Bell operator and Gaussian squeezed states in noncommutative quantum mechanics
Catarina Bastos, Alex E. Bernardini, Orfeu Bertolami, Nuno Costa Dias, João Nuno Prata
aa r X i v : . [ qu a n t - ph ] D ec Bell operator and Gaussian squeezed states in noncommutativequantum mechanics
Catarina Bastos ∗ GoLP/Instituto de Plasmas e Fus˜ao Nuclear,Instituto Superior T´ecnico, Universidade de Lisboa,Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal
Alex E. Bernardini † Departamento de F´ısica, Universidade Federal de S˜ao Carlos,PO Box 676, 13565-905, S˜ao Carlos, SP, Brasil.
Orfeu Bertolami ‡ Departamento de F´ısica e Astronomia,Faculdade de Ciˆencias da Universidade do Porto,Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
Nuno Costa Dias § Departamento de Matem´atica, Universidade Lus´ofona de Humanidadese Tecnologias Avenida Campo Grande, 376, 1749-024 Lisboa, Portugal
Jo˜ao Nuno Prata ¶ Escola Superior N´autica Infante D. Henrique,Av. Eng. Bonneville Franco2770-058 Pa¸co de ArcosPortugal (Dated: August 21, 2018) ∗ E-mail: [email protected] † E-mail: [email protected] ‡ Also at Centro de F´ısica do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal. E-mail:[email protected] § Also at Grupo de F´ısica Matem´atica, UL, Avenida Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal.E-mail: [email protected] ¶ E-mail: [email protected] bstract One examines putative corrections to the Bell operator due to the noncommutativity in the phase-space. Starting from a Gaussian squeezed envelop whose time evolution is driven by commutative(standard quantum mechanics) and noncommutative dynamics respectively, one concludes that,although the time evolving covariance matrix in the noncommutative case is different from thestandard case, the squeezing parameter dominates and there are no noticeable noncommutativecorrections to the Bell operator. This indicates that, at least for squeezed states, the privilegedstates to test Bell correlations, noncommutativity versions of quantum mechnics remains as non-local as quantum mechanics itself.
PACS numbers: . INTRODUCTION In their well known “EPR” paper, Einstein, Podolsky and Rosen [1] discussed the dynam-ical properties of a composite quantum system with two interacting subsystems. Despite theoriginality of their approach, they controversially concluded that quantum mechanics wasnot complete and that some underlying (hidden) variables would be necessary for a completephysical description of reality. They further argued that as, in order to predict the evolutionof a physical quantity at a given precision, the system should not be disturbed. However,it is known that if two physical quantities are described by noncommuting operators, thenthe knowledge of one observable precludes the knowledge of the other with an arbitraryprecision. Thus, they concluded, either quantum mechanics does not describe reality in acomplete fashion, or the two noncommuting quantities can be simultaneously described withan arbitrary precision.In the 1950s, after the work of Bohm and Aharonov [2], spin-like systems have beenstudied, and considered as particularly suitable for testing the assumptions underlying ahidden variable hypothesis. However, a crucial new element was introduced by John Bell[3], who showed that quantum correlations are essentially non-local. This allowed for atheoretical analysis of the hidden variable scenario, leading eventually to Bell’s inequalities[3]. However, the original formulation of the EPR discussion was based on continuousvariable systems. Actually, quantum correlations for position-momentum variables can bestudied in the phase-space using the the Wigner function formulation of quantum mechanics.Bell argued that, due to the positivity of the Wigner function, the original EPR states wouldbe necessarily non-local [3]. Recently, it has been shown that Wigner functions of two-modesqueezed vacuum states, being positive definite, provide a direct evidence of the non-localityof states [4]. The Wigner function can be interpreted as a correlation function for the jointmeasurement of the parity operator. Clearly, the connection of the Wigner function to theparity operator, in this case, is linked to the fact that a dichotomic operator is needed inorder to study the quantum systems. The key point is that a state does not have to violateall possible Bell’s inequalities to be non-local, but rather to violate just one of the Bell’sinequalities.In addition, the definition of entanglement is that a composite system cannot be writtenas a product of states of individual subsystems. Thus, there exists a relation between en-3anglement and EPR correlations. However, there are Gaussian states that are entangled,but are compatible with a local hidden variable theory, and there are entangled Gaussianstates that are non-local [5, 6]. This confirms the existence of different types of quantumcorrelations and non-locality. Indeed, the role of quantum entanglement has been acknowl-edged for its wide range of applications in quantum information protocols [7] and quantumcommunication [8]. One of the key results is the so-called positive partial transposed (PPT)separability criterion [9], which provides a necessary and, in some cases, a sufficient conditionfor distinguishing between separable and entangled states in discrete quantum systems. Thiscriterion was extended to continuous variable systems by implementing the partial transposeoperation as a mirror reflection of the Wigner formulation in phase-space [10]. The “con-tinuous” PPT criterion has important applications in the theory of quantum information ofGaussian states [11, 12], which is at the core of testing procedures for estimating quantumcorrelations [13]. In these cases, the PPT criterion yields both a necessary and sufficientseparability condition [10, 12]. Gaussian states are also quite useful for investigating theentanglement exclusively induced by a noncommutative (NC) deformation of phase-space[14–16].Before discussing these issues in the context of phase-space noncommutative quantummechanics (NCQM), let us point out that configuration space noncommutatitvity is be-lieved to be an intrinsic feature of quantum gravity and string theory [17, 18]. Phase-spacenoncommutativity, on its turn, has shown to have striking features, with implications forquantum cosmology, black hole singularity and thermodynamics [19], and the equivalenceprinciple [20]. Furthermore, the quantum mechanical aspects of NC theories have focusedon studies of the quantum Hall effect [21, 22], the gravitational quantum well for ultra-cold neutrons [23], the Landau/2 D -oscillator problem in the phase-space [24, 25], for thegraphene [26], and as a probe of quantum beating and missing information effects [27].Furthermore, the phase-space noncommutativity was shown to give rise to possible viola-tions of the Robertson-Schr¨odinger uncertainty principle [28]. It has also effects for theOzawa’s measurement-disturbance uncertainty principle, as phase-space noncommutativityintroduces extra terms and turns, for instance, a backaction evading measurement into anon backaction evading one [29].Phase-space NCQM are based on extensions of the Heisenberg-Weyl algebra [30, 31]and can be formulated in terms of phase-space Wigner functions [32]. Here, a different4pproach is considered to introduce the noncommutativity. Instead of noncommutativitybeen considered as a feature of phase-space, it is a consequence of the interaction betweenthe two modes, “Alice” and “Bob”. In Ref. [14] it was shown that the entanglement arisesfrom the interaction between the two modes and that this entanglement has inherent NCfeaures. Thus, it is both theoretically and experimentally interesting to test if an interactionbetween modes can be related to some form of noncommutativity between them. To studythese correlations between modes, one considers NC version of the Bell operator defined, forcontinuous variable systems, in terms of the Wigner function of the system [4].In this framework the particular cases of Gaussian squeezed envelopes for time evolvingGaussian states [14] deformed by the NCQM are considered. On one hand, one considersa departure Gaussian squeezed envelop with time evolution driven either by commutative(standard quantum mechanics) or by NC dynamics. Although the covariance matrix of theNC two mode time evolving Gaussian squeezed state is different from the commutative one,it is found that the squeezing parameter is the unique driving vector of quantum correlationsand non-locality and it is not influenced by noncommutativity. II. NONCOMMUTATIVE BELL OPERATOR
Let one considers a bipartite quantum system (2 − dimensional system) described in termsof a subsystem A (Alice) and a subsystem B (Bob). One may write the collective degreesof freedom of the composite system b z = ( b z A , b z B ), where b z A = ( b x, b p x ) and b z B = ( b y, b p y ) asthe corresponding generalized variables of the two subsystems, which obey the commutationrelations [14] (cid:2)b z Ai , b z Bj (cid:3) = i Ω ij , i, j = 1 , , (1)where the associated matrix is given by Ω = [Ω ij ], being a real skew-symmetric non-singular4 × Ω = J Υ − Υ J , (2)where Υ measures the noncommutativity of the position and momentum, Υ = θ η , (3)5 is the standard symplectic matrix J = − , (4)and one uses natural units, where ~ = 1.The NC structure can be formulated in terms of commuting variables by considering alinear Darboux transformation (DT), b z = D b ξ , where b ξ = ( b ξ A , b ξ B ), with b ξ A = ( b x c , b p xc ) and b ξ B = ( b y c , b p yc ), satisfy the usual commutation relations [32]: hb ξ i , b ξ j i = iJ ij , i, j = 1 , . (5)where [ J ij ] = J from Eq. (4). The linear transformation D ∈ Gl (4) is such that Ω = DJD T .Notice that the map D is not uniquely defined. If one composes D with block-diagonalcanonical transformations one obtains an equally valid DT. The matrix form of the DT is,therefore, D = λ − θ λ µ η µ θ λ λ − η µ µ , (6)where λ and µ are arbitrary real parameters.In order to obtain the non-locality of a state through the Clauser, Horne, Shimony andHolt (CHSH) inequality [33], one must, in general, have an explicit definition for the Belloperator. This operator usually represents a combination of dichotomy measurements. Theresult is that, if the expectation value of such a Bell operator violates the correspondinginequality, then the system is non-local. In continuous variable quantum information, inorder to mimic the dichotomic behavior, the parity operator is used. The later can bedetermined, on the photon number, by assigning +1 or −
1, depending on whether an evenor an odd number of photons are registered. For the connection between the Wigner functionof the state and the joint measurement of the parity operator on the bipartite quantum state,see e.g. Ref. [4].Following Ref. [4], the Bell operator in the commutative case is given by the linearcombination of four expectation values, B = hM (0 , i + hM ( α , i + hM (0 , α ) i − hM ( α , α ) i , (7)6here α , are the phase-space variables (or symbols) associated with the operators ˆ α =ˆ x c + i ˆ p xc and ˆ α = ˆ y c + i ˆ p yc which are complex amplitudes that carry two degrees of freedom,ˆ α and ˆ α ∗ for each mode, 1 and 2, and hM ( α , α ) i ≡ π W ( α , α ) , (8)with W ( α , α ) being the commutative Wigner function of the state calculated in ( α , α ).Local theories admit a description in terms of local hidden variables [4], i.e. α and α (inEq.(7)).This variables are the ones that will be used to paramterize the dichotomic behaviourof the Bell operator. Let us consider a free parameter I that will represent this dichotomicbehaviour. In that sense, α and α , can only correspond to I or −I . On the other hand,non-locality is identified by |B min | > . (9)From Eqs. (7) and (8), one notices that the Bell operator is a functional depending onthe Wigner function of the state. In order to understand what is the real effect of NC in theBell functional, let one considers an analogy with what happens to the expectation values ofthe position, h x i , which are also functionals of the Wigner function. One begins the analysiswith a Wigner function that could be either commutative (i. e. standard QM), W ( x, p ), orNC, W NC ( x, p ) [32]. That is, due to the fact that one actually does not know if the stateobeys a NC dynamics or not. The only requirement here is that the initial Wigner function, W ( x, p ), will give the expectation values that are going to be observed in the laboratory atthe initial time. The expectation value < x > is also given by a functional of the Wignerfunction, that is h x i = R dx dp x F ( x, p ), which is the same in the commutative and in theNC case. Thus, it is expected that at the initial time, W C ( x, p ) = W NC ( x, p ). Analogously,one should have B NC ≡ B ( W NC ( x, p )) = B ( W C ( x, p )) ≡ B C .Now, to understand what is the influence of NC in the expectation value, h x i , the evolu-tion of the system with the two Hamiltonians, the commutative one and the NC one, shouldbe studied. Then, one should look for discrepancies between the values of the expectationvalue h x t i in the two cases. This clearly can be done in the laboratory in order to determineif the state is either NC or commutative, once one lets the system evolve in time. In thiscase, B NC ( t ) = B ( W NCt ( x, p )) and one should compare it with B ( t ) = B ( W t ( x, p )).In what follows, we considere a NC Gaussian squeezed state and compute the Bell func-tionals for the NC and commutative case for the two states. For Gaussian states, in cases7n which the CHSH inequality can be rewritten in terms of Gaussian symplectic invariants[33, 35], the noncommutativity may change the Gaussian covariant matrix without changingtheir invariants.Of course, it might exist other ways to implement NC effects into the Bell operator,however we shall not venture here into this path and will constrain ourselves to examinejust squeezed states. III. NC TWO MODE SQUEEZED STATE
One considers now a bipartite Gaussian state [5], W ( R ) = 1 π p Det [ σ ] exp (cid:18) − R T σ − R (cid:19) , (10)where R ≡ ( x, p x , y, p y ) is the vector of a set of orthogonal quadratures, for modes a and b ,respectively. The covariance matrix, σ , is given by a 4 × σ = α γγ T β , (11)where α and β represent the self-correlation of each single subsystem and γ describes thecorrelation between the two subsystems. Then, any covariance matrix that describes aphysical state can be written, through a local symplectic transformation, in the standardform: σ = n c n c c m c m , (12)where n, m, c and c are determined by the four local symplectic invariants I ≡ Det [ α ] = n , I ≡ Det [ β ] = m , I ≡ Det [ γ ] = c c and I ≡ Det [ σ ] = ( nm − c )( nm − c ).When considering Bell inequalities, most often a squeezed state is used. By consideringthe case of a squeezed state in standard commutative quantum mechanics, a departureHamiltonian can be given by [34] H = ik ( ζ ∗ a † b † − ζ ab ) , (13)8here a † , b † , a, b are the creation and annihilation operators of the two modes, A and B,respectively. The only non-vanishing commutators are [ a, a † ] = [ b, b † ] = 1. Moreover, ζ = re − iφ , where r is the squeezing parameter and φ a phase associated to the “pump” thatproduces the squeezed state. If the phase is null, φ = 0, then the equations of motion aresolved for the operators, ˙ O = − i [ O, H ], and one obtains the way the operators transform[34]: a = a cosh( r ) + b † sinh( r ) , a † = a † cosh( r ) + b sinh( r ) ,b = b cosh( r ) + a † sinh( r ) , b † = b † cosh( r ) + a sinh( r ) . (14)This is the how creation/annihilation operators transform when one applies a squeezingtransformation, given by the operator S = exp ( ζ ∗ ab − ζ a † b † ), where the quadratures aredefined as x = 1 √ a + a † ) , p x = − i √ a − a † ) ,y = 1 √ b + b † ) , p y = − i √ b − b † ) . (15)Applying the Bogoliubov transformation, x = x cosh( r ) + y sinh( r ) , p x = p x cosh( r ) − p y sinh( r ) ,y = y cosh( r ) + x sinh( r ) , p y = p y cosh( r ) − p x sinh( r ) , (16)one is able to define S as the matrix that transforms the quadratures into the quadraturesof a squeezing state: S ( r ) = cosh( r ) 0 sinh( r ) 00 cosh( r ) 0 − sinh( r )sinh( r ) 0 cosh( r ) 00 − sinh( r ) 0 cosh( r ) , (17)where the time evolution is given in terms of the squeezing parameter r , and which enablesone to construct the covariance matrix of the squeezed state, Σ ( r ) = S T ( r ) S ( r ).The same strategy can be applied for the NC case, but now one has to take into accountthat the position and momenta obey the NC algebra:[ x, y ] = iθ , [ p x , p y ] = iη , [ x, p x ] = [ y, p y ] = i , (18)9here θ and η are new fundamental constants of Nature. The NC variables can be relatedwith the commutative ones by a Darboux transformation (or Seiberg-Witten map), Eq. (6),ˆ z = D ˆ ξ . In particular, in this case the NC variables are given by, x = λx c − θ λ p y c , p x = µp x c + η µ y c ,y = λy c + θ λ p x c , p y = µp y c − η µ x c , (19)where λ and µ are dimensionless constants, such that 2 λµ = 1 + √ − θη so to ensure thatthe map is invertible, and the index c denotes commutative variables. Through the definitionof creation/annihilation operators in terms of position and momenta a = (1 / √ x + ip y ), a † = (1 / √ x − ip y ) (analogous to the B mode), one obtains a = 1 √ (cid:20) λx c − θ λ p y c + i (cid:18) µp x c + η µ y c (cid:19)(cid:21) , (20) a † = 1 √ (cid:20) λx c − θ λ p y c − i (cid:18) µp x c + η µ y c (cid:19)(cid:21) , (21) b = 1 √ (cid:20) λy c + θ λ p x c + i (cid:18) µp y c − η µ x c (cid:19)(cid:21) , (22) b † = 1 √ (cid:20) λy c + θ λ p x c − i (cid:18) µp y c − η µ x c (cid:19)(cid:21) . (23)Then, one finds the NC quadrature operators solving the previous system of equations, x c = µ √ λµ − µ (cid:20) ( a + a † ) − i θ λµ ( b − b † ) (cid:21) , (24) p x c = 12 √ λµ − µ (cid:2) i ( a † − a ) − η (2 λµ )( b + b † ) (cid:3) , (25) y c = µ √ λµ − µ (cid:20) ( b + b † ) + i θ λµ ( a − a † ) (cid:21) , (26) p y c = 12 √ λµ − µ (cid:2) i ( b † − b ) − η (2 λµ )( a + a † ) (cid:3) . (27)Since now the positions and the momenta do not commute in the phase space, one evaluatesall commutators for the creation and annihilation operators in terms of the commutativevariables [ a, a † ] = [ b, b † ] = 1, [ a, b ] = [ a † , b † ] = i ( θ − η ), [ a, b † ] = [ a † , b ] = i ( θ + η ), and therest of the commutators vanish. The equations of motion for the creation and annihilation10perators become dadt = 1 i [ a, H ] = k (cid:20) b † + i (cid:0) θ ( a † − a ) + η ( a † + a ) (cid:1)(cid:21) , (28) da † dt = 1 i [ a † , H ] = k (cid:20) b + i (cid:0) θ ( a † − a ) − η ( a † + a ) (cid:1)(cid:21) , (29) dbdt = 1 i [ b, H ] = k (cid:20) a † − i (cid:0) θ ( b † − b ) + η ( b † + b ) (cid:1)(cid:21) , (30) db † dt = 1 i [ b † , H ] = k (cid:20) b † + i (cid:0) θ ( a † − a ) + η ( a † + a ) (cid:1)(cid:21) . (31)Solving the system of equations, considering kt = r , where r is the squeezing parameter,and using the definitions of the quadratures operators, one has the following Bogoliubovtransformation: x = cosh ( p θηr ) [ x cosh( r ) + y sinh( r )] + s θη sinh ( p θηr ) [ p x cosh( r ) − p y sinh( r )] , (32) p x = cosh ( p θηr ) [ p x cosh( r ) − p y sinh( r )] + r ηθ sinh ( p θηr ) [ x cosh( r ) + y sinh( r )] , (33) y = cosh ( p θηr ) [ y cosh( r ) + x sinh( r )] − s θη sinh ( p θηr ) [ p y cosh( r ) − p x sinh( r )] , (34) p y = cosh ( p θηr ) [ p y cosh( r ) − p x sinh( r )] − r ηθ sinh ( p θηr ) [ y cosh( r ) + x sinh( r )] . (35)Thus, in order to have the NC covariance matrix one needs S NC ( r ) = cosh ( ξr ) cosh( r ) q θη sinh ( ξr ) cosh( r ) cosh ( ξr ) sinh( r ) − q θη sinh ( ξr ) sinh( r ) p ηθ sinh ( ξr ) cosh( r ) cosh ( ξr ) cosh( r ) p ηθ sinh ( ξr ) sinh( r ) − cosh ( ξr ) sinh( r )cosh ( ξr ) sinh( r ) q θη sinh ( ξr ) sinh( r ) cosh ( ξr ) cosh( r ) − q θη sinh ( ξr ) cosh( r ) − p ηθ sinh ( ξr ) sinh( r ) − cosh ( ξr ) sinh( r ) − p ηθ sinh ( ξr ) cosh( r ) cosh ( ξr ) cosh( r ) . (36) Hence, likewise the commutative case, one can obtain the covariance matrix, Σ NC ( r ) = S NCT ( r ) S NC ( r ). Besides having the same departure Gaussian states (envelops), i. e. Σ NC (0) = Σ (0) since S NC (0) = S (0), which guarantees that the NC effects are purelydue to the dynamics driven by the corresponding (non)commutative Hamiltonian, one no-tices that the symplectic invariants from Eq. (36) are the same as those from Eq. (17),such that since the covariance matrix, Σ NC ( r ) = S NCT ( r ) S NC ( r ) is identified with σ fromEq. (10), the second moments of the Gaussian state (c.f. Eq. (10)) are not affected by theNC map. 11 V. BELL OPERATOR OF A NC SQUEEZING STATE IN NC QUANTUM ME-CHANICS
Knowing the covariance matrix for the NC squeezed state one is now able to evaluate theWigner function for the quantum state. The inverse of the covariance matrix is written as( Σ NC ) − = n d t − cd m c t t c n − d − c t − d m . (37)where the parameter r was suppressed from the notation, such that n = cosh(2 r )2 (cid:20) (1 + cosh (2 ξr )) − θη (1 − cosh (2 ξr )) (cid:21) ,m = cosh(2 r )2 h (1 + cosh (2 ξr )) − ηθ (1 − cosh (2 ξr )) i ,d = − ( θ + η )2 ξ cosh(2 r ) sinh (2 ξr ) ,c = ( θ + η )2 ξ sinh(2 r ) sinh (2 ξr ) ,t = − sinh(2 r )2 (cid:20) (1 + cosh (2 ξr )) − θη (1 − cosh (2 ξr )) (cid:21) t = sinh(2 r )2 h (1 + cosh (2 ξr )) − ηθ (1 − cosh (2 ξr )) i . (38)After some algebraic manipulation, the NC Wigner function Eq. (10) can be written as W ( α , α ) = 1 π exp { − (cid:2) ( n − m )( | α | + | α | ) + 2 c ( α α ∗ − α ∗ α )++ 12 (cid:0) ( n + m + 2 d )( α + ( α ∗ ) ) + ( n + m − d )(( α ∗ ) + α ) (cid:1) ++ ( t + t )( α α + α ∗ α ∗ ) + ( t − t )( α α ∗ + α ∗ α )] } , (39)where α = x + ip x and α = y + ip y are complex amplitudes. Starting from Eq. (7) andusing the definitions, Eqs. (8) and (39), one otains after some algebraic manipulation that B = 14 h (cid:16) − n I (cid:17) − exp ( − ( n − t ) I ) i , (40)where I is some complex displacement amplitude. Thus, one has a Bell operator, B ( I , n, t ),that depends on a free parameter I , and on the state properties n and t . Looking for the12aximum violation for a given state, one should maximize the Bell operator in terms of thefree parameter I [5], ∂ B ∂ I | I =˜ I = 0 ⇔ ˜ I = 2 n − t ln (cid:18) n − t n (cid:19) . (41)which leads to B ( ˜ I ) = 14 " n − t ) (cid:0) − t n (cid:1) − nn − t n − t . (42)Substituting n and t by the respective functions, Eqs. (38), one finally obtains˜ I = ˜ I ( θ, η ) = 4 η log (1 + tanh(2 r )) η − θ + ( η + θ ) cosh(2 ξr )(cosh(2 r ) + 2 sinh(2 r )) (43)where the NC dynamics engenders the dependence of ˜ I on θ and η .The result for the Bell operator is given by B ( ˜ I ( θ, η )) = 14 " − r )(3 cosh(2 r ) + 2 sinh(2 r ))(1 + tanh(2 r )) r ) , (44)which implies that, it is exactly identical to the Bell operator in the standard QM (commu-tative) limit, ˜ I ( θ, η ) → ˜ I (0 , I (0 ,
0) = 2 log (1 + tanh(2 r ))cosh(2 r ) + 2 sinh(2 r ) , (45)In order to admit local theories in terms of hidden variables this Bell operator shouldsatisfy the condition |B ( ˜ I ) | ≤ . (46)Thus, for a set of departure Gaussian states for which B NC ≡ B ( W NC ( x, p )) = B ( W C ( x, p )) ≡ B C , one concludes that the NC Bell operator for the squeezed state thatdepends on the NC parameters, B ( ˜ I ( θ, η ), is the same as the Bell operator in the commu-tative case, B ( ˜ I (0 , V. CONCLUSIONS
In this work it is shown that for squeezed states phase noncommutativity does not in-troduce corrections to the Bell operator. In order to show that one considers, in particular,13 noncommutativity between modes as this is a way to strengthen the interaction betweenthem and has been considered earlier when considering an entanglement state [14]. However,when considering a NC squeezed state, it is shown that, despite the NC corrections in theevolution of the startes, the squeezing parameter is dominant and the Bell operator doesnot change when the noncommutativity is introduced. This result seems to indicate thatthe non-localities that the Bell operator reveals for quantum mechanics are unaltered bynoncommutativity. One way to interpret this result is that noncommutativity affects theway a system evolves, through the changes in the Hamiltonian, although not necessarilyintroducing further non-localities to the quantum mechanical problem. Of course, one couldconjecture that this result might be changed by going beyond squeezed states or by consid-ering more general (non canonical) NC structures.. For sure, this would open interestingexperimental opportunities to test noncommutativity and certainly deserve some furtherstudy.
Acknowledgments
This work is supported by the COST action MP1405. The work of CB is supported bythe FCT (Portugal) grant SFRH/BPD/62861/2009. The work of AEB is supported by theBrazilian Agencies FAPESP (grant 15/05903-4) and CNPq (grant 300809/2013-1 and grant440446/2014-7). [1] A. Einstein, B. Podolsky and N. Rosen, Phys. Rev. , 777 (1935) .[2] D. Bohm and Y. Aharonov, Phys. Rev., , 1070 (1957).[3] J.S. Bell, Physics , 195 (1964).[4] K. Banaszek and K. W´odkiewicz, Phys. Rev. A58 , 4345 (1998); Phys. Rev. Lett. , 2009(1999).[5] D. Buono, G. Nocerino, S. Solimeno, and A. Porzio, Laser Phys. , 074008 (2014).[6] D. Buono, G. Nocerino, V.D’ Auria, A. Porzio, S. Olivares, and M.G.A. Paris, J. Opt. Soc.Am. B27 , A110 (2010).[7] M. Horodecki, J. Oppenheim, and A. Winter, Nature , 6 (2005); Comm. Math. Phys. 268,
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