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Hybrid non-locality test in a microscopic-macroscopic system

Nicolo Spagnolo,1,2 Chiara Vitelli,1 Mauro Paternostro,3 Francesco De Martini,4, 1 and Fabio Sciarrino1, 5,∗

1Dipartimento di Fisica, Sapienza Universita di Roma, piazzale Aldo Moro 5, 00185 Roma, Italy2Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, piazzale Aldo Moro 5, 00185 Roma, Italy

3School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom4Accademia Nazionale dei Lincei, via della Lungara 10, I-00165 Roma, Italy

5Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (INO-CNR), largo E. Fermi 6, I-50125 Firenze, Italy

We investigate the feasibility of a non-locality test in a bipartite system composed of a single-photon anda multi-photon field that embodies the paradigm of a microscopic-macroscopic entangled state. We run anintrinsically hybrid test requiring polarization measurements on the microscopic part of the state and phase-space ones on the macroscopic counterpart. Our analysis shows that, under ideal conditions, maximal violationof a Bell-CHSH inequality is achievable regardless of the number of photons in the macroscopic part of the state.The difficulty in observing non-locality when losses and detection inefficiency are included can be overcomeusing a hybrid entanglement witness that allows to efficiently correct for losses in the few photon regime.

PACS numbers:

The possibility to observe quantum features at the macro-scopic level stands out as one of the most challenging yet fas-cinating open questions in modern quantum physics. The dif-ficulties inherent in such a quest are manifold. First, the un-avoidable interaction with the surrounding environment liesdeeply at the roots of the loss of coherence and corruptionof quantum correlations in a quantum state [1]. Such effectsare commonly believed to become more severe as the sizeof the system being studied grows [2–5]. Second, one facesthe debated problem of achieving a measurement-precisionthat is sufficient to observe quantum effects at such macro-scales. Recent studies have shown that, for certain Hamilto-nian models, classicality emerges from the coarse-graininessof measurements performed over a quantum system [6]. Onthe other hand, examples of genuinely macroscopic states vi-olating non-locality constraints under quite coarse measuringapparatuses have been provided [7]. In this context, it hasbeen experimentally proven that a dichotomic measurementperformed upon a multi-photon entangled state is not suffi-cient to catch quantumness [8]: the accuracy of the measure-ment is crucial for the observation of quantum features andshould be put on the same footing as the use of proper en-tanglement and non-locality criteria for macroscopic quantumsystems [7, 9–15].

In this paper we investigate on the possibility to observequantumness beyond the microscopic domain by studying aparadigmatic microscopic-macroscopic system, dubbed fromnow on as micro-macro system, obtained from a fully mi-croscopic entangled system through an amplification process[16–18]. More precisely, our study addresses a situation lyingbetween the quantum (microscopic) and the classical (macro-scopic) worlds by looking at the occurrence of quantum phe-nomena in increasing-size systems. The nature of the stateat hand implies the use of a carefully designed strategy forthe inference of quantumness. Here, we identify in the useof a hybrid non-locality test a valuable tool for our goals.While the microscopic part of the state is measured using spin-1/2 projection operators, the macroscopic counterpart under-

goes phase-space measurements based on the properties of itsWigner function [11]. We analyze the effects of losses on aBell-CHSH-like inequality test [19] and show that maximumviolation is achieved when losses are absent, regardless ofthesize of the macroscopic part of the state. This is not the caseunder non-ideal conditions. However, we show how lossescan be efficiently taken into account so as to infer entangle-ment of our multi-photon state. Our study draws a viable path-way towards the experimental demonstration of non-localityat the boundary between microscopic and macroscopic world.

Hybrid non-locality test.–We now describe the test inspiredby the proposal put forward in Ref. [11] that allows us to re-veal the non-local properties of our micro-macro state. Letus consider a general micro-macro state with its microscopicpart embodied by a single-photon polarization state (a qubit).We take the macroscopic part, on the other hand, as encodedin the multi-photon state of a continuous-variable (CV) sys-tem. The two subsystems are supposed to be entangled by amechanism whose details are inessential for our tasks here.A benchmark state of such situation will be provided later

FIG. 1: (Color online) Hybrid non-locality and entanglement teston an optical microscopic-macroscopic state generated by a“black-box”. The single-photon modekA is measured by a polarization de-tection apparatus, while the multi-photon modekB undergoes bothpolarization and homodyne measurements.

2

on. Polarization measurements performed over state of thesingle-photon modekA are described by the Pauli spin oper-ator σA(φ) = |φ〉A〈φ| − |φ⊥〉A〈φ⊥|, whereφ is the direc-tion identifying the polarization state in the Poincare sphereandφ⊥ is its orthogonal direction. The CV measurements, onthe other hand, are given byΠBχ,χ⊥

(αχ, χ)=ΠBχ (αχ)⊗1Bχ⊥,

where ΠBi (αi)=DBi (αi)(−1)n

Bi DB †

i (αi) is the displacedparity operator built from the displacementDB

i (αi) (αi∈C)and the photon-number operatornBi (i=χ, χ⊥ stands forthe polarization state). The average value of the mea-surement operator on a stateρBi of the multi-photon modeis directly related to the value of its Wigner functionat αi WB

Φ (αi)=(2/π)Tr[ΠBi (αi)ρBi ]. The latter can be

straightforwardly reconstructed experimentally using homo-dyne measurements. We define the qubit-CV correlatorC(αχ, χ;φ)=〈σA(φ)⊗ΠBχ,χ⊥

(αχ, χ)〉, where the average isevaluated on a general micro-macro stateρAB, and the Bell-CHSH parameter

B=C(α′χ, χ

′;φ′)+C(α′χ, χ

′;φ)+C(αχ, χ;φ′)−C(αχ, χ;φ).(1)

As the outcome of theσA(φ) and ΠBχ,χ⊥(αχ, χ) measure-

ments can only be±1, the use of a local hidden variable(LHV) model imposes the bound|B|≤2 [19] on the Bell-CHSH parameter. A violation of this bound confutes all LHVtheories. The measurement settings for the single-photonmodekA [multi-photon modekB] are given by the measuredpolarizations (φ, φ′) [measured polarizations (χ, χ′) and thechosen phase-space points (αχ, α′

χ)]. This requires a stan-dard polarization detection system for the microscopic modeand a homodyne detection system for the multi-photon one,as shown in the scheme presented in Fig. 1.Hybrid entanglement witness with losses.–The non-localitytest above can be modified so as to embody a witness ableto reveal entanglement when the state at hand is affected bylosses. This is modeled by inserting a beam-splitter of trans-mittivity η∈[0, 1] in the path of the modes at hand, “tapping”the corresponding signal [7]. The choiceη=1 (η=0) corre-sponds to a lossless (fully-lossy) process. To this end, themeasurement performed on the~πχ polarization of the multi-photon part is replaced by the operator [15]

OBχ (αχ; η)=

1η Π

Bχ (αχ)+

(

1− 1η

) 1Bχ if η∈]0.5, 1],2ΠBχ (αχ)−1Bχ if η∈]0, 0.5].

(2)

In this way, the overall measurement on the macroscopic sub-system readsOBχ,χ⊥

(αχ, χ; η)=OBχ (αχ; η)⊗1Bχ⊥

. We havethat for any separable state undergoing measurement afterthe lossy process occurs,|〈OBχ,χ⊥

(αχ, χ; η)〉η|≤1 [14, 15,20, 21]. Hence, by introducing the micro-macro correlatorCη(αχ, χ, φ)=σA(φ)⊗OBχ,χ⊥

(αχ, χ; η), we define

Wη=Cη(α′χ, χ

′, φ′)+Cη(α′χ, χ

′, φ)

+Cη(αχ, χ, φ′)−Cη(αχ, χ, φ).(3)

Any separable state undergoing a lossy process on modekB

is bound to satisfy|Wsepη |≤2 [22]. Violation of this inequal-

ity witnesses entanglement in the system. It is important tonotice that, in virtue of the assumption that the macrostateofmodekB undergoes lossesη before (rather thanat) detection,this entanglement witness reveals the presence of entangle-mentwithoutany assumption on the micro-macro source [seeFig.1]. On the other hand, the lossy mechanism can be shiftedto occur just before measurement, thus modeling the effectsofa non-ideal detection efficiency. Forη=1, Wη coincides withthe Bell-CHSH parameter in Eq. (1).

Experimental benchmark.–As a benchmark for the hy-brid Bell-CHSH test and entanglement witness describedabove, we analyze the micro-macro state-source ad-dressed in Refs. [16, 17]. A layout of the systemis reported in Fig. 2. The polarization singlet state|ψ−〉AB=(|H〉A|V 〉B−|V 〉A|H〉B)/

√2 of a photon pair is

generated in a nonlinear crystal through a spontaneous para-metric down-conversion (SPDC) process. Here|H〉 (|V 〉)stands for horizontal (vertical) polarization state. The pho-ton populating modekB is then injected into an optical para-metric amplifier (OPA) in a collinear configuration. Sincethe OPA implements a unitary operation, the symmetryof |ψ−〉AB is preserved by the amplification process andthe overall state|Ψ−〉AB=(|φ〉A|Φφ⊥〉B−|φ⊥〉A|Φφ〉B)/

√2

maintains rotational invariance form for any polarizationba-sis [16]. Here,|Φφ〉 are the multi-photon states generated byamplification of a single photon polarization state|φ〉 accord-ing to the unitary evolution operator resulting from the Hamil-

tonianHOPA=(ı/2)~χe−ıφ(

a†2φ −a†2φ⊥

)

+ H.c., whereχ is

the interaction strength andg=χt is the nonlinear gain [22].The average number of photons generated by the amplifierwhen a single photon in the polarization state|φ〉B is injectedare respectively〈nφ〉=3n+1 and〈nφ⊥

〉=nwith n = sinh2 g.Quantum entanglement between micro- and macro-part of|Ψ−〉AB has been demonstrated [17] under a supplementaryassumption on the source [18]. The optical parametric ampli-fier performs the optimal cloning process only for equatorialpolarization~πφ=(~πH+eıφ~πV )/

√2. We thus restrict our at-

tention to this subset of polarization states, which motivates

FIG. 2: (Color online) Layout of the micro-macro source based onthe process of optical parametric amplification of a single photon be-longing to an entangled pair. On the single-photon modekA, the fieldis analyzed in the~πφ, ~πφ⊥

equatorial polarization basis throughmeasurements of the Pauli operatorσA(φ). On the multi-photonmodekB, the field is probed by a homodyne detection apparatusafter projection in the~πχ equatorial polarization state.

3

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

2.0

2.5

LHV

g 3g 2g 1g 0.4g 0

FIG. 3: (Color online) Bell-CHSH parameterBη as a function of thenumber of lost photons(1− η)〈n〉 for different values of the gaing.We show the local realistic boundaryBLHV=2.

our choice forσA(φ) performed above.Results.–We now discuss the results of the Bell-CHSH testand the application of the entanglement witness to the micro-macro state given in Fig. 2. We begin analyzing the Bell-CHSH inequality (1) in the lossless case (η = 1). In thiscondition, the correlation operator evaluated on the|Ψ−〉ABstate takes the form [22]

C(Xχ, Pχ, χ;φ)=(1−Z) cos[2(χ−φ)]e−Z , (4)

where Z=2(e−2gX2

χ+e2gP

2

χ) is a function of the ro-tated variables Xχ=Xχ cos(χ/2)−Pχ sin(χ/2) andPχ=Xχ sin(χ/2)+Pχ cos(χ/2), (Xχ, Pχ) are the fieldquadratures andαχ=Xχ+ıYχ. The correlator in Eq. (4)is maximized at the origin of the phase space, whereC(0, 0, χ;φ)= cos[2(χ − φ)], which is independent of thegain g and, in turn, of the number of generated photons.The correlator has the same form as for a Bell-CHSH testperformed on an entangled polarization photon-pair wherespin-1/2 operators are measured. The Bell-CHSH parameterB is then maximized by choosing the measurement settingsfor (φ, φ, χ, χ′) corresponding to such simple case, whichensures the maximum degree of violation of the local realisticboundary, i.e.B=2

√2.

We are now in a position to address the possibilityto observe micro-macro non-locality under realistic ex-perimental conditions. We thus analyze the effects ofdetection efficiency at the homodyne apparatus, whileother sources of experimental imperfections are discussedin the accompanying Supplementary Material [22]. Themeasurement of the generalized parity operator on themulti-photon modekB can be performed using homodynedetection. Although the homodyne measurements neededin order to infer the Wigner function of the macro-partof our state are typically performed with a high quantumefficiency (η∼70%−90%), they are far from ideal and weshould include in the qubit-CV correlator the possibilityof a non-unitary detection efficiency on modekB . This isdone by taking the convolution of the lossless correlatorwith a Gaussian function according toCη(Xχ, Pχ, χ;φ) =

2π(1−η)

∫

dX ′χdP

′χC(X ′

χ, P′χ, χ;φ)Kη(Xχ, Pχ, X

′χ, P

′χ),where Kη is the Gaussian kernel of the transforma-

tion [22, 24]. By restricting our attention to the origin ofthe phase space, where maximum non-classical effects areachieved, we getCη(0, 0, χ;φ)= cos[2(χ−φ)]L(g, η), where

L(g, η)= η[1 + 2n(1 − η)]

(1 + 4η(1− η)n)3/2(5)

is a loss-function for the test. Hence, the maximum amountof violation is directly determined by the loss-function asBη=BL(g, η). In Fig. 3 we show the value ofBη as a func-tion of the average number of lost photons(1−η)〈n〉. Ev-idently, the Bell-CHSH inequality Eq. (1) is satisfied whenonly a moderate number of photons is lost. A lower boundηlim=1/

√2 for the detection efficiency can be found below

which non-locality is not observed anymore. On the otherhand, at set values ofη there is a minimum gainglim(η) abovewhich the micro-macro correlations can be explained by a lo-cal hidden-variable model. Such threshold value decreaseswith the reduction of the efficiencyη. The behavior ofBηin the (η, g)-plane is shown by the contour plot in Fig. 4 (a),where we see that the sensitivity to losses in the multi-photonstate increases when the number of photons progressively be-comes larger. In order to relate the violation of the Bell-CHSHinequality to intrinsically non-classical features enforced atthe level of the macro-part of the state, in Fig. 4 (b) we showthe negativity of the Wigner function of an amplified single-photon state againstη andg [23]. We observe that the tran-sition ofBη to the region below the classical limit is directlylinked to the decrease in the negativity of the Wigner functionitself. Indeed, the value of the micro-macro correlatorCη cal-culated at the origin of the phase space is determined by theexcursion of the Wigner function inXχ=Pχ=0, as a functionof the polarization of the injected photon. When losses are in-troduced, the negativity is damped and such an excursion nolonger allows for the violation of the Bell-CHSH inequality.

We complement the analysis of our micro-macro systemby discussing the use of the entanglement witness describedabove. The evaluation of the correlation operator over state|Ψ−〉AB after losses leads toCη(αχ, χ, φ)=h(η)Cη(αχ, χ;φ),

FIG. 4: (Color online) (a) Contour plot of the shifted loss functionL(g, η)−2−1/2 as a function of the gaing and the detection effi-ciencyη. (b) Contour plot of the negativity of the Wigner functionof an amplified single-photon state [23] againstg andη, evaluated inthe origin of the phase space. In both panels the solid line divides theregion of non-locality (|Bη|>2, above the line) from the LHV one(|Bη| ≤ 2, below the line).

4

FIG. 5: (Color online) (a)Wη against the detection efficiencyη and the nonlinear gaing. (b) Contour plot of the effective loss functionh(η)L(η, g). Entanglement can be revealed in the region above the black line. (c) Summary of the results obtained from our tests. We identifythree regions in the(η, g) space, depending on whether non-locality or entanglement can be demonstrated with our technique.

whereh(η)=1/η (h(η)=2) for 1/2<η≤1 (0≤η≤1/2) [22].Therefore, the entanglement witness can be directly obtainedfrom the Bell-CHSH parameter asWη=h(η)Bη. In Fig. 5(a) we report the dependence ofWη as a function ofη andg: for single-photon states (i.e. atg = 0), the correctionof losses introduced by the factorh(η) allows one to ob-serve micro-micro entanglement up toη∼0.35. As the num-ber of photons in the macro-state increases, the damping inthe negativity of the Wigner function induced by losses (that,as we have seen above, is directly related to the extent ofBell-CHSH inequality violation) scales more rapidly thanηand theh(η)-correcting term becomes less effective. Fig. 5(b) shows the behavior of the effective overall loss-functionh(η)L(η, g), highlighting the thresholds ing and η abovewhich non-locality is observed.Conclusions and perspectives.–We have proposed an exper-imentally oriented approach to detect non-locality and entan-glement in a micro-macro entangled state involving a single-photon and a multi-photon bipartite system, where correla-tions are encoded in the polarization degree of freedom. Wehave used a hybrid Bell-CHSH inequality and an entangle-ment witness whose use against such a class of states is effec-tive. As an experimental benchmark, we applied the proposedinequalities to the bipartite state obtained by amplification ofan entangled single-photon singlet state. We have found ex-tensive regions in the parameter-space of the system wherenon-classical features can be reliably observed. Our findingsare quite effectively summarized as in Fig. 5 (c), where we seethe existence of three interesting regions. In the inconclusivepart of the(g, η) space, entanglement and nonlocality cannotbe inferred by the proposed tests, while a gradual transition to-wards full affirmation of non-classicality is observed asη→1.While our study spurs further interest in the identificationofsuitable tests in the high-loss and large-photon-number region,it paves the way to an experimentally feasible demonstrationof non-local properties in an interesting class of states lying atthe very border between quantum and classical domains.Acknowledgements.–We acknowledge support by the “Fu-turo in Ricerca” Project HYTEQ and Progetto d’Ateneo ofSapienza Universita di Roma. MP is grateful to the Diparti-mento di Fisica, Sapienza Universita di Roma, for the kind

hospitality and acknowledges financial support from EPSRC(EP/G004579/1).

∗ Electronic address: fabio.sciarrino@uniroma1.it[1] W. H. Zurek, Rev. Mod. Phys.75, 715 (2003).[2] P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin,Na-

ture (London)409, 1014 (2001).[3] J.-W. Pan andet al., Nature (London)423, 417 (2003).[4] N. A. Peterset al., Phys. Rev. Lett.92, 133601 (2004).[5] F. Sciarrino, E. Nagali, F. De Martini, M. Gavenda, and R.Filip,

Phys. Rev. A79, 060304 (2009).[6] J. Kofler and C. Brukner, Phys. Rev. Lett.101, 090403 (2008).[7] H. Jeong, M. Paternostro, and T. C. Ralph, Phys. Rev. Lett. 102,

060403 (2009).[8] C. Vitelli and et al., Phys. Rev. A81, 032123 (2010).[9] R. Simon, Phys. Rev. Lett.84, 2726 (2000).

[10] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev.Lett. 84, 2722 (2000).

[11] K. Wodkiewicz, New J. Phys.2, 21 (2000).[12] G. Giedke, B. Kraus, M. Lewenstein, and J. I. Cirac, Phys. Rev.

Lett. 87, 167904 (2001).[13] M. Stobinska, H. Jeong, and T. C. Ralph, Phys. Rev. A75,

052105 (2007).[14] S.-W. Lee, H. Jeong, and D. Jaksch, Phys. Rev. A80, 022104

(2009).[15] S.-W. Lee, H. Jeong, and D. Jaksch, Phys. Rev. A81, 012302

(2010).[16] E. Nagali, T. De Angelis, F. Sciarrino, and F. De Martini, Phys.

Rev. A76, 042126 (2007).[17] F. De Martini, F. Sciarrino, and C. Vitelli, Phys. Rev. Lett.100,

253601 (2008).[18] N. Spagnolo, C. Vitelli, F. Sciarrino, and F. De Martini, Phys.

Rev. A82, 052101 (2010).[19] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys.

Rev. Lett.23, 880 (1969).[20] K. E. Cahill and R. J. Glauber, Phys. Rev.177, 1857 (1969).[21] K. E. Cahill and R. J. Glauber, Phys. Rev.177, 1882 (1969).[22] Details on the calculations, are reported in the onlineEPAPS

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Supplementary Material: Hybrid non-locality test in a microscopic-macroscopic

system

Nicolo Spagnolo,1, 2 Chiara Vitelli,1 Mauro Paternostro,3 Francesco De Martini,4, 1 and Fabio Sciarrino1, 5, ∗

1Dipartimento di Fisica, Sapienza Universita di Roma, piazzale Aldo Moro 5, 00185 Roma, Italy2Consorzio Nazionale Interuniversitario per le Scienze Fisiche della Materia, piazzale Aldo Moro 5, 00185 Roma, Italy

3School of Mathematics and Physics, Queen’s University, Belfast BT7 1NN, United Kingdom4Accademia Nazionale dei Lincei, via della Lungara 10, I-00165 Roma, Italy

5Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche (INO-CNR), largo E. Fermi 6, I-50125 Firenze, Italy

In this supplementary-material section we report the details of the calculations sketched in themain letter. We discuss the hybrid nonlocality test and the hybrid entanglement witness introducedin the paper. We report the full calculation of the correlators for our benchmark micro-macro systemby considering, ab initio, the presence of detection losses and non-unitary injection efficiency.

PACS numbers:

HYBRID POLARIZATION-CONTINUOUS

VARIABLES NONLOCALITY TEST

In this section we review the nonlocality test per-formed in the paper. Our test is the extension of theBell’s inequality test proposed by Wodkiewicz in Ref.[1].We begin by focusing our attention on the multiphotonmode kB. Our microscopic-macroscopic system, which isgenerated by amplification of an entangled polarizationphoton-pair, is strongly correlated in such degree of free-dom. To exploit it, we define the measurement operatorof the multiphoton state as

ΠBχ,χ⊥(αχ, χ) = ΠBχ (αχ)⊗ 1Bχ⊥ (1)

Here ΠBi (αi) = DBi (αi)(−1)n

Bi DB†

i (αi) is the general-

ized parity operator, where DB(αi) is the displacementoperator and the subscript i = χ, χ⊥ describes the po-larization mode. This definition of the measurement op-erator corresponds to the application of a displacementoperator DB

i (αi) followed by a parity measurement. Theoverall detection operator is then dichotomic since it canassume only values ±1. The average of such operator ona state of mode B is directly related to the value of itsWigner function at the phase space point αχ.In order to detect the correlations present in the sys-

tem in the polarization degree of freedom, we perform themeasurement of the Pauli operator σA(φ) on the single-photon mode. Here, σ(φ) is the Pauli σz operator alongthe direction of the Bloch sphere identified by the equa-torial polarization state ~πφ = 2−1/2(~πH + eıφ~πV ). Thecorrelation of the joint system is then defined as

C(αχ, χ;φ) = 〈σA(φ) ⊗ ΠB(αχ, χ)〉, (2)

where the averages are evaluated on the investigated|Ψ〉AB micro-macro state. Since this correlation oper-ator corresponds to a set of dichotomic measurements,we can use the Bell-CHSH inequality [2]

B=C(α′χ, χ

′;φ′)+C(α′χ, χ

′;φ)+C(αχ, χ;φ′)−C(αχ, χ;φ),(3)

Here, the measurement settings for the single-photonmode kA are given by the measured polarizations (φ,φ′), while the measurement settings for the multi-photonmode kB are given by the measured polarizations (χ, χ′)and the chosen phase space points (αχ, α

′χ).

HYBRID POLARIZATION-CONTINUOUS

VARIABLES ENTANGLEMENT WITNESS WITH

INEFFICIENT DETECTORS

In this section we discuss in details the hybrid entangle-ment witness defined in the paper. Such inequality is anextension of the Bell-CHSH test of Eq. (3) where differentmeasurement operators are exploited in the multiphotonmode. The main idea of this extension is to take intoaccount detection losses in order to build measurementoperators apt for witnessing entanglement with an ineffi-cient detection apparatus. To this end, the measurementperformed on the ~πχ polarization of the multiphoton fieldcan be replaced by the operator [3, 4]

OBχ (αχ; η) =

1η Π

Bχ (αχ) +

(

1− 1η

) 1Bχ if 12 < η ≤ 1,

2ΠBχ (αχ)− 1Bχ if η ≤ 12 ,

(4)where η is the detection efficiency of the apparatus. Suchdefinition of the measurement operator is performed inorder to correct the detrimental effect of losses on theproperties of the detected state. Let us consider a gen-eral state |Φ〉Bχ on spatial mode kB and polarization ~πχ(although we illustrate our argument using pure states ofmode B, our arguments apply equally to mixed states).After losses occur, the state evolves into a density ma-trix ρη BΦχ. The average value of O

Bχ (αχ; η) on such density

matrix gives [4]

〈OBχ (αχ; η)〉η =

π2ηW

η BΦ (αχ) +

(

1− 1η

)

if 12 < η ≤ 1,

2W η BΦ (αχ)− 1 if 0 ≤ η ≤ 1

2 .(5)

2

Here, W η BΦ (αχ) is the Wigner function of the detected

state, which is related to the Wigner function of the ini-tial state before losses |Φ〉Bχ by the Gaussian convolution

W η BΦ (Xχ, Pχ) =

2

π(1− η)

∫ ∞

−∞

∫ ∞

−∞dX ′

χdP′χ

×WBΦ (X ′

χ, P′χ)e

−2

[

(Xχ−√ηX′

χ)2

1−η+

(Pχ−√ηP ′

χ)2

1−η

]

.

(6)

The measured Wigner function given in Eq. (6) corre-sponds to the s-parametrized quasi-probability distribu-

tionWBΦ , (αχ, s), of |Φ〉Bχ with s = − (1−η)

η [5, 6]. Exploit-ing the properties of such distributions, it is straightfor-ward to prove that [4]

|〈OBχ (αχ; η)〉η| ≤ 1 (7)

for all values of η. We can then define the overall mea-surement performed on the multiphoton state as

OBχ,χ⊥(αχ, χ; η) = OBχ (αχ; η)⊗ 1Bχ⊥ (8)

with average values bounded by |〈OBχ,χ⊥ (αχ, χ; η)〉η| ≤ 1.The two-mode correlation operator for the entanglementwitness is then defined as:

ˆC (αχ, χ;φ; η) = σA(φ) ⊗ OBχ,χ⊥(αχ, χ; η) (9)

where σA(φ) is the Pauli operator for mode kA alongthe direction φ in the Bloch sphere. Starting from thesedefinitions, we construct the witness operator

W = ˆCη(α′χ, χ

′;φ′) + ˆCη(α′χ, χ

′;φ)+

+ ˆCη(αχ, χ;φ′)− ˆCη(αχ, χ;φ).

(10)

In order to define the bounds on 〈W〉 satisfied by sep-arable states, we consider a generic micro-macro sep-arable state described by the density matrix ρsep =∑

i piρAi ⊗ ρBi . After detection losses on the multiphoton

mode kB , such state evolves into ρsep =∑

i piρAi ⊗ ρη Bi ,

which gives

|〈W 〉sepη | =∣

∣

∣

∑

i

pi(

〈A′〉i〈B′〉iη + 〈A′〉i〈B〉iη+

+ 〈A〉i〈B′〉iη − 〈A〉i〈B〉iη)

∣

∣

∣

(11)

where

〈B〉iη=Tr[

OBχ,χ⊥ (αχ, χ; η)ρη Bi

]

,

〈A〉i=Tr[

σA(φ)ρAi]

,

〈A′〉i=Tr[

σA(φ′)ρAi]

,

〈B′〉iη=Tr[

OBχ′,χ′⊥(α′χ, χ

′; η)ρη Bi

]

.

(12)

As all these terms satisfy |〈X〉i| ≤ 1 with X =A, A′, B, B′, we get

|Wsepη | ≤ 2, (13)

which is the desired witness condition. We conclude bydiscussing the features of this inequality. On one side,we note that the derivation of this bound is performedunder the assumption that the state is measured withefficiency η. Hence, such witness operator permits todemonstrate the entanglement before detection losses.On the other side, no assumption is necessary on themicro-macro source due to the generality of the derivedcriterion. Finally, we note that for the case η = 1 thisentanglement witness coincides with the Bell-CHSH in-equality Eq. (3), given that no assumption is made onthe efficiency of the detection apparatus.

CORRELATOR FOR THE NONLOCALITY TEST

IN IDEAL CONDITIONS

In this section we report the full calculation of the cor-relator C(Xχ, Pχ, χ;φ) reported in the main letter. We

begin with the two-mode correlation Q, defined as

Q(αχ, αχ⊥ , χ;φ) = σA(φ)⊗(

ΠBχ (αχ)⊗ ΠBχ⊥(αχ⊥))

(14)This operator corresponds to the measurement of thegeneralized parity operator on both polarization modes~πχ, ~πχ⊥ of the macro-part of our state. The average

Q(αχ, αχ⊥ , χ;φ) = AB〈Ψ−|Q|Ψ−〉AB is related to thecorrelator of the Bell-CHSH inequality by

C(αχ, χ;φ) =2

π

∫

d2αχ⊥Q(αχ, αχ⊥ , χ;φ). (15)

This expression holds by considering the closure relation2π

∫

d2αχ⊥Πχ⊥(αχ⊥) ≡ 1χ⊥ , which in turns comes fromthe normalization of the Wigner function.

Two-mode correlator

We now calculate the two-mode correlatorQ(αχ, αχ⊥ , χ;φ). Let us recall the expression ofthe micro-macro state under investigation

|Ψ−〉AB =1√2(|φ〉A|Φφ⊥〉B − |φ⊥〉A|Φφ〉B) (16)

where the state has been expressed in a generic equa-torial polarization basis ~πφ, ~πφ⊥. The value ofQ(αχ, αχ⊥ , χ;φ) is obtained by exploiting the rela-

tion between the two-mode ΠBχ (αχ) ⊗ ΠBχ⊥(αχ⊥) oper-

ator and the two-mode Wigner function B〈Φ|ΠBχ (αχ) ⊗ΠBχ⊥(αχ⊥)|Φ〉B=π2

4 WΦ(αχ, αχ⊥). We get

Q(αχ, αχ⊥ , χ;φ) =π2

8

[

WBφ⊥(αχ, αχ⊥)−WB

φ (αχ, αχ⊥)]

.

(17)

3

Here, WBφ⊥ and WB

φ stand for the two-mode Wigner functions of an amplified |φ⊥〉 and |φ〉 single-photon states

respectively, evaluated at the rotated phase-space variables αχ, αχ⊥. The correlator QAB(αχ, αχ⊥ , χ;φ) is thenderived starting from the expression of the Wigner functions [7] [ S = sinh g and C = cosh g]

WBφ⊥(αχ, αχ⊥) =

4

π2

4[

|αφ⊥ |2(1 + 2S2) + 2CS Re(α2φ⊥e

ıφ)]

− 1

e−2[(|αφ⊥ |2+|αφ|2)(1+2S2)+2CSRe(α2φ⊥e

ıφ−α2φe

ıφ)]

WBφ (αχ, αχ⊥) =

4

π2

4[

|αφ|2(1 + 2S2) + 2CS Re(α2φeıφ)

]

− 1

e−2[(|αφ⊥ |2+|αφ|2)(1+2S2)+2CSRe(α2φ⊥e

ıφ−α2φe

ıφ)]

(18)

by rotating the polarization of the phase-space variablesαφ, αφ⊥ as

αφ = eı(χ−φ)/2[αχ cos(χ− φ)− ıαχ⊥ sin(χ− φ)],

αφ⊥ = eı(χ−φ)/2[αχ⊥ cos(χ− φ)− ıαχ sin(χ− φ)].(19)

Finally, we replace the complex phase-space variableswith the real quadratures (Xχ, Pχ, Xχ⊥ , Pχ⊥) and finallyobtain the full expression for Q(Xχ, Pχ, Xχ⊥ , Pχ⊥ , χ;φ).However, this is too lengthy and rather uninformativeand will not be reported here.

Single-mode correlator

We now calculate the single mode correlatorC(Xχ, Pχ, χ;φ). This choice of the measurement op-erator allows to capture the nonlocal features of themicro-macro state generated by amplification of an en-tangled photon pair. To evaluate this quantity we exploitEq. (15)

C(Xχ, Pχ, χ;φ)=2

π

∫ ∫

dΩQ(Xχ, Pχ, Xχ⊥ , Pχ⊥ , χ;φ)

(20)where the integral in d2αχ⊥ has been replaced by the in-tegral in the quadrature variables dΩ=dXχ⊥dPχ⊥ . Afterstraightforward algebra, we obtain the following expres-sion for the correlator

C(Xχ, Pχ, χ;φ) = cos[2(χ− φ)]e−2(e−2gX2χ+e

2gP2χ)

×[

1− 2(e−2gX2

χ + e2gP2

χ)] (21)

where Xχ, Pχ define a set of rotated variablesXχ = Xχ cos(χ/2) − Pχ sin(χ/2), Pχ = Xχ sin(χ/2) +Pχ cos(χ/2). The maximum of such correlation operatoris obtained at the origin of the phase-space and readsC(0, 0, χ;φ) = cos[2(χ− φ)].

CORRELATOR FOR THE NONLOCALITY TEST

UNDER DETECTION LOSSES AND

NONUNITARY INJECTION EFFICIENCY

Here we report in details the calculation of the correla-tor Cp,η, when detection losses and a nonunitary injection

efficiency are taken into account. These two effects repre-sent the two main issues for an experimental observationof nonlocal correlations in a micro-macro system.

The model for the effect of losses at the detection stageis performed by inserting a beam-splitter of transmittiv-ity η along the transmission path of the field on modekB. The other port of this beam-splitter is injected witha vacuum state, thus introducing vacuum-noise fluctua-tions in the system. Here we demonstrate that the cor-relator Cη in presence of detection losses η can be evalu-ated as the convolution of the lossless correlator C witha Gaussian function of the form:

Cη(Xχ, Pχ, χ;φ) =2

π(1 − η)

∫ ∫

dX ′χdP

′χ C (X ′

χ, P′χ, χ;φ)

× e−2

[

(Xχ−√ηX′

χ)2

1−η+

(Pχ−√ηP ′

χ)2

1−η

]

.(22)

We begin by writing the density matrix ρΨ−

η of the micro-macro state after losses occur at the detection stage

ρΨ−

η =1

2

|φ〉A〈φ|⊗L[

|Φφ⊥〉B〈Φφ⊥ |]

+ |φ⊥〉A〈φ⊥|⊗L[

|Φφ〉B〈Φφ|]

− |φ〉A〈φ⊥|⊗L[

|Φφ⊥〉B〈Φφ|]

−|φ⊥〉A〈φ|⊗L[

|Φφ〉B〈Φφ⊥ |]

,

(23)

where L[·] is the map that describes the action of detec-tion losses. The evaluation of the correlation operator Qon this density matrix leads to

Qη(αχ, αχ⊥ , χ;φ)=π2

8

[

WBη,φ⊥(αχ, αχ⊥)−WB

η,φ(αχ, αχ⊥)]

(24)where WB

η,φ and WBη,φ⊥

are the Wigner functions of the

macrostates |Φφ〉 and |Φφ⊥〉 after losses. The action ofdetection losses in the phase-space can be written in theform of a Gaussian convolution [8]

Wη(X,P ) =

∫ ∫

dX ′dP ′ W (X,P )Kη(X,P,X′, P ′),

(25)

where Kη(X,P,X′, P ′)= 2

π(1−η) exp−2[(X−√

ηX′)2

1−η +

4

(P−√ηP ′)2

1−η ]. The correlator Cη is obtained from Qη as

Cη(Xχ, Pχ, χ;φ) =2

π

∫ ∫

dΩQη(Xχ, Pχ, Xχ⊥ , Pχ⊥ , χ;φ).

(26)By writing explicitly the Wigner function after losses asa Gaussian convolution we obtain

Cη(Xχ, Pχ, χ;φ) =2

π

∫ ∫

dXχ⊥dPχ⊥I(X ′χ, P

′χ)

(27)where

I(X ′χ, P

′χ) =

∫ ∫

dX ′χ⊥dP

′χ⊥Q(X ′

χ, P′χ, X

′χ⊥ , P

′χ⊥ , χ;φ)

×∫ ∫

dXχ⊥dPχ⊥Kη(Xχ⊥ , Pχ⊥ , X′χ⊥ , P

′χ⊥).

(28)By changing the integration variables as Xχ⊥ → Xχ⊥ =Xχ⊥−√

ηX′χ⊥√

1−η , Pχ⊥ → Pχ⊥ =Pχ⊥−√

ηP ′χ⊥√

1−η we have the

explicit function

I(X ′χ, P

′χ) =

2|J |π(1− η)

∫ ∫

dXχ⊥dPχ⊥e−2(X2

χ⊥+P 2χ⊥ )

×∫ ∫

dX ′χ⊥dP

′χ⊥Q(X ′

χ, P′χ, X

′χ⊥ , P

′χ⊥ , χ;φ),

(29)where |J | = 1 − η. Eq. (22) is found by integratingover dXχ⊥dPχ⊥ , using Eq. (15) to have I(X ′

χ, P′χ) =

C(X ′χ, P

′χ, χ;φ) and replacing this in Eq. (27).

We now proceed with the explicit calculation ofEq. (22). As a first step, we rotate the quadratures(Xχ, Pχ) and the integration variables (X ′

χ, P′χ) as

X χ=Xχ cos(χ/2)−Pχ sin(χ/2),Pχ=Xχ sin(χ/2)+Pχ cos(χ/2)

(30)

with X = (X,X ′) and P = (P, P ′) and the conventionthat only primed (unprimed) variables are involved in theequations above. The correlator Cη can be then expressedas a function of the rotated variables. After replacing theexpression of Kη in the correlator Cη, it is matter of somestraightforward (although tedious) algebra to find that

Cη(Xχ, Pχ, χ;φ) =cos[2(χ− φ)]e

−2

[

X2χ

M +P2

χN

]

√

1 + 4η(1− η)n

1− (1 − η)(1 + 2ηn)

1 + 4η(1− η)n− 2η

[

e2gX2

χ

M2+e−2gP

2

χ

N 2

]

(31)with M = ηe2g + (1− η) and N = ηe−2g + (1− η). Thisexpression is maximized at the origin of the phase space,reading

Cη(Xχ, Pχ, χ;φ) = cos[2(χ− φ)]L(η, g) (32)

where the loss function L(η, g) has the form:

L(η, g) = η + 2η(1− η)n

(1 + 4η(1− η)n)3/2. (33)

In typical experimental conditions, the injection of thesingle photon of the entangled pair |ψ−〉AB into the opti-cal parametric amplifier, occurs with an efficiency p < 1because of the imperfect matching between the opticalmodes of the amplifier and the single-photon one. Such anon-ideality can be modeled by allowing for a probabilityp of correct single-photon injection and a complementaryprobability (1 − p) that just vacuum state is injected inthe amplifier and no correlations between the two outputmodes are set. This modifies the density matrix of theoutput modes as

ρψ−

p = p|ψ−〉AB〈ψ−|+ (1− p)1A2

⊗ |0〉B〈0|, (34)

where 1A = |H〉A〈H | + |V 〉A〈V | is a completely mixedsingle-photon polarization state, and |0〉B〈0| is the vac-uum state. The bipartite state after the amplificationprocess then reads

ρΨ−

p = p|Ψ−〉AB〈Ψ−|+(1−p) 1A2

⊗(

UOPA|0〉B〈0|U †OPA

)

.

(35)We can now proceed with the calculation of C(αχ, χ;φ)as

C(αχ, χ;φ) = pAB〈Ψ−|σA(φ)⊗ ΠB(αχ, χ)|Ψ−〉AB

+(1−p)Tr[ 1A

2⊗(

UOPA|0〉B〈0|U †OPA

)

σA(φ)⊗ΠB(αχ, χ)

]

.

(36)As the second term factorizes (due to the lack of quantum

correlations) and Tr[ 1A

2 σA(φ)

]

= 0, such contribution

is null. Therefore, under non-ideal injection efficiency,the correlator is related to the ideal one according toCp(Xχ, Pχ, χ;φ) = p C(Xχ, Pχ, χ;φ). This result can beextended to the case of nonunitary detection efficiency,leading to

Cη,p(Xχ, Pχ, χ;φ) = p Cη(Xχ, Pχ, χ;φ). (37)

CORRELATOR FOR THE ENTANGLEMENT

WITNESS AFTER DETECTION LOSSES AND

NONUNITARY INJECTION EFFICIENCY

Here we sketch the steps needed for the calculation ofthe correlator Cp,η entering the entanglement test basedon the witness operator of Eq. (10) under losses and non-ideal photon injection. By using arguments similar to

5

those put forward in the previous sections, we have

Cη(αχ, χ;φ) =1

2

Tr[

L[

|Φφ⊥〉B〈Φφ⊥ |]

OBχ,χ⊥(αχ, χ; η)]

− Tr[

L[

|Φφ〉B〈Φφ|]

OBχ,χ⊥(αχ, χ; η)]

,

(38)where L[·] is the map describing the lossy process.We focus on the case η ≥ 1

2 . By exploiting re-sults that have been previously obtained here, we have

Cη(αχ, χ;φ)= π4η

∫

d2αχ⊥

(

W ηφ⊥

(αχ, αχ⊥)−W ηφ (αχ, αχ⊥)

)

.

We now exploit the chain of relations

π

4

∫

d2αχ⊥

(

W ηφ⊥

(αχ, αχ⊥)−W ηφ (αχ, αχ⊥)

)

=2

π

∫

d2αχ⊥π2

8

(

W ηφ⊥

(αχ, αχ⊥)−W ηφ (αχ, αχ⊥)

)

=2

π

∫

d2αχ⊥Qη(αχ, αχ⊥ , χ;φ) = Cη(αχ, χ;φ)(39)

so as to get Cη(αχ, χ;φ; η) = 1ηCABη (αχ, χ;φ). With an

analogous procedure, we obtain

Cη(αχ, χ;φ; η) = 1

η Cη(αχ, χ;φ) if 12 < η ≤ 1,

2 Cη(αχ, χ;φ) if η ≤ 12 .

(40)

We can further generalize this result so as to take intoaccount the effect of a nonunitary injection efficiency andfinally get Cη,p(αχ, χ;φ; η) = p Cη(αχ, χ;φ; η).

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