Royal Swedish Academy of Sciences Physica Scripta

Phys. Scr. T160 (2014) 014035 (4pp) doi:10.1088/0031-8949/2014/T160/014035

Detecting quantum non-Gaussianity ofnoisy Schrödinger cat states

Mattia L Palma1,2, Jimmy Stammers2, Marco G Genoni2,Tommaso Tufarelli2, Stefano Olivares1, M S Kim2 and Matteo G A Paris1

1 Dipartimento di Fisica dell’Università degli Studi di Milano, I-20133 Milano, Italy2 QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK

E-mail: m.genoni@imperial.ac.uk

Received 14 September 2013Accepted for publication 30 October 2013Published 2 April 2014

AbstractHighly quantum nonlinear interactions between different bosonic modes lead to the generationof quantum non-Gaussian states, i.e. states that cannot be written as mixtures of Gaussianstates. A paradigmatic example is given by Schrödinger’s cat states, that is, coherentsuperpositions of coherent states with opposite amplitude. We here consider a novel quantumnon-Gaussianity criterion recently proposed in the literature and prove its effectiveness onSchrödinger cat states evolving in a lossy bosonic channel. We prove that the quantumnon-Gaussianity can be effectively detected for high values of losses and for large coherentamplitudes of the cat states.

Keywords: quantum information, quantum optics, non-classical states, non-Gaussianity

(Some figures may appear in colour only in the online journal)

1. Introduction

The achievement and control of optical nonlinearities atthe quantum level (QNL) is arguably one of the centralgoals of modern quantum optics3. Highly Hamiltoniannonlinear processes such as Kerr interactions, or conditionaloperations such as photon addition/subtraction and Fockstate generation [1] have indeed proven to be powerful toolsfor investigating and exploiting the quantum nature of theelectromagnetic field. In light of this, techniques that reliablyverify the successful experimental implementation of QNLare highly desirable. Closely linked to the concept of opticalnonlinearity is that of the set of Gaussian states [2], whichcan be seen as a collection of states that can be obtainedby applying quadratic Hamiltonians to thermal states ofradiation. It is easy to realize that, without the use of QNL,one is invariably limited to the preparation of Gaussian statesand their convex combinations. Conversely, the successfuldetection of a state that cannot be written in this form,a quantum non-Gaussian state, can only be explained bythe presence of a QNL during the preparation stage. The

3 In this paper, we use the term ‘nonlinearity’ to indicate any process thatcannot be realized with Hamiltonians that are second-degree polynomialsin the bosonic operators. More technically, a nonlinear process cannot beobtained as a convex combination of Gaussian operations.

detection and characterization of quantum non-Gaussianity(QNG) thus acquires fundamental importance in the studyof continuous-variable quantum states. The literature presentsa number of methods for detecting non-classical states [3],defined as states that cannot be written as mixtures ofcoherent states, or quantifying the deviation of a quantumstate from a Gaussian [4–7]. However these methods are,respectively, not able to discriminate between quantumnon-Gaussian states and squeezed states, and not suitablefor distinguishing between quantum non-Gaussian states andmixtures of Gaussian states. In fact, excluding the case ofstates with negative Wigner function, which are certainlyquantum non-Gaussian, no general method for distinguishingbetween the two sets is known. This state of knowledgetriggered the development of sufficient methods to detectQNG in noisy setups, where no negativity of the Wignerfunction can be observed [8, 9], allowing one to witness thesuccessful implementation of QNL processes despite the highlevels of noise [10, 11]. In this paper we apply the methodintroduced in [9], to investigate QNG of Schrödinger catstates [12–14] undergoing severe optical loss, such that itsWigner function becomes positive everywhere. Focusing onthe so-called odd and even cat states, we find that QNG can bewitnessed for any value of the model parameters in the formercase, and for a significant but finite range of parameters in the

0031-8949/14/014035+04$33.00 1 © 2014 The Royal Swedish Academy of Sciences Printed in the UK

Phys. Scr. T160 (2014) 014035 M L Palma et al

latter. In what follows, we start by briefly summarizing theresults of [9] and the basics of the employed physical model,and then use these tools to carry out a detailed analysis of theproblem of interest.

2. Quantum non-Gaussianity criteria

We here review QNG criteria based on the Wigner functionwhich have been proposed in [9]. We will restrict ourselveshere to single-mode systems, described by bosonic operatorssatisfying the commutation relation [a, a†] = 1. Any singlemode quantum state % can be equivalently described byits characteristic function or its Wigner function, definedrespectively as

� [%](� )=Tr[%D(� )], W [%](↵)=Z

d2�

⇡2e�

⇤↵��↵⇤� [%](� ),

(1)where D(� ) = exp{� a† � � ⇤a} represents the displacementoperator. A quantum state is called Gaussian if and only ifits Wigner function is a Gaussian function.

The Gaussian convex hull is the set of states

G =⇢% 2H | % =

Zd� p(�) |

G

(�)ih G

(�)|�

, (2)

where H denotes the Hilbert space of continuous-variablequantum states, p(�) is a proper probability distribution and|

G

(�)i are pure Gaussian states. We define a quantum statequantum non-Gaussian iff it is not possible to express it as aconvex mixture of Gaussian states, that is, iff % /2 G. In [9] itis proved that for any % 2 G, the following inequality holds

W [%](0)> 2⇡

e�2n̄(n̄+1) , (3)

where n̄ = Tr [%a†a]. Together with the observation that theset G is closed under any Gaussian map E

G

, inequality (3)leads to the following generalized QNG criterion.

Criterion 1. Given a quantum state % and a Gaussian mapE

G

, define the QNG witness

1 [%, EG

] = W [EG

(%)](0) � 2⇡

exp{�2n̄E(n̄E + 1)} , (4)

where n̄E = Tr[EG

(%)a†a]. Then,

9 EG

s.t. 1[%, EG

] < 0 ) % /2 G. (5)

In the following section, we will apply this criterionto Schrödinger’s cat states evolving in a lossy channel. Asadditional Gaussian maps E

G

we will consider the simplestexamples, that is, displacement operations D(�), squeezingoperations S(⇠) = exp

� 12⇠(a

†)2 � 12⇠

⇤a2

and combinationsof the two.

3. Detecting quantum non-Gaussianity ofSchrödinger’s cat states

Schrödinger’s cat states are defined as

| ↵,⇠ i = | �↵i + ⇠ |↵iN , (6)

where, without losing generality, ↵ 2 R and N =p1 + ⇠ 2 + 2⇠e�2↵2 denote the normalization constant. By

considering the parameter ⇠ = 1 and �1, one obtainsrespectively the so-called even | eveni and odd | oddi catstates. In the following we will restrict our analysis to theseparticular classes of states, whose Wigner functions areplotted in figure 1 for ↵ = 1. We will consider their evolutionin a lossy bosonic channel described by the Markovian masterequation

d%dt

= � a%a† � �

2(a†a% + %a†a). (7)

The resulting time evolution is characterized by a singleparameter ✏ = 1 � e�� t and it models both the incoherentloss of photons in a zero temperature environment, and theperformances of detectors having an efficiency parameter ⌘ =1 � ✏. We will denote the evolved state as E✏(%0). In theWigner function picture, the evolution can be analyticallysolved by means of the formula

W [E✏(%0)](�) = 2⇡✏

Zd2�0 W [%](�0)

⇥ exp

(

�2���� �0p1 � ✏

��2

✏

)

. (8)

Also, the average values of the operators needed to computethe QNG witnesses 1[E✏(%0), EG

] can be analyticallyevaluated as

n̄✏ = Tr [E✏(%0)a†a] = (1 � ✏)n̄0 ,

ha2i✏ = Tr [E✏(%0)a2] = (1 � ✏)ha2i0 , (9)

where for an initial Schrödinger’s cat state %0 = | ↵,⇠ ih ↵,⇠ |the initial averages read

n̄0 = ↵2(1 + ⇠ 2 � 2⇠e�2↵2)

N 2, ha2i0 = ↵2 . (10)

We will focus on large noisy parameters, i.e. ✏ > 0.5 such thatno negativity of the Wigner function can be observed [15]. Inparticular we will determine the maximum values for whichwe observe a violation

✏max [%] = max {✏ : 9EG

s.t. 1 [E✏(%), EG

]6 0} . (11)

The quantity ✏max is a relevant figure of merit to assessour criterion. In fact, having values of ✏max close to unitycorresponds to situations where the criterion is able to detectQNG even in a highly noisy channel or, equivalently, by usinghighly inefficient detectors.

3.1. Odd cat states

We will start here by considering a odd cat state | oddi. TheWigner function of the initial pure state, plotted in figure 1(left), is squeezed along the P quadrature and presents aminimum (negative) value at the origin of the phase space.Then we can consider the QNG witness optimized over anadditional squeezing operation, i.e.

1odd(s) =1[E✏(| oddih odd|, S(s)]. (12)

2

Phys. Scr. T160 (2014) 014035 M L Palma et al

Figure 1. Contour plots of the Wigner functions W [%](�) of the odd cat state (left) and even cat state (right) for ↵ = 1.

Figure 2. (Left) Optimized QNG witness 1odd

(sopt

) for odd cat states evolving in a lossy channel, as a function of ✏ and for different valuesof ↵: red dotted line: ↵ = 0.5; green dashed line: ↵ = 1.0; blue solid line: ↵ = 1.5. (Right) maximum value of the noise parameter ✏max suchthat the optimized QNG witness 1odd(sopt) takes negative values, as a function of the coherent states amplitude ↵ (blue solid line).The dashed red line corresponds to the maximum value of the noise parameter ✏max obtained without considering an additional squeezing(i.e. for s = 0).

The average photon number of the squeezed evolved odd catstate, needed to determine 1odd(s), can be evaluated as

n̄(odd)(s) = (µ2s + ⌫2

s )n̄✏ + 2µs⌫sha2i✏ + ⌫2s , (13)

where µs = cosh s, ⌫s = sinh s, and the values of n̄✏ andha2i✏ can be obtained from equations (9) and (10) by setting⇠ = �1. We will then look for values of the additionalsqueezing parameter s, such that the criterion 1 is fulfilled.In particular one can then try to optimize over the additionalsqueezing s, for each value of ↵ and ✏. By exploitingthe invariance of the Wigner function in the origin undersqueezing operation, this optimization corresponds to theminimization of the average photon number in equation (13).An analytic solution can be obtained, yielding

sopt

= �14

log1 � e2↵2 � 4↵2e2↵2

(1 � ✏)1 � e2↵2 � 4↵2(1 � ✏)

. (14)

The behaviour of the resulting optimized witness1odd(sopt) is plotted as a function of ✏ for different valuesof ↵ in figure 2 (left): for the values of ↵ considered, QNGof odd cat states can be detected by the criterion for all thevalues of ✏. As pictured in figure 2 (right), it is possible toprove numerically that ✏max is equal to one, that is, QNG can

also be detected for all values of noise, for larger values of↵, in cases without an additional squeezing, by considering1odd(0), ✏max ⇡ 0.5.

3.2. Even cat states

We now consider the problem of detecting QNG for even catstates | eveni evolving in the lossy channel E✏ . By inspectingthe plot in figure 1 (right), we notice that the Wigner functionof the initial pure state is squeezed and that its minimumis along the P quadrature axis. As a consequence wewill consider a combination of displacement and squeezingoperations, in order to construct the following QNG witness:

1even(�, s) =1[E✏(| evenih even|, D(i�)S(s)]. (15)

The average photon number n̄(even)�,s to be used in the

calculation of 1even(�, s) reads

n̄(even)�,s = (µ2

, + ⌫2s )n̄✏ + 2µs⌫sha2i✏ + ⌫2

s +�2, (16)

where in this case the values of n̄✏ and ha2i✏ can be obtainedfrom equations (9) and (10) by setting ⇠ = 1.

We will look for the optimal values {�opt, sopt} whichminimize 1even(�, s) for given values of the noise parameter

3

Phys. Scr. T160 (2014) 014035 M L Palma et al

Figure 3. (Left) Optimized QNG witness 1even(�opt, sopt) for even cat states evolving in a lossy channel, as a function of ✏ and for differentvalues of ↵: red dotted line: ↵ = 0.4; green dashed line: ↵ = 0.6; blue solid line: ↵ = 1.0. (Right) maximum value of the noise parameter✏max such that the optimized QNG witness 1even(�opt, sopt) takes negative values, as a function of the coherent states amplitude ↵. Noticethat for � = 0 and s = 0, one would obtain ✏max = 0 for all values of ↵.

✏ and the coherent states amplitude ↵. Unfortunately, inthis case an analytical approach cannot be pursued, sincethe displacement operation changes both the value of theWigner function in the origin of the evolved state and theaverage photon number in equation (16). As a consequencethe optimal values will be obtained numerically for eachcouple of values of ✏ and ↵. The corresponding optimizedQNG witness 1even(�opt, sopt) has been evaluated and plottedin figure 3 (left). We can clearly observe that, thanks tothe additional Gaussian operations, we are able to detectQNG for non-trivial values of the noise parameter, that is for✏ > 0.5. The maximum value of the noise parameter ✏max forwhich we observe negative values of 1even has been obtainednumerically and it is plotted in figure 3 (right). For smallvalues of the coherent states amplitude ↵, one can detect QNGfor practically all the possible values of noise: in particularfor ↵ 6 0.1 we have ✏max ⇡ 1 (up to numerical precision),and for ↵ < 0.6 we still have ✏max > 0.99. Unfortunately, byfurther increasing the amplitude up to ↵ = 1, the witnessperformances are drastically reduced and ✏max approaches itslimiting value ✏max ⇡ 0.5. Notice that if we do not consideradditional operations, that is by setting � = 0 and s = 0, oneobtains ✏max = 0 for all values of ↵, that is QNG cannot bedetected.

4. Conclusions

We have applied a recently proposed QNG criterion toSchrödinger’s cat states evolving in a lossy bosonic channel.We observe that by optimizing the witness by additionalGaussian operations, one can detect QNG, and thus a QNLprocess, for non-trivial values of the noise parameter, that isfor severe optical loss yielding a positive Wigner function.

In particular, the criterion works really well for odd catstates, while it is effective only for small amplitude even catstates.

Acknowledgments

MGG acknowledges support from UK EPSRC(EP/I026436/1). TT and MSK acknowledge the supportfrom the NPRP 4- 426 554-1-084 from Qatar NationalResearch Fund. SO and MGAP acknowledge the supportfrom MIUR (FIRB ‘LiCHIS’ no. RBFR10YQ3H).

References

[1] Kim M S 2008 J. Phys. B: At. Mol. Opt. Phys. 41 133001[2] Ferraro A, Olivares S and Paris M G A 2005 Gaussian States

in Quantum Information (Napoli: Bibliopolis)[3] Glauber R 1963 Phys. Rev. 131 2766[4] Genoni M G, Paris M G A and Banaszek K 2007 Phys. Rev. A

76 042327[5] Genoni M G, Paris M G A and Banaszek K 2008 Phys. Rev. A

78 060303[6] Genoni M G and Paris M G A 2010 Phys. Rev. A 82 052341[7] Barbieri M et al 2010 Phys. Rev. A 82 063833[8] Filip R and Mista L Jr 2011 Phys. Rev. Lett. 106 200401[9] Genoni M G, Palma M L, Tufarelli T, Olivares S, Kim M S and

Paris M G A 2013 Phys. Rev. A 87 062104[10] Jezvek M et al 2011 Phys. Rev. Lett. 107 213602[11] Jevzek M et al 2012 Phys. Rev. A 86 043813[12] Ourjoumtsev A, Tualle-Brouri R, Laurat J and Grangier P

2006 Science 312 83[13] Wakui K, Takahashi H, Furusa A and Sasaki M 2007 Opt.

Express 15 3568[14] Neergaard-Nielsen J S et al 2006 Phys. Rev. Lett.

97 083604[15] Paavola J et al 2011 Phys. Rev. A 84 012121

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