Gaussian error correction of quantum states in a correlated noisy channel

Mikael Lassen*,1 Adriano Berni,1 Lars S. Madsen,1 Radim Filip,2 and Ulrik L. Andersen1

1Department of Physics, Technical University of Denmark, Fysikvej, 2800 Kongens Lyngby, Denmark2Department of Optics, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic

(Dated: February 20, 2018)

Noise is the main obstacle for the realization of fault tolerant quantum information processingand secure communication over long distances. In this work, we propose a communication protocolrelying on simple linear optics that optimally protects quantum states from non-Markovian or corre-lated noise. We implement the protocol experimentally and demonstrate the near ideal protection ofcoherent and entangled states in an extremely noisy channel. Since all real-life channels are exhibit-ing pronounced non-Markovian behavior, the proposed protocol will have immediate implicationsin improving the performance of various quantum information protocols.

PACS numbers: 03.67.Hk, 03.67.Dd, 42.50.Dv

A future quantum information network will consist ofquantum communication channels that connect differentnodes of the network [1]. These quantum links could ei-ther be used for establishing a secret key between nodes,and thereby allowing for unconditional secure commu-nication, or they could be used for communication ofquantum information between quantum processors. Thetransmission of quantum information can be carried outeither by sending the quantum states directly through thequantum links, or, by establishing entanglement betweenthe nodes and subsequently use teleportation for transfer-ring the quantum states [2]. The transmitted quantuminformation can be conveniently described by quantumstates of two-level systems, that is, qubits, but a vastnumber of real world realizations rely on modes of theelectro-magnetic fields described by quantum systems ofcontinuous variables [35].

All these quantum communication schemes, however,will be ultimately limited in their performance by thenoise that inevitably invades all realistic communicationchannels. Such noise may eventually lead to a lack of se-curity in quantum key distribution (QKD) and to errorsin directly transmitted quantum states. To combat thedetrimental noise of the channel, various strategies havebeen proposed including noise-robust QKD protocols [68], entanglement distillation protocols and error correct-ing codes [2, 917]. The complexity of these schemesstrongly depends on the type of noise in the channel. Ithas been shown that if the noise is additive Gaussian,and the information carrying states are Gaussian, thenneither entanglement distillation nor quantum error cor-recting codes can be realized by simple Gaussian opera-tions [1821]. On the other hand, for non-Gaussian errormodels (such as random attenuation and phase diffusion)simple Gaussian operations suffice to correct the errors inCV systems [17, 2226]. However, in many conventionalcommunication systems, the error model is Gaussian, andthus it appears that one is faced with the complexity ofimplementing experimentally challenging non-Gaussianoperations for enabling fault-tolerant quantum commu-

nication [2730].

In the above mentioned No-Go theorems, the Gaussiannoise is assumed to be uncorrelated. However, with theminiaturization of solid state systems and the increas-ing speed of optical communication, the noise in todayscommunication systems inevitably exhibit correlations intime and space [31, 32], and thus it will be relevant toconsider channels with correlated noise. In this case, theNo-Go theorem does not apply. Here we propose a simpleencoding and decoding technique based on linear opti-cal transformations that ideally protects arbitrary quan-tum states from Gaussian noise in correlated quantumchannels. The protocol works in particular for Gaussianquantum states, and thus, perfect Gaussian error correc-tion with Gaussian transformations is possible due to thecorrelations of the channel noise. We implement the pro-tocol for coherent and entangled states of light, therebycharacterizing the protocol for the two main communi-cation approaches; direct communication and teleporta-tion based communication. Correlated noise in quantumcommunication were initially considered for qubits lead-ing to the concept of decoherence free subspace [3335]but recently a few theoretical studies have also addressedcorrelated noise in bosonic channels using similar strate-gies [3639].

Our error-protecting scheme is depicted in Fig. 1 forthe case of two partially correlated channels with classicalnoise which could either represent two spatially separatedchannels with spatial correlations or two consecutive usesof a single channel with temporal correlations (corre-sponding to a non-Markovian channel). The channels arenoisy and a part of the noise is described by the perfectlycorrelated complex random variables 1 and 2 [40]. Todescribe an asymmetry of the correlated classical noiseamong the two channels, we assume 1 =

g1vC and

2 =g2vC , where the magnitude of the classical excess

noise contributions are given by the factors g1 > 0 andg2 > 0, and vC is the complex random variable corre-sponding to the classical fluctuations from the environ-ment. Assuming the transmission of the channels to be

arX

iv:1

308.

2102

v2 [

quan

t-ph

] 1

0 D

ec 2

013

2

FIG. 1. Schematic of the proposed error correcting schemefor the protection of an arbitrary quantum state against cor-related noise. The scheme is divided into three different stepsassociated with the encoding, noise addition and decoding.The insets serve as an illustration of the function of the pro-tocol on a coherent state. They are ensemble measurementsof the coherent state quadratures as a function of time atdifferent positions of the protocol. The input state exhibitsquantum noise limited fluctuations but after the noisy chan-nel, the state evidently contains excess noise. This is thenremoved at the correction stage, where the noise is clearlyseparated from pure quantum state.

, the Bogoliubov transformations for the channels are

a1 =a1 +

1 v1 +

g1vC (1)

a2 =a2 +

1 v2 +

g2vC (2)

where a1,2 and a1,2 are the annihilation field operators as-sociated with the input and output modes, and v1,2 rep-resent the uncorrelated thermal fluctuations (which for azero-temperature channel is identical to the loss-inducedvacuum fluctuations). For circumventing the noise, wefollow the encoding and decoding strategy illustrated inFig. 1. The channel inputs are prepared by combin-ing the input signal (bin) with an auxiliary vacuum state

(baux) on a beam splitter, and subsequently introducing arelative phase shift of between the two resulting states.The encoding transformation can be written as (a1, a2) =

(Tebin

1 Tebaux,

1 Tebin

Tebaux) where Te

is the transmissivity of the encoding beam splitter. Thedecoding transformation is the reverse of the encodingtransformation, and thus represented by the transforma-tions; (bout, b

aux) = (

Tda

1

1 Tda2,

1 Tda1 Tda

2) where Td is the transmissivity of the decoding

beam splitter. By choosing Te = Td = g2/(g1 + g2), theinput-output relation for the entire scheme is

bout =bin +

1

(g2

g1 + g2v1

g1

g1 + g2v2

),

(3)

which corresponds to a purely lossy but noiseless chan-nel for any values of g1 and g2 (for the zero-temperature

channel). The correlated classical noise of the environ-ment has therefore been completely separated from thesignal; the noise will leave one output of the beam split-ter whereas the signal will leave the other output. Even ifthe two channels are partially uncorrelated, our scheme isperfectly removing the correlated part of the noise with-out amplifying the uncorrelated part. For a general treat-ment of the protocol, see Supplemental Material [41].

SH Pump

SH Pump

IR Seed

IR Seed

OPA 2

Diff

LO

HD 1 BBS

OPA 1

BS 99/1

Diff

DAQ/SP

AM PM

Auxiliary beam

LO

Diff

LO

HWP

PBS

HD 2

HD 3

HWP

BBS

BBS

BBS

HWP

Beam A

Beam B

AM PM

100 100

Preparation of Thermal States

Preparation of Coherent States

Squeezed beam

Squeezed beam

q

q

q

FIG. 2. Schematic of the experimental setup. The error cor-recting part is inside the shaded box which contains two inputand two outputs: The inputs are for the input quantum stateand for noise addition whereas the two outputs are associatedwith the two output of the protocol. The quantum statesat the inputs are either coherent states or entangled states.Coherent states are prepared at the sideband frequencies of4.9MHz using a pair of modulators; amplitude (AM) andphase modulators (PM), whereas the entangled states are gen-erated by interfering two squeezed beams on a balanced beamsplitter (BBS). The squeezed beams are produced by opticalparametric amplifiers (OPA1 and OPA2) which are pumpedby laser beams at 532nm (denoted Pump) and stabilizedwith seed beams at 1064nm. When coherent states areused as inputs to the protocol, the OPAs are not operationaland thus the coherent states produced in front of OPA2 arebypassing the squeezing operation and injected directly intothe protocol. Correlated noise in the two channels is producedby an auxiliary beam that traverses a pair of noise-controlledmodulators and subsequently injected into the scheme. Forthe verification, high-efficiency homodyne detectors (HD) areused; HD2 and HD3 are measuring the two outputs of theprotocol whereas