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Different operational meanings of continuous variable Gaussian entanglement criteria and Bell inequalities

Entanglement, one of the most intriguing aspects of quantum mechanics, marks itself into different features of quantum states. For this reason different criteria can be used for verifying entanglement. In this paper we review some of the entanglement criteria casted for continuous variable states and link them to peculiar aspects of the original debate on the famous EPR paradox. Moreover, we give a handy expression for valuating Bell-type non-locality on Gaussian states. We also present the experimental measurement of a particular realization of the Bell operator over continuous variable entangled states produced by a sub-threshold type-II OPO.


I. INTRODUCTION
Since the first reply by Schrödinger [1] to the famous EPR paper [2] the word "entanglement" has been primarly used for indicating a class of quantum states that shows non-local features. Discussing the dynamical properties of a composite system made of two subsystems that, after mutual interaction, move away one from the other Einstein Podolsy and Rosen concluded that quantum mechanics was not-complete and that some more local (hidden) dynamical variables would have been necessary for a correct description of the physical reality. Essentially, they pointed to two quantum-mechanical aspects that they found counter-intuitive. In primis, the possible ambiguity of the wave function so that ". . . as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions . . . ". The second aspect was later indicated, by Einstein itself, as a spooky action at distance ". . . since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system . . . ". We nowadays know that they were wrong and that quantum mechanics gives, so far, a complete representation of this strange phenomenon.
Up to the late fifties of last century the debate on entanglement was mostly confined to the fundamental aspect of quantum mechanics and the word itself hadn't any particular operational meaning. In 1957 a paper by Bohm and Aharonov [3] moved the focus from the original Gedankenexperiment toward more feasible and intuitive physical implementations and, in particular, to spin-like systems. This paved the way to a more complete theoretical analysis of the hidden variables scenario that leads to the famous Bell inequalities[? ] [4] for dichotomic quantum variables. This made spin-like systems the preferential candidates for proving the failure of any hidden variables hyphotesis. Single photons have been then widely used in several experimental tests of the Bell inequalities (see for a review Ref. [6]). Very recently, a novel experiment made the photon the first physical system for which each of the main loopholes has been closed [7].
On the other hand, the original formulation of the EPR paradox was based on continuous variable systems. So that, in 1986 Reid and Walls proposed the first translation of Bell inequalities into the continuous varaible language [8] and in 1992 the first experimental realization of an EPR-like system appeared [9]. Since then, a few more attempts have been carried out for translating the Bell argument into the language of continuous variables (CV) [10]. Among them the one proposed by Banaszek and Wódkiewicz [11] considers the relation between the Wigner function of the state and non-locality.
Entanglement, in its orignal formulation, states the existence of global states of a composite system which cannot be written as a product of states of individual subsystems. While this definition set an univoque border between separable and entangled states, entanglement gives rise to different features of quantum systems and can be seen under different perspectives [12]. On one hand, mathematically defining entanglement as a property of the composite system wavefunction, make it intrinsically related to pure states [13]. On the other hand, we all know that experimentally acessible states are mixed, so that feasible entanglement tests have to be related to density matrices rather than wave functions [14].
In this paper we aim at discussing the operational meanings of different criteria usually employed for assessing CV entanglement. In particular, we will link each of them to different facets of the original entanglement debate. We will apply them to entangled Gaussian states (GS) [15,16] produced by a type-II sub-threshold frequency degenerate OPO. By experimentally analysing the properties of experimentally generated CV entangled states we will express all these criteria in terms of the arXiv:1407.3965v1 [quant-ph] 15 Jul 2014 covariance matrix elements. Moreover, we give a novel handy relation that describes in a simple way the connection among entanglement, purity and Bell's non-locality. The experimental analysis prove that some entanglement features are strongly hold against decoherence while Bell inequality is violated only for nearly pure states.
The paper is organized as follows. In Sect. II the properties of Guassian states are reviewed. Then, in Sect. III, a summary of different entanglement criteria is given. Each of the presented criteria is related to a particular feature of entanglement. In Sect. IV a Bell-type inequality is given in terms of the Gaussian states properties. While in Sect. V a discussion on the effects of decoherence on states violating Bell inquality is presented. In Sect. VI we report an overview on some experimental results and, in particular, an a posteriori Bell test on an effective bipartite CV entangled state. Eventually, in Sect. VII conclusions are drawn.

II. GAUSSIAN STATES
A continuous-variable bi-partite GS is a two-mode state, on the Hilbert space H = H a ⊗ H b , whose characteristic function or, equivalently, Wigner function in phase space is Gaussian: where K ≡ X a,ϑ , X a,ϑ+π/2 , X b,ϑ , X b,ϑ+π/2 is the vector of a set of orthogonal quadratures, for mode a and b respectively. (being X k,ϑ ≡ ae iϑ + a † e −iϑ √ 2 ). We remind that the pair X 0 = X, and X π/2 = Y ([X k , Y k ] = i) associated to a single e.m. mode is the analogue to the position/momentum pair for a mechanical oscillator. This makes optical mode a good candidate for replicating EPR states in their original fashoin. All the features of GS are embedded in the second order momenta of the joint quadrature distribution, namely the covariance matrix[? ] (CM) σ in Eq. (1). A pure GS can be seen as the action of a displacement and a squeezing operator onto the vacuum state. While, the most general mixed GS can be obtained replacing the vacuum with a thermal field at finite temperature. For a bipartite state the σ is a 4 × 4 matrix, where α and β represent self-correlation of the single subsystem and γ describes the cross correlation between the two subsystems. Remarkably, a GS represents any quantum system whose evolution can be described by a at least bilinear Bosonic Hamiltonian. In particular this is the case of optical parametric oscillators (OPO).
Any CM, representing a physical state, can be transformed into the so-called standard form [17] by means of local symplectic transformations [? ] where n, m, c 1 and c 2 are determined by four local symplectic invariants As a matter of fact, a sub-threshold type-II OPO, due to the symmetry of its Hamiltonian, can only produce states whose CM is a standard form [18]. Hereafter, whenever we refer to CMs we will mean such standard form. Moreover, at the time of birth inside the non-linear crystal, the bipartite state we would analyse in the following shows n = m and c 1 = −c 2 = c.
From the symplectic invariants it is possible to give a criteria for disitnguishing among physical and nonphysical CMs. σ describes a physical state iff We also note that a pure GS is a minimum uncertainty state and that the CM relative to a pure state necessary has det(σ) = I 4 = 1/16 so that for a pure state while, for mixed symmetric states, c < n 2 − 1/4. In general, for a bipatite GS, the purity reads

III. ENTANGLEMENT CRITERIA
A quantitative measure of entanglement for a mixed state is, so far, an unsolved issue. This is probably due to the different operational implications that different levels of quantum correlation open. At the same time, there exist different necessary and/or sufficient conditions to asses whether a given state is entangled or not. These criteria are easily translated into experimental tests for entanglement. Here we aim to look at the different criteria and connect them, logically, to the debate on the original EPR paper [2].
A. Un-separability criteria: PHS and Duan The first criterion was developed by considering the definition of entangled states: a state of a composite system whose wavefunction cannot be given as product of sub-systems wavefunction. Or, in the case of mixed states, following the Werner extension to the relative density matrix [14]. In the bi-partite case a density matrix represent a separable state iff its can be written as a convex combination of the tensor product of density operators relative to the two sub-systems where j p j = 1 while ρ ji i = 1, 2 are the density matrices of subsystems 1 and 2. The criterion can be casted considering that if one performs a partial transposition (i.e. transposition of the density matrix with respect to only one of the two Hilbert subspaces) ρ transform into ρ P T that, for a state written in form given in Eq.
(7) will still represent a physical state of the composite system. Conversely, if the state un-separable, the tranformed density operator ρ P T would have no more a physical counterpart. This criterion, is sometime referred to as the ppt criterion (positivity under partial transposition) or PHS from the names of the people that proposed it for discrete (Peres [19] and Horodecki [20]) and continuous variables (Simon [21]). Translated into the CM language a bi-partite Gaussian state is separable iff and it is entangled otherwise. We also note that the PHS criterion is invariant under symplectic transformations and that for pure states the inequality is saturated. The PHS criterion set, then, is roots into the fact that two systems that have interacted cannot, even if the split apart after the interaction, be described independently.
A second crierion, the Duan one [17], has been derived considering that in presence of an entangled state also the Heisenberg uncertainty principle, written for the joint system and a pair of EPR-like operators, has to take into account the inherent quantum correlation. The Duan criterion, a necessary condition for entanglement, for a CM in the usual standard form reads: Based on the calculation of the total variance of a pair of Einstein-Podolsky-Rosen (EPR) type operators. It relies on the fact that the inherent correlation reduces the total variance that, in separable states, is greater than the sum of the standard quantum limit applied to the single subsystem. In the case of frequency degenerate type-II OPO, this result in the squeezing of the modes obtained by letting the two entagled companions interfere [22].

B. The EPR "Reid" criterion
A stronger bound can be found by considering the original EPR Gedankenexperiment where the paradox was found in the possibility of determining the state of a faraway system by measuring its entangled companion. For this reason this criterion is usually indicated as the EPR criterion and was firstly introduced by Reid in 1989 [23], in the very early days of quantum information. It describes the ability to infer the expectation value of an observable on a sub-system by performing a suitable measurement on the second sub-system. This criterion sets only a sufficient condition for assessing entanglement being a stricter condition on the strenght of quantum correlation. It can be easily given in terms of CM elements: While the criterion is asymmetric under the exchange of the two sub-systems so that the two definitions can make it ambiguous if one of the relations is not satisfied. This is not the case of balanced systems (m = n) where this one-side violation is not possible. The asymmetry of the criterion allows to use it for the so called steering capability: the state of a far away system can be steered by a suitable measurement on its entangled companion [24].

IV. BELL-LIKE INEQUALITY (NON-LOCALITY) IN PHASE SPACE
Generally, if we want to evaluate the non-locality of a state through a CHSH inequality [5], the operative form of the Bell one, we should build a Bell operator representing a combination of dichotomic (true-false) measurements. Then, if the expectation value of such a Bell operator violates the corresponding inequality, the system is not considered local, otherwise it would admit a classical description in terms of hidden variables. A parity operator is dichotomic. It can be constructed, on the photon number, for assigning +1 or −1 depending on whether an even or an odd number of photons has been registered. In Refs. [11,25] a connection between the Wigner function of the state and the joint measurement of the parity operator performed on the bi-partite quantum state has been shown.
Here we want to give a handy expression that relates such a measurement to the CM of a generic GS. We consider the Bell operator in the form given in Eq. (7) of Ref. [11]. The Bell type function B is then, given by the linear combination of four expectation values B= W(0, 0) + W( where with W (α 1 , α 2 ) the Wigner function of the state calculated in (α 1 , α 2 ) and where α k are complex amplitudes (and so is √ I is Eq. (11)). Local theories, admitting a description in terms of local hiddden variables, set the bound |B| ≤2.
On one hand, any Bell inequality concerns the analysis of joint probabilities measured at space-time-separated locations. So that, to actually perform a Bell measure we should need to make repeated simultaneous measurements at different space-time-separated locations stochastically changing the detector settings (in this case the amounts of displacement). Then by statistical analysis we could conclude or not the violation of the CHSH type inequality.
On the other hand, Eq. (12) show that the knowledge of the Wigner function, i.e. the full reconstruction of the quantum state gives an insight to the local/nonlocal character of the state. Without running into delicate questions we wish to show that, being a GS fully described by a rather simple object, the CM, it is possible to evaluate, a posteriori, B on the state so to asses whether or not it is Bell correlated without the need of reconstructing the whole Wigner function. This paves the way to a handy experimental procedure to disciminate among different levels of quantum correlations.

A. Bipartite Gaussian state case
Now, we consider the bipartite GS generated by a type-II OPO described by the covariance matrix (3) σ, with n = m and c 1 = −c 2 = c.
It can be esaily found that the quantity (11) becomes B (I, n, c) = 1 + 2 exp − n n 2 −c 2 I − exp − n+c n 2 −c 2 2I 4 (n 2 − c 2 ) . (16) This gives the expectation value of the maximum value of the Bell operator B as a function of the Gaussian state parameters. So that, being possible to experimentally retrieve the CM of such a state [22], this formula can be used to perform an a posteriori test on the non-local property of the state.
Moreover, it is possible to relate B to the purity of the single subsystem µ s ≡ µ a = µ b = 1/ (2n). Having in mind the interconnection between entanglement and the purity of the constituent sub-systems [26], we have: is the correlation coefficient whose limit for a pure state is C 2 It can also be proved that, when the Gaussian state is pure, B I can be considered an entanglement witness: any entangled state violates the Bell inequality and viceversa.

B. Purity, entanglement and non-locality
For mixed states the above equivalence does not hold. Given a mixed system, one has µ 2 s < 1 − C 2 ab . So that, for a given correlation coefficient C ab , it is possible to set three boundaries for the values of µ s so that µ s < µ D denotes separable states, µ D < µ s < µ B indicates entangled states that do not violate the Bell inequality, and, finally, for µ B < µ s < µ P the states are entangled and violate the Bell inequality. For µ s > µ P the relative CM would be not physical. In this way it is possible to distinguish three (physical) regions in the plane (µ s , C ab ) (see Fig. 1) Region I): Separable states compatible with the local hidden variables theory ( B (µ s , C ab ) < 2).
Region II): Entangled states compatible with the local hidden variables theory ( B (µ s , C ab ) < 2).
Region III): Entangled states not compatible with the local hidden variables theory ( B (µ s , C ab ) > 2).
It is clear that there aren't separable GS that violate the Bell's inequality. Instead, a state compatible with a local theory (i.e. compatible with a theory in hidden variables) can also be entangled. This confirms the existence of different forms of quantum correlations and non-locality.

V. GAUSSIAN NOISE DOES NOT BREAK THE ENTANGLEMENT, BUT IT BREAKS THE BELL'S NONLOCALITY
In this Section we want to analyze the behaviour of the Bell's nonlocality subjected to passive Gaussian noise. We will see that when a pure (c = n 2 − 1/4), entangled and Bell's non-local (i.e.B I > 2) state evolves through a Gaussian channel, retains its entanglement, but looses its Bell's nonlocality. This means that although, at the time of its birth, the state is pure, so that it violates Bell's inequality and breaks the Duan bound, decoherence highlights the different nature of the two markers: the Gaussian state becomes local (in according to Bell), i.e. it would admit a description in terms of local hidden variables, although it remains entangled.
A bipartite state, described by the CM Eq. (3), subjected to the action of a passive Gaussian channel, undergoes a transformation such that: [18] where n 1 and c 1 are the CM elements of the initial pure state and T is the trasmittivity of a fictituous beam splitter mimecking a lossy transmission [27]. We can calculate the evolution of the Bell function B (n 1 , c 1 ) (16) starting from an initially pure state, described by the CM elements n 1 and c 1 and analyzing B (n T , c T ) as a function of the coefficient of transmissivity T (0 < T < 1). So the Bell's function B becomes a function B T depending on the initial (pure) state and transmissivity T of the channel. The relative expression, ideed rather long and complicate, will be used for evaluating the correspondence among experimental results and theory in the next section.

VI. EXPERIMENTAL RESULTS
In a recent paper (see Ref. [18]) we have analysed how different entanglement and quantum signatures evolves under decoherence. In this paper we wish to include the experimental analysis of the non-local character under decoherence. We have to stress that, in view of the restrict region (T > 90%) where one could expect a Bell inequality violation, we have not observed any Bell inequality violation. This is essentially due to the maximum overall transmission we can get from the OPO cavity to the homodyne detector (63%). Moreover, we stress that this is rather an a posteriori check of the non-local character of the state than a Bell measure.
The analysed state is the one outing a sub-threshold type-II OPO [28]. The block scheme of the experiment is given in Fig. 2. The full covariance matrix is retrieved by a single homodyne detector [29] following the procedure described in details in Ref. [30]. The generation stage is a type-II OPO operating below the oscillation threshold. At the OPO output a neutral adsorber mimicks the transmission over a real channel. The state is reconstructed exploiting data collected by a single homodyne detector.
In Fig. 3 we have plotted the experimental value obtained for B T . The continuous line represent the theoretical expections for the pure ancestor state.
As it can be seen experimental data are in good agreement with the expected evolution.

VII. CONCLUSIONS
Different bounds have been, so far, discussed in the literature for discriminating continuous variable separable and entangled states. Each of them looks at slightly different facets of the EPR paradox. So that, in this paper, they are presented in connection to the original EPR arguments.
Moreover, for the first time, we express by a handy and direct formula a Bell-type inequaly, written for CV states, in terms of the covariance matrix of a Gaussian state. We discuss its relation with the purity of the entangled sub-systems and analyse, also experimentaly, its behaviour under decoherence. So doing we have proved, experimentally, that, also in CV regime, there exists mixed entangled states that do not violate the Bell inequality. While, for pure states, any entangled state is Bell nonlocal.