⟩ |H⟩|−⟩ |H⟩|+⟩x = 3 |H⟩|l⟩ − |V ⟩|r⟩ |H⟩|l⟩+ |V ⟩|r⟩ |H⟩|r⟩ − |V ⟩|l⟩ |H⟩|r⟩+ |V ⟩|l⟩

Bobb1=−1, b2=−1 b1=−1, b2=1 b1=1, b2=−1 b1=1, b2=1

y = 1 |V ⟩|r⟩ |H⟩|r⟩ |V ⟩|l⟩ |H⟩|l⟩y = 2 |−⟩|−⟩ |−⟩|+⟩ |+⟩|−⟩ |+⟩|+⟩y = 3 |+⟩|r⟩ − |−⟩|l⟩ |+⟩|r⟩+ |−⟩|l⟩ |−⟩|r⟩ − |+⟩|l⟩ |−⟩|r⟩+ |+⟩|l⟩

Table 5.2. Measurement settings. Each row represents a measurement (context). Thefour states in each row represent the four projectors of each measurement. a1,2 and b1,2 arethe two-bit outcomes of Alice and Bob respectively. In each state, the first ket refers topolarization, while the second one to path. |±⟩ correspond to 1√

2 (|H⟩±|V ⟩) or 1√2 (|r⟩±|l⟩),

for polarization or path respectively.

5.2.4 Experimental results

The nine terms of expression (5.11) correspond to the different combinations be-tween one of Alice’s three contexts and one of Bob’s three contexts listed in Table5.2. In the settings x = 1, 2 (y = 1, 2) Alice (Bob) must project into states thatare separable between path and polarization (eigenstates of Pauli operators X andZ). To project into |r⟩, |l⟩ the modes are detected without BS. On the otherhand, the BS is used to project into 1√

2(|r⟩ ± |l⟩). PBSs and waveplates have been


Correlation Experimental result⟨a1 · b1|1, 1⟩ 0.9968± 0.0032⟨a1 · b2|2, 1⟩ 0.9759± 0.0058⟨a2 · b1|1, 2⟩ 0.9645± 0.0068⟨a2 · b2|2, 2⟩ 0.941± 0.010⟨a1 · a2 · b1|1, 3⟩ 0.9705± 0.0048⟨a1 · a2 · b2|2, 3⟩ 0.9702± 0.0049⟨a1 · b1 · b2|3, 1⟩ 0.9688± 0.0073⟨a2 · b1 · b2|3, 2⟩ 0.890± 0.013⟨a1 · a2 · b1 · b2|3, 3⟩ −0.888± 0.018

Table 5.3. Experimental results. Errors were calculated by propagating Poissonianerrors of the counts.

exploited to project into |H⟩, |V ⟩ or 1√2(|H⟩ ± |V ⟩). More details are needed for

contexts x, y = 3, corresponding to the projection into single photon Bell states (thetwo entangled qubits of the Bell state are encoded in polarization and path of thesingle particle) [see Table 5.2]. For instance, let us consider the projection on thestates |H⟩|l⟩± |V ⟩|r⟩ and |V ⟩|l⟩± |H⟩|r⟩ for Alice. By inserting a HWP oriented at45 on the mode |l⟩A before the BS, the previous states become |V ⟩|±⟩ and |H⟩|±⟩respectively. The two BS outputs allow to discriminate between |r⟩+ |l⟩ and |r⟩−|l⟩while the two outputs of the PBSs discriminate |H⟩ and |V ⟩.

Table 5.3 provides the experimental values of all nine correlations in the Bellinequality.

The obtained Bell violation βexpQ = 8.564 ± 0.028 provides the upper bound

pLmax = 0.218± 0.014.

5.3 Linear optics C-Phase gateMany efforts have been made in the last years to experimentally implement sev-eral basic quantum gates, such as the CNOT or C-Phase gate. The latter was inparticular realized by exploiting the polarization DOF of a photonic system [203]and, more recently, was implemented within a quantum dot scenario [20]. In thissection I report the experimental implementation of the C-Phase gate based onlyon the path DOF of a single photon [204]. This work provides a further exampleof how to exploit the multiDOF quantum states in the photonic experiments and itdemontrates the power of this approach when applied in the quantum informationfield.

The unitary transformation corresponding to the C-Phase, is defined as follows:

Uphase =

1 0 0 00 eϕ1 0 00 0 1 00 0 0 eϕ2

(5.12)

The optical setup of Fig. 5.4 shows the high stability closed-loop displacedSagnac scheme used in the experiment. It represents a modified version of the

5.3 Linear optics C-Phase gate 83

Figure 5.4. C-Phase gate experimental setup based only on the path DOF of a singlephoton. The control qubit is identified by the different paths followed by the photon af-ter the BS1 (i.e. |r⟩ or |ℓ⟩), while the target qubit is given by the clockwise (|C⟩) orcounterclockwise (|A⟩) path followed by the photon after BS2 within the displaced Sagnacinterferometer. The phase shift performed by the gate has been obtained by using the twothin glass plates ϕℓ and ϕr, both on the counterclockwise paths |A⟩. Two delayers ϕd allowto compensate the temporal delay introduced by ϕℓ and ϕr. The insertion of ϕ′d is neededto avoid interference between the modes coming back from the displaced Sagnac systemand impinging on BS1.

ones adopted for the experiments described in Sec. 2.2.1,2.4,3.1.3. Here the secondbeam splitter (BS2) intercepts only the optical path of lower photon and enablesthe realization of a diplaced Sagnac interferometer [See Sec. 2.4].

Let us now describe how the implemented gate works. In the HE source, de-scribed in Sec. 1.5, only one polarization cone, namely the H − cone, is consideredand only one mode, corresponding to the lower photon, is taken into account. Inorder to explain the experiment let us consider only the |r⟩B mode coming out ofthe holed mask, as reported in Fig. 1.4. The BS1 acts as follows:

|r⟩BBS1−−−→ 1√

2(|r⟩B + |ℓ⟩B). (5.13)

The photon, arriving at the BS2, can go clockwise (|C⟩B) or counterclockwise (|A⟩B)within the diplaced Sagnac. This corresponds to add a further qubit, encoded inthe path DOF, hence the state in Eq. (5.13) becomes:

1√2

(|r⟩B + |ℓ⟩B) BS2−−−→ 1√2

(|r⟩B|ϕr⟩B + |ℓ⟩B|ϕℓ⟩B) (5.14)

where |ϕr⟩B = 1√2(|C⟩B + eiϕr |A⟩B), |ϕℓ⟩B = 1√

2(|C⟩B + eiϕℓ |A⟩B). By consideringthe following relations between logical states and physical qubits:

|0⟩1, |1⟩1 → |r⟩B, |ℓ⟩B|0⟩2, |1⟩2 → |C⟩B, |A⟩B (5.15)

the state (5.14) reads:12

[|0⟩1 ⊗ (|0⟩+ eϕr |1⟩)2 + |1⟩1 ⊗ (|0⟩+ eϕℓ |1⟩)2] =12

(|00⟩12 + eϕr |01⟩12 + |10⟩12 + eϕℓ |11⟩12) (5.16)


Figure 5.5. Measured oscillations of the single counts with dots representing the exper-imental data and the solid line corresponding to the fitting curve. The dark counts havebeen subtracted. The uncertainties have been determined by associating Poissonian fluc-tuations to the single counts. The red (black) data have been measured by projecting thestate reported in Eq. (5.16) on |0⟩1⟨0| (|1⟩1⟨1|) and varying ϕr (ϕℓ) with the thin glassplates in the displaced Sagnac interferometer. For ϕr = π and ϕℓ = 0 I performed theQuantum State Tomography of qubit 2. The fidelity have been calculated with respect tothe theoretical ones, i.e. |+⟩2⟨+| for the state in the box I and |−⟩2⟨−| for the state in thebox II.

The phases ϕr and ϕℓ can be indipendently varied by using two thin glass platesplaced within the interferometer. This corresponds to implement the transformationreported in Eq. (5.12) with ϕr = ϕ1 and ϕℓ = ϕ2. It is worth to remember that boththe control and target qubits of the quantum gate are encoded in the path DOF ofphoton B. Precisely, the control qubit is encoded in the longitudinal momentum ofthe photon before BS2 (i.e. |r⟩B,|ℓ⟩B) while the target qubit is encoded in thepath followed in the Sagnac scheme (i.e. |C⟩B,|A⟩B). I report in Table 5.4 the“truth table” of the engineered gate.

Table 5.4. “Truth table” of the realized C-phase gate. In the first column I report thelogical qubits while in the second column there are the corresponding physical qubits.

Logical qubit Physical qubitControl Target Control Target|0⟩1⟨0| 1√

2(|0⟩2 + eiϕr |1⟩2) |r⟩B⟨r| 1√2(|C⟩B + eϕr |A⟩B)

|1⟩1⟨1| 1√2(|0⟩2 + eiϕℓ |1⟩2) |ℓ⟩B⟨ℓ| 1√

2(|C⟩B + eϕℓ |A⟩B)

The second passage through BS2 allows to perform the measurement of thePauli operators.

The obtained experimental results are shown in Fig. 5.5. The oscillations of thesingle counts have been measured by projecting the state (5.16) on |0⟩1⟨0| (|1⟩1⟨1|)and varying ϕr (ϕℓ). The projection on |r⟩B⟨r| (|ℓ⟩B⟨ℓ|) was performed by inter-cepting the input mode |ℓ⟩B (|r⟩B).

In the experiment, ϕr = ϕℓ + π, thus there is a particular phase factor betweenϕr and ϕℓ, however it is important to underline that they can assume any generalvalue with this setup. In the case ϕℓ = 0, ϕr = π, I have performed the tomo-

5.3 Linear optics C-Phase gate 85

graphic reconstruction [148] of the density matrix related to the state |ϕr⟩B⟨ϕr| and|ϕℓ⟩B⟨ϕℓ|. These values correspond to realize a C −NOT gate. As already pointedout, the second passage through BS2 allows to measure the Pauli operators σx andσy. The third Pauli operator σz has been measured by intercepting the mode in thedisplaced Sagnac (i.e. |C⟩ or |A⟩). This corresponds to make a projection on thecomputational basis. The fidelities of the measured states, calculated with respectto the theoretical states, are larger than 98%.

Appendix A

Decoherence introduced in thePhased Dicke states

Let us now describe in more detail the considered decoherence sources. First of all,in the engineered setup, we must consider a polarization decoherence at the levelof the |ξ⟩ generation due to a non perfect superposition between the |HH⟩ and|V V ⟩ emission. Since |ξ⟩ is given by the superposition of the |HH⟩ and |V V ⟩ terms,our decoherence partially erases the coherence between them but cannot introduceterms containing |V H⟩ or |HV ⟩. This decoherence can be modeled by a phasedecoherence channel acting on polarization qubit 2:

ρ→ (1− q1)ρ+ q1Z2ρZ2 (A.1)

By exploiting the same arguments used to obtain Eq. (2.13), the channel (A.1)can be interpreted as a decoherence channel on |D(ph)

4 ⟩. Since UZ2U† = Z1Z2, thepolarization decoherence (A.1) can be written as

|D(ph)4 ⟩⟨D(ph)

4 | →2∑

j=1Aj |D(ph)

4 ⟩⟨D(ph)4 |A†j (A.2)

with A1 =√

1− q111 and A2 = √q1Z1Z2. By measuring the visibility of polarizationinterference I estimated q1 ≃ 0.05.

The second decoherence affects the path degree of freedom and corresponds tothe following channel [See Eq. (2.12)]:

ρ→ (1− q2)2ρ+ q2(1− q2) [Z1ρZ1 + Z3ρZ3] + q22Z1Z3ρZ1Z3 (A.3)

I could change the parameter q2 by varying the delay ∆x in the first interferometer.By considering the path terms in the |HH⟩ contribution in |ξ⟩, namely |ψ−⟩ =

1√2(|rℓ⟩ − |ℓr⟩), this second decoherence acts by (partially) spoiling the coherence

between the |rℓ⟩ and |ℓr⟩ term giving the state

ρdec2 =12

(|ℓr⟩⟨ℓr|+ |rℓ⟩⟨rℓ|)

− 12

(1− 2q2)2(|ℓr⟩⟨rℓ|+ |rℓ⟩⟨ℓr|).(A.4)

87

88 A. Decoherence introduced in the Phased Dicke states

We measured the coincidences for different values of ∆x and fitted them with thefollowing function

C(∆x) = B(1− V0e−4 log 2 (∆x)2

(cτ)2 ) (A.5)

where V0 is the maximum visibility and where B are the coincidences measured outof interference (i.e. measured for ∆x much longer than the single photon coherencelenght). I obtained cτ = 24.84± 0.01µm and V0 = 0.9313± 0.0005. We defined themeasured visibility as

Vexp(∆x) = B − C(∆x)B

= V0e−4 log 2 (∆x)2

c2τ2 (A.6)

By assuming that for |ψ−⟩ the decoherence (2.12) is the main source of imperfections,the measured visibility Vexp(∆x) of first interference on BS may be compared withthe value calculated from Eq. (A.4), V = (1− 2q2)2: then, the relation between ∆xand q2, shown in Fig. 2.2, can be obtained:

q2(∆x) = 12

(1−

√V0e−4 log 2 (∆x)2

c2τ2 ) (A.7)

A third decoherence effect, again in the path degree of freedom, is related to thesecond interference on the BS. The non-perfect interference can be modeled as adecoherence channel acting exactly as (2.12). Written in the Kraus representationit reads:

|D(ph)4 ⟩⟨D(ph)

4 | →4∑

k=1Ck|D

(ph)4 ⟩⟨D(ph)

4 |C†k (A.8)

with C1 = (1 − q3)11, C2 =√q3(1− q3)Z1, C3 =

√q3(1− q3)Z3 and C4 = q3Z1Z3.

By measuring the interference visibility I estimated q3 ≃ 0.05.The three decoherence channels can be summarized as follows

ρ(q1, q2, q3) ≡4∑

k=1

4∑j=1

2∑i=1

CkBjAi|D(ph)4 ⟩⟨D(ph)

4 |A†iB†jC†k (A.9)

From the previous expression, it is possible to calculate the theoretical expectationvalues of the operators appearing in the witness as a function of the q’s parameters:

⟨Sxx(π)⟩ =4− 83q3(3− q3)

− 163

(1− q3)2[q1(1− 2q2)2 + 2q2(1− q2)]

⟨Syy(π)⟩ =4− 163q1(1− q3)2 + 8

3(q3 − 3)q3]

⟨Szz(0)⟩ =− 2 + 163q2(1− q2)

(A.10)

For q1 = 0.05 and q3 = 0.05 I obtain the following expectation value for W:

⟨W⟩ = −0.455 + 2.333 q2 − 2.333 q22 (A.11)

This expression is used for the theoretical curve in Fig. 2.3.

89

The witness used to detect multipartite entanglement (see Eq. 36 of [119]) is

Wmult = 18

[21− 2Sxx(π)− 2Syy(π) + Szz(0)− 2X1X2X3X4 − 2Y1Y2Y3Y4 − 7Z1Z2Z3Z4](A.12)

By following [119] it is possible to obtain a bound for the fidelity:

F >23− 1

3⟨Wmult⟩. (A.13)

Appendix B

On the Quantum Protocols

In this Appendix I provide further details about the QTC and ODT protocols [SeeSec. 2.4].

B.1 Optimal quantum cloning machine

In this section I want just to recall the main properties of a quantum cloning machine.It can be described as a Unitary Trasformation on the input state |ϕ⟩X⊗|A⟩, where|A⟩ is an ancillary qubit and |ϕ⟩X is the qubit to be cloned: |ϕ⟩X = a|0⟩X + b|1⟩X(i.e. a general state on the Bloch Sphere). I can describe this process on the wholesystem as follows:

U1M (|ϕ⟩X ⊗ |A⟩) = a|ϕ0⟩AC + b|ϕ1⟩AC (B.1)

where |ϕ0⟩AC and |ϕ1⟩AC are entangled states between the ancillary qubit and thecopies. The reduced density matrix for each qubit, which is a copy of the input one,is expected to be:

ρc = γ|ϕ⟩X⟨ϕ|+ (1− γ)|ϕ⊥⟩X⟨ϕ⊥| (B.2)

where γ = M(N+1)+NM(N+2) is the best possible fidelity for the cloned qubit. As predicted

by the No-cloning Theorem [205], it is not possible to have optimal copies of anarbitrary unknown quantum state.

Figure B.1. Scheme of an optimal quantum cloning Machine. The input qubitX enters thecloning machine represented by the gray box. The output copies are given by the incoherentsuperposition of the states |ϕ⟩X⟨ϕ| and |ϕ⊥⟩X⟨ϕ⊥|. Here γ is best possible fidelity for thecloned qubit.

91

92 B. On the Quantum Protocols

B.2 Quantum Telecloning ProtocolTelecloning combines quantum teleportation and optimal quantum cloning of oneinput qubit into M outputs (i.e. M receivers).

Figure B.2. Scheme of the Quantum Telecloning 1→2 protocol via |ψTC⟩. Initially, qubits2,3,4,5 share the state |ψTC⟩ while the qubit 1 represents a general single qubit state to betelecloned. Alice performes a Bell measurement between the qubit 2 and 1 and the receiversare informed about the implemented operation. The qubits 4 and 5 encode two copies ofthe input qubit (i.e. qubit 1) up to a single qubit unitary transformation U.

The general protocol for the telecloning is reported in Fig. B.2. Qubits 2345share a particular entangled state, i.e. the quantum resource of the protocol, |ψT C⟩.

• The input state of the protocol is given by the tensor product of the quantumresource and the qubit to be cloned (qubit 1 in Fig. B.2).

• Alice performs a generalized Bell measurement on two qubits of the inputstate: qubits 1 and 2 in Fig. B.2

• Charly and Claire read the output states ρ4 and ρ5

• This scheme is completed by a classical communication of Alice, in fact shehas to inform the receivers about which measure has been done (i.e. a classicalcommunication)

• ρ4 and ρ5, the reduced density matrices associated to each qubit are copies ofthe input qubit within the No-Cloning theorem limits (see Eq. (B.2) for thedensity matrix of a cloned qubit).

B.3 QTC 1 → 2 via |ψTC⟩ 93

A complete description of Quantum Telecloning is reported in [142]. In this paper,the authors showed that the optimal quantum resource for the Telecloning 1→2 isrepresented by the following state:

|ψT C⟩ = 1√3|0000⟩+ 1√

3|1111⟩+ 1√

12|1001⟩+ 1√

12|0110⟩+ 1√

12|0101⟩+ 1√

12|1010⟩

(B.3)so the input state in Murao’s scheme is given by the tensor product |ϕ⟩1⊗|ψT C⟩2345.In Fig. B.3 I report a scheme of the entanglement between the qubits before andafter the Bell measurement (i.e. each link represents entanglement). It’s importantto point out that in this framework, one qubit of the quantum resource is an An-cillary Qubit which plays a fundamental role since, by implementing this protocol,only two qubits can receive an optimal copy of the initial state.

Figure B.3. Quantum telecloning 1 → 2 of an unknown one-qubit state. The qubits2,3,4,5 initially share a multiparticle entangled state which involves also the “receivers” (i.e.Claire and Charly) and the ancillary qubit (i.e. qubit 3). The solid lines indicate theexistence of entanglement between pairs of qubits when the remaining ones are traced out.Alice performs a Bell measurement between the qubits 1 and 2, subsequently, the receiversperform appropriate rotations on the output qubits, obtaining two optimal quantum clones.These receivers could be, in principle, at different locations.

It is worth noting that |ψT C⟩ is not a Dicke State. A state equivalent to |ψT C⟩has been experimentally engineered and characterized by exploiting a source of fourentangled photons [146] and encoding the qubits only in the polarization DOF.Precisely, in this paper they have generated the state |ψ′T C⟩ = X1X2|ψT C⟩ andthey have studied its entanglement properties which have allowed the experimentalrealization of several quantum protocols [206, 207].

B.3 QTC 1 → 2 via |ψTC⟩

In the following lines I report the demonstration that the state |ψ′T C⟩ can be usedfor a Telecloning experiment. Let us consider the following general input state:

|ϕ⟩1 ⊗ |ψ′T C⟩2345 = (cos(θ2

)|0⟩+ sen(θ2

)eiϕ|1⟩)1 ⊗ |ψ′T C⟩2345 (B.4)

By measuring the Bell state |ψ+⟩ on the qubits 1 and 2 I obtain:

12√

6[cos(θ

2)(2|100⟩+ |001⟩+ |010⟩) + sen(θ

2)eiϕ(2|101⟩+ |011⟩+ |110⟩)]345 (B.5)


Let us call ρ345 the density matrix related to this state. The reduced density matri-ces for the qubits 2,3 and 4, obtained by tracing out the other two qubits, are:

ρ4 = ρ5 = 16

[|c|2(5|0⟩⟨0|+ |1⟩⟨1|) + |s|2(|0⟩⟨0|+ 5|1⟩⟨1|) + 4cseiϕ|1⟩⟨0|+ 4cse−iϕ|0⟩⟨1|]

ρ3 = 16

[|c|2(4|1⟩⟨1|+ 2|0⟩⟨0|) + |s|2(2|1⟩⟨1|+ 4|0⟩⟨0|) + 3cseiϕ|1⟩⟨0|+ 3cse−iϕ|0⟩⟨1|](B.6)

where c : cos( θ2), s : sen( θ

2).ρ4 and ρ5 represent two optimal clones of the input state, in fact their density

matrices can be written as follows:

56|ϕ⟩1⟨ϕ|+

16|ϕ⊥⟩1⟨ϕ⊥| (B.7)

This result does not depend on the qubit |ϕ⟩1 to be cloned, thus the value of thefidelity of the cloned qubit is always 5

6 . It is worth noting that I obtained theoptimal value of the fidelity for a cloning machine 1→2.

For the other three Bell measurements we obtain:

- |ϕ±⟩

12√

6[cos(θ

2)(2|011⟩+ |110⟩+ |101⟩)± sen(θ

2)eiϕ(2|100⟩+ |010⟩+ |001⟩)]345 (B.8)

and

ρ4 = ρ5 = 16

[|c|2(5|1⟩⟨1|+ |0⟩⟨0|) + |s|2(|1⟩⟨1|+ 5|0⟩⟨0|)± 4cseiϕ|0⟩⟨1| ± 4cse−iϕ|1⟩⟨0|]

ρ3 = 16

[|c|2(4|0⟩⟨0|+ 2|1⟩⟨1|) + |s|2(2|0⟩⟨0|+ 4|1⟩⟨1|)± 2cseiϕ|0⟩⟨1| ± 2cse−iϕ|1⟩⟨0|](B.9)

- |ψ−⟩

12√

6[cos(θ

2)(2|100⟩+ |001⟩+ |010⟩)− sen(θ

2)eiϕ(2|101⟩+ |011⟩+ |110⟩)]345 (B.10)

and

ρ4 = ρ5 = 16

[|c|2(5|0⟩⟨0|+ |1⟩⟨1|) + |s|2(|0⟩⟨0|+ 5|1⟩⟨1|)− 4cseiϕ|1⟩⟨0| − 4cse−iϕ|0⟩⟨1|]

ρ3 = 16

[|c|2(4|1⟩⟨1|+ 2|0⟩⟨0|) + |s|2(2|1⟩⟨1|+ 4|0⟩⟨0|)− 3cseiϕ|1⟩⟨0| − 3cse−iϕ|0⟩⟨1|](B.11)

We can conclude that, by projecting on the state |ψ+⟩, no further Unitary Tran-formations on the qubit 4 and 5 are necessary in order to obtain the optimal cloneof the input state. In the following table I summerize the necessary Unitary Trans-formations when the other three Bell measurements are performed [See Fig. B.2]:

B.4 Phase-Covariant QTC 1 → 3 via Dicke State 95

Bell measurement Unitary Transformation|ψ+⟩ 11|ψ−⟩ Z|ϕ+⟩ X|ϕ−⟩ ZX

Figure B.4. Scheme of the Quantum Telecloning 1→3 protocol via Dicke states. Thequbits 1,2,3,4 share the state |D(2)

4 ⟩ while the qubit X represents a general single qubitstate to be telecloned. Alice performes a Bell measurement between the qubit 1 and Xand the three receivers are informed about the implemented operation. The qubits 2,3and 4 encode three copies of the input qubit (i.e. qubit X) up to a single qubit unitarytransformation U.

B.4 Phase-Covariant QTC 1 → 3 via Dicke State

In order to implement the Optimal Quantum Telecloning 1 → 3, the Dicke statecan be used as quantum resource for the protocol. In Fig. B.4 I report the schemefor the Optimal Quantum Telecloning 1 → 3 via Dicke state while in the followinglines I want to demonstrate the feasibility of this protocol in the case of the Phase-Covariant Quantum Telecloning (i.e. |ψ⟩X = 1√

2(|0⟩X + eiϕ|1⟩X)).Let us start from the input state:

|χ⟩ = 1√2

(|0⟩X + eiϕ|1⟩X)⊗ |D(2)4 ⟩1234 (B.12)

Following the general scheme for the telecloning protocol presented in the previoussections, I perform a Bell measurement by projecting the qubits 1 and X on the


state |ψ+⟩ :

1√2

(X1⟨01|+X1 ⟨10|)|χ⟩ = 1√6

(|100⟩+|010⟩+|001⟩)234+ eiϕ

√6

(|110⟩+|011⟩+|101⟩)234

(B.13)and we can define

|a⟩ = 1√6

(|100⟩+ |010⟩+ |001⟩)

|b⟩ = 1√6

(|101⟩+ |110⟩+ |011⟩)(B.14)

It can be straightforwardly obtained that ⟨ϕ±|χ⟩ = |b⟩ ± eiϕ|a⟩ and ⟨ψ±|χ⟩ = |a⟩ ±eiϕ|b⟩.The density matrix associated to the state (B.13) is :

ρ234 = 16

(|a⟩⟨a|+ |b⟩⟨b|+ e−iϕ|a⟩⟨b|+ eiϕ|b⟩⟨a|) (B.15)

where the subscript 234 indicates the involved qubits.The state of the single qubits 2,3 and 4 is:

ρ2 = ρ3 = ρ4 =⟨00|ρ234|00⟩+ ⟨01|ρ234|01⟩+ ⟨10|ρ234|10⟩+ ⟨11|ρ234|11⟩ =

= 12

(|0⟩⟨0|+ |1⟩⟨1|) + 13

(eiϕ|1⟩⟨0|+ e−iϕ|0⟩⟨1|) =

= 512

(|0⟩⟨0|+ |1⟩⟨1|+ e−iϕ|0⟩⟨1|+ eiϕ|1⟩⟨0|)+112

(|0⟩⟨0|+ |1⟩⟨1| − e−iϕ|0⟩⟨1| − eiϕ|1⟩⟨0|) =

= 56|ϕ⟩X⟨ϕ|+

16|ϕ⊥⟩X⟨ϕ⊥|

(B.16)

where |ϕ⊥⟩X = 1√2(|0⟩X−eiϕ|1⟩X). I have demonstrated that we can generate three

optimal copies of the input state. In fact ρ2, ρ3 and ρ4 are equivalent to the statereported in Eq. (B.2) with γ = 5

6 = Fph which is the optimal fidelity for a PhaseCovariant Cloning 1 → 3 qubits. Even in this case, by projecting on the state |ψ+⟩no further Unitary Tranformations on the qubit 2,3 and 4 are necessary in orderto obtain the optimal clone of the input state. In the following table I report thenecessary Unitary Transformations in the case of the other three Bell measurements:

Bell measurement Unitary Transformation|ψ+⟩ 11|ψ−⟩ Z|ϕ+⟩ X|ϕ−⟩ ZX

B.5 General QTC 1 → 3 via Dicke StateLet us start from the general input state |ψ⟩X = c|0⟩X + seiϕ|1⟩X :

|χ⟩ = (c|0⟩X + seiϕ|1⟩X)⊗ |D(2)4 ⟩1234 (B.17)

B.6 ODT protocol 97

Figure B.5. Fidelity between the qubit X and each telecloned qubit when the QTC 1 →3 protocol is performed via Dicke states. Here the angle θ characterizes the initial state|ψ⟩X = (cos( θ2 )|0⟩+ sen( θ2 )eiϕ|1⟩)X . In this calculation the quantum resource representedby the Dicke state and the qubit X are considered as completely pure states.

We perform a Bell Measurement by projecting the qubits 1 and X on the state |ψ+⟩(i.e. no further transformations are necessary):

1√2

(X1⟨01|+X1⟨10|)|χ⟩ = c√6

(|100⟩+|010⟩+|001⟩)234+ seiϕ

√6

(|110⟩+|011⟩+|101⟩)234

(B.18)The state of the qubits 2,3 and 4 is:

ρ2 = ρ3 = ρ4 = 13

[c2(|1⟩⟨1|+2|0⟩⟨0|)+s2(|0⟩⟨0|+2|1⟩⟨1|)+2sceiϕ|1⟩⟨0|+2sce−iϕ|0⟩⟨1|(B.19)

and the fidelity between this state and the input one is given by:

F =X ⟨ψ|ρ4|ψ⟩X = 23

(1 + c2s2) (B.20)

In Fig. B.5 I report the behaviour of the fidelity between its maximum, achievedfor a Phase Covariant Telecloning, and its minimum, achieved when the input stateis |0⟩ or |1⟩.

B.6 ODT protocol

Let us consider the Dicke state in the computational basis:

|D(2)4 ⟩ = 1√

6(|1100⟩+ |0011⟩+ |1010⟩+ |0101⟩+ |0110⟩+ |1001⟩)1234 (B.21)

The implemented ODT protocol can be described with few simple passages:


1. The qubit X to be teleported has to be added to the quantum resource andthe five qubit state reads: |D(2)

4 ⟩1234 ⊗ (a|0⟩+ b|1⟩)X

2. The CNOTX2 gate has to be realized. Here the qubit X (2) is the control(target).

3. A two-qubit projection has to be performed. I choosed to project the qubits 3and 4 but each couple of qubits is totally equivalent. Precisely the proectionhas to be performed onto the states |01⟩34 or |10⟩34.

4. Let us consider the case in which the qubit 3 and 4 have been projected ontothe state |10⟩34. In this case the three remaining qubits share the followingstate: a(|010⟩+ |001⟩)X12 + b(|111⟩+ |100⟩)X12.

5. In order to have the teleportation on the qubit 1 (2), the last operation consistsof the projection |1+⟩2X (|1+⟩1X).

The symmetric nature of the quantum resource (i.e. the Dicke state) allows tovary the qubits involved in the projections just described.

It is worth noting that even in this case, the Bell projection has been decomposedinto the CNOT gate and single qubit projections, as explained in the Sec. 2.4.1.

Appendix C

On the non-Markoviandynamics

C.1 Controlled-rotation gates from quantum Ising mod-els

In this Section I sketch the procedure for the achievement of a controlled rotationfrom a quantum Ising model and determine the conditions to achieve a desired rota-tion angle. For convenience of notation, I will use the correspondences (already intro-duced in the Sec. 3.1.3) |0⟩, |1⟩S,A → |H⟩, |V ⟩S,A and |0⟩, |1⟩E → |r⟩, |ℓ⟩E .

The transverse Ising-like model whose associated propagator has been simulatedin the presented experiment is

HIsing = ϵEx σEx + ϵEz σ

Ez +

∑p=x,y,z

ϵSpσSp + JσE

z σSz . (C.1)

We take ϵEx ≫ J , so that the dynamics of the E subsystem as induced by thiscoupling is effectively frozen 1. This Hamiltonian can be re-written as follows:

HSE = ϵEz σEz +

(ϵSz ϵSx − ıϵSy

ϵSx + ıϵSy −ϵSz

)+ JσE

z σSz = ϵEz (|0⟩⟨0|E − |1⟩⟨1|E)11S +(

ϵSz ϵSx − ıϵSyϵSx + ıϵSy −ϵSz

)+ J(|0⟩⟨0|E − |1⟩⟨1|E)σS

z (C.2)

In the computational basis, this model can be written as

HSE = |0⟩⟨0|E ⊗[(

ϵSz + J ϵSx − ıϵSyϵSx + ıϵSy −ϵSz − J

)+ ϵEz 11S

]

+ |1⟩⟨1|E ⊗[(

ϵSz − J ϵSx − ıϵSyϵSx + ıϵSy −ϵSz + J

)− ϵEz 11S

].

(C.3)

The diagonal terms proportional to ϵEz only give rise to a shift in the energy ofthe two subspaces that have been identified by this formal splitting. We can thus

1In Eq. (C.1) we can neglect the term ϵEy σE

y for the environment because it is not necessary forthe gate we have simulated.

99

100 C. On the non-Markovian dynamics

step

Figure C.1. Tomographic reconstruction of the state shared by S and A at each step ofthe Non-Markovian dynamics. The initial state is represented by a |ϕ+⟩πSA. It is worthnoting that in the fourth step the coherence terms are null, in fact in this case we obtainedthe lowest value of EOFSA. In the fifth step there is a revival of the coherence terms butwith smaller values with respect to the third step. The imaginary matrices are negligiblefor each reconstructed state.

safely neglect them. On the other hand, each non-diagonal matrix acting in the Ssubspace gives rise to a non-trivial single-qubit evolution of the form

US0 =

cos(ν0t)− ı(ϵSz +J)ν0

sin(ν0t) − ϵSy +ıϵS

x

ν0sin(ν0t)

ϵSy−ıϵS

x

ν0sin(ν0t) cos(ν0t) + ı(ϵS

z +J)ν0

sin(ν0t)

,US

1 =

cos(ν1t)− ı(ϵSz−J)ν1

sin(ν1t) − ϵSy +ıϵS

x

ν1sin(ν1t)

ϵSy−ıϵS

x

ν1sin(ν1t) cos(ν1t) + ı(ϵS

z−J)ν1

sin(ν1t)

,(C.4)

where νk = 2π√

(ϵSx )2 + [ϵSz + (−1)kJ ]2 + (ϵSy )2 (k = 0, 1). In order to satisfy theequivalence between GES = |0⟩⟨0|E⊗11S + |1⟩⟨1|E⊗RS(φ) and the conditional-gate|0⟩⟨0|E ⊗ US

0 + |1⟩⟨1|E ⊗ US1 , which has been derived from the transverse Ising-like

model, we want US0 = 11S and US

1 = RS(φ) for a time τ which the conditional gateis performed. The first condition is straightforwardly fulfilled for ν0τ/(2π) = n ∈ Z.As for the second condition, using the definition of RS(φ) given in the Chapter 3,we should enforce

cosφ = cos(ν1τ), sinφ = (ϵSy /ν1) sin(ν1τ),0 = [(ϵSz − J)/ν1] sin(ν1τ) = −(ϵSx/ν1) sin(ν1τ).

(C.5)

These can be solved, at a chosen value of J , for ϵSz = J , which give ν0 = 2π√ϵSy + 4J2

and ν1 = 2πϵSy . In turn, these give us

ϵSy = φ

2πτ,

√4J2τ2 + φ2

4π2 = n. (C.6)

As the value of J is determined by the experimental setting chosen to implementthe gate, these conditions fully specify the values of the parameters in the Isingmodel needed in order to implement a controlled rotation gate with an arbitraryvalue of the angle φ.

C.2 Experimental system-ancilla density matrices 101

C.2 Experimental system-ancilla density matricesIn the Figure C.1 I report the S-A states that have been created experimentallyat each step of the non-Markovian dynamics that we have simulated. We haveused the operative approach put forward in Ref. [148], which is the standard in thetomographic reconstruction of multi-qubit states, to determine the density matricesof the S-A system, which have then been expressed in the Bell basis spanned by|ϕ⟩± = 1√

2(|00⟩ ± |11⟩) and |ψ⟩± = 1√2(|01⟩ ± |10⟩).

Appendix D

Peres-Mermin proof of the KStheorem

The KS theorem proves that noncontextual deterministic outcomes cannot be as-signed to projective quantum measurements. noncontextual assignments are suchthat a projector should get the same value for all the measurements (contexts) inwhich it appears.

The Peres-Mermin (PM) KS proof is based on the set of observables of Table D.1,also known as the PM square, which can take two possible values, ±1. This proof interms of observables can be mapped into a proof in terms of 24 rank-one projectors[198]. The key point is that in the PM square each operator appears in two differentcontexts, one being a row and the other a column. This allows one to distribute thecontexts between Alice and Bob in such a way that Alice (Bob) performs the mea-surements corresponding to the rows (columns). The corresponding Bell scenario,then, is such that Alice and Bob can choose among three different measurementsx, y ∈ 1, 2, 3 of four different outcomes, a, b ∈ 1, 2, 3, 4. Consistently with thePM square, I denote in what follows Alice and Bob’s observables by r and c, forrows and columns, and divide the 4-value outputs in two bits, a = (a1, a2) andb = (b1, b2), respectively, which can take value ±1.

103

104 D. Peres-Mermin proof of the KS theorem

Table D.1. The Peres-Mermin square. One of the simplest KS proofs was derived by Peresand Mermin [198, 199] and is based on the nine observables of this table. The observablesare grouped into six groups of three, arranged along columns and rows. Xn, Yn, and Znrefer to Pauli matrices acting on qubits n = 1 and n = 2, which span a four-dimensionalHilbert space. Each group constitutes a complete set of mutually commuting (and thereforecompatible) observables, defining thus a context. This way, there are six contexts and everyobservable belongs to two different ones. The product of all three observables in each contextis equal to the identity 11, except for those of the third row, whose product gives −11. Itis impossible to assign numerical values 1 or −1 to each one of these nine observables in away that the values obey the same multiplication rules as the observables.

c = 1 c = 2 c = 3r = 1 Z2 X1 X1Z2 = 11r = 2 Z1 X2 Z1X2 = 11r = 3 Z1Z2 X1X2 Y1Y2 = −11

= 11 = 11 = 1∏

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List of Publications andPreprints

• G. Vallone, G. Donati, N. Bruno, A. Chiuri, P. Mataloni, “Experimental Real-ization of the Deutsch-Jozsa Algorithm with a Six-Qubit Cluster State”, Phys.Rev. A 81, 050302(R) (2010).

• A. Chiuri, G. Vallone, N. Bruno, C. Macchiavello, D. Bruß, P. Mataloni, “Hy-perentangled mixed phased Dicke states: optical design and detection”, Phys.Rev. Lett. 105, 250501 (2010).

• A. Chiuri, G. Vallone, M. Paternostro, and P. Mataloni, “Extremal QuantumCorrelations: Experimental Study with Two-qubit States ”, Phys. Rev. A 84,020304(R) (2011).

• A. Chiuri, V. Rosati, G. Vallone, S. Pádua, H. Imai, S. Giacomini, C. Macchi-avello, P. Mataloni, “Experimental Realization of Optimal Noise Estimationfor a General Pauli Channel ”, Phys. Rev. Lett. 107, 253602 (2011).

• L. Aolita, R. Gallego, A. Acín, A. Chiuri, G. Vallone, P. Mataloni, A. Cabello,“Fully nonlocal quantum correlations ”, Phys. Rev. A 85, 032107 (2012).

• A. Chiuri, L. Mazzola, M. Paternostro, P Mataloni, “Tomographic characteri-sation of correlations in a photonic tripartite state ”, New J. Phys. 14, 085006(2012).

• A. Chiuri, C. Greganti, M. Paternostro, G. Vallone, and P. Mataloni, “Experi-mental Quantum Networking Protocols via Four-Qubit Hyperentangled DickeStates”, Phys. Rev. Lett. 109, 173604 (2012).

• A. Chiuri, C. Greganti, and P. Mataloni, “Engineering a C-Phase quantumgate: optical design and experimental realization”, Eur. Phys. J. D 66, 195(2012).

• A. Chiuri, C. Greganti, L. Mazzola, M. Paternostro, P. Mataloni, “LinearOptics Simulation of Non-Markovian Quantum Dynamics”, Nature ScientificReports 2, 968 (2012).

• A. Chiuri, S. Giacomini, C. Macchiavello, P. Mataloni, “Experimental achieve-ment of the entanglement assisted capacity for the depolarizing channel”,arXiv:1206.6881 (2012).

117

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Exploring High-dimensional Hilbert spaces by Quantum Optics · that quantum systems, prepared by a number of diﬀerent approaches, represent a great resource for several tasks, ﬁrst