Generalized squeezing operators, bipartite Wigner functions and entanglement via Wehrl's entropy functionals
aa r X i v : . [ qu a n t - ph ] J a n Generalized squeezing operators, bipartite Wigner functions, and entanglement viaWehrl’s entropy functionals
Marcelo A. Marchiolli ∗ and Di´ogenes Galetti † Instituto de F´ısica Te´orica, Universidade Estadual Paulista,Rua Pamplona 145, 01405-900, S˜ao Paulo, SP, Brazil (Dated: October 30, 2018)We introduce a new class of unitary transformations based on the su (1 ,
1) Lie algebra that genera-lizes, for certain particular representations of its generators, well-known squeezing transformationsin quantum optics. To illustrate our results, we focus on the two-mode bosonic representation andshow how the parametric amplifier model can be modified in order to generate such a generalizedsqueezing operator. Furthermore, we obtain a general expression for the bipartite Wigner functionwhich allows us to identify two distinct sources of entanglement, here labelled by dynamical andkinematical entanglement. We also establish a quantitative estimate of entanglement for bipartitesystems through some basic definitions of entropy functionals in continuous phase-space represen-tations.
PACS numbers: 02.20.Qs, 03.65.Ca, 03.67.Mn
I. INTRODUCTION
In the last few years, physics has experienced the ap-pearance of two relatively young branches with strongappeal in both theoretical and experimental aspects. La-belled by Quantum Information and Quantum Computa-tion, these branches apparently entangled have attractedsince then a lot of attention from researchers workingin well-established areas in physics (such as, for ins-tance, solid state physics, nuclear physics, high energyphysics, general relativity, and cosmology [1]). The in-terdisciplinarity provided by Quantum Information andQuantum Computation is basically focussed upon fun-damental physical concepts that constitute the corner-stones of quantum mechanics. Hence, concepts relatedto nonclassical states, superposition principle, entangle-ment, phase-space representations, quantum teleporta-tion, quantum key distribution, among others, representnowadays common words in many scientific papers cove-ring different areas of knowledge in physics.In particular, let us restrict our attention to quan-tum information theory and its description by quantumcontinuous variables [2], where entanglement effects canbe efficiently produced in laboratory through the ade-quate manipulation of continuous quadrature amplitudesof the quantized electromagnetic field. In this promi-sing scenario, the squeezed light [3, 4] has a prominentrole in the experimental implementation of continuous-variable entanglement, since the degree of imperfection inentanglement-based quantum protocols depends on ‘theamount of squeezing of the laser light involved’. From aconceptual point of view, some questions related to theentanglement measures (or separability criteria) still re- ∗ Electronic address: [email protected] † Electronic address: [email protected] main open in the specialized literature [5]. For instance,entanglement of formation [6] and concurrence [7] arenow widely accepted as entanglement measures for thetwo-qubit case. However, it is worth noticing that diffe-rent approaches to this problem exist which are outlinedby means of information-theoretic arguments [8].The main goal of this paper is to present some contri-butions to certain specific topics in quantum informationtheory that allow us to go further in our comprehen-sion on the entanglement process in ideal bipartite sys-tems (the interaction with any dissipative environment[9] is discarded in a first moment). For this purpose,we first construct a general family of unitary transfor-mations associated with the su (1 ,
1) Lie algebra genera-tors that generalizes – if one considers the one- and two-mode bosonic representations of its generators – two well-known expressions of squeezing operators [3]. Following,we study a modified version of the parametric ampli-fier [10] with emphasis in obtaining the solutions of theHeisenberg equations and its respective time-evolutionoperator. In particular, we show the efficacy of this modelin generating the generalized two-mode squeezing opera-tor through its connection with the time-evolution ope-rator.The next step then consists in performing a prelimi-nary study via Wigner function on the qualitative as-pects of entanglement for a general class of bipartite sys-tems whose dynamics is governed by arbitrary quadraticHamiltonians. As a by-product of this investigation, weobtain a general integral representation for the bipartiteWigner function that leads us to identify two distinctsources of entanglement (here labelled by dynamical andkinematical entanglement). The last contribution refersto a direct application of the results obtained by Pi¸atekand Leo´nski [11] for the intermode correlations in con-tinuous phase-space representations, which are based onthe Wehrl’s approach [12] regarding some definitions ofentropy functionals and their inherent properties. In fact,we introduce a correlation functional that permits us tomeasure the degree of entanglement between both partsof the joint system for any initial conditions associatedwith the Hamiltonian operator. It is important to em-phasize that the sequence of topics covered in this workpresents an inherent logical consistency that improves ourcomprehension on the subtle mechanisms associated withthe entanglement effects in bipartite systems.This paper is structured as follows. In Section II, wepresent a mathematical statement that leads us to cons-truct a general family of unitary transformations associ-ated with the su (1 ,
1) Lie algebra where, in particular,the two-mode bosonic representation is emphasized. InSection III, we obtain the solutions of the Heisenbergequations for the parametric amplifier model, and showthe subtle link between time-evolution operator and ge-neralized two-mode squeezing operator. The results ob-tained are then applied in Section IV, within the con-text of Wigner functions, in order to establish an initialdiscussion on entanglement for certain groups of bipar-tite states of the electromagnetic field. Section V is de-voted to establish a reasonable measure of entanglementwhich is based on some specific information-theoretic ar-guments. Besides, we illustrate our results through twodifferent examples of initially uncoupled bipartite statesfor the model under investigation. Finally, Section VIcontains our summary and conclusions.
II. SQUEEZING OPERATORS ASSOCIATEDWITH THE su (1 , LIE ALGEBRA
Let us initially introduce the generators of the su (1 , K ± and K , which satisfy the following com-mutation relations:[ K − , K + ] = 2 K and [ K , K ± ] = ± K ± . The Casimir operator is defined within this contextthrough the mathematical identity K := K − (1 / { K − , K + } with [ K , K ± ] = [ K , K ] = 0, where { K − , K + } repre-sents the anticommutation relation between the opera-tors K − and K + . Furthermore, let us also consider theabstract operator T (Ω ± , Ω ) := e Ω + K + +Ω K +Ω − K − (1)written in terms of the arbitrary c-number parametersΩ ± and Ω , whose generalized normal- and antinormal-order decomposition formulas have already been establi-shed in literature [13, 14]. These preliminary considera-tions on the su (1 ,
1) Lie algebra lead us to demonstratean important composition formula involving the productof two abstract operators, each one being defined as inequation (1) and characterized by a particular set of ar-bitrary c-number parameters. In fact, this section will deal with some applications of the composition formulawith emphasis on unitary transformations which resem-ble, for certain representations of the generators K ± and K , the squeezing transformations in quantum optics. Lemma II.1
Let T (Ω ± , Ω ) and T (Λ ± , Λ ) be two abs-tract operators whose functional forms obey equation(1). For a given set of arbitrary c-number parameters { Ω ± , Ω , Λ ± , Λ } it is always possible to verify the gene-ral composition formula T (Ω ± , Ω ) T (Λ ± , Λ ) = T (Σ ± , Σ ) , (2) where Σ ± and Σ are solutions of the coupled set of non-linear equations A + + B B + [ A − A + ( A − + B − )] B ( A − + B − ) = Σ + Σ − , (3a) p B /A ( A − + B − ) = (Σ − /β ) sinh( β ) , (3b) with β = [(Σ / − Σ + Σ − ] / . The c-number functions A ± , A , B ± , and B , present in the lhs of equations (3a)and (3b), are connected with { Ω ± , Ω , Λ ± , Λ } throughthe identities [14]: A ± = (Ω ± /φ ) sinh( φ )cosh( φ ) − (Ω / φ ) sinh( φ ) , (4a) A = [cosh( φ ) − (Ω / φ ) sinh( φ )] − , (4b) B ± = (Λ ± /θ ) sinh( θ )cosh( θ ) + (Λ / θ ) sinh( θ ) , (4c) B = [cosh( θ ) + (Λ / θ ) sinh( θ )] , (4d) for φ = [(Ω / − Ω + Ω − ] / ,θ = [(Λ / − Λ + Λ − ] / . Proof II.1
Firstly, we apply the generalized normal- andantinormal-order decomposition formulas established byBan [14] for exponential functions of the generators of su (1 , Lie algebra to the abstract operators T (Ω ± , Ω ) and T (Λ ± , Λ ) , that is, T (Ω ± , Ω ) = e A + K + e ln( A ) K e A − K − (5) and T (Λ ± , Λ ) = e B − K − e ln( B ) K e B + K + , (6) where the c-number functions A ± , B ± , A , and B weredefined by means of the identities (4a)-(4d). In this way,the product T (Ω ± , Ω ) T (Λ ± , Λ ) can be expressed as T (Ω ± , Ω ) T (Λ ± , Λ ) = e A + K + e ln( A ) K e ( A − + B − ) K − × e ln( B ) K e B + K + = e A + K + e ln( A ) K e C + K + × e ln( C ) K e C − K − , (7) with C ± and C given by [14]: C + = B B + − B B + ( A − + B − ) ,C − = B ( A − + B − )1 − B B + ( A − + B − ) ,C = B [1 − B B + ( A − + B − )] . The second step consists in using the relationship e ln( A ) K e C + K + = e A C + K + e ln( A ) K for the second and third exponentials in the second equa-lity on the rhs of equation (7), with the aim of establishingthe intermediate result T (Ω ± , Ω ) T (Λ ± , Λ ) = e ( A + + A C + ) K + e ln( A C ) K e C − K − . The rhs of this equation represents the normal-order de-composition of an abstract operator defined as T (Σ ± , Σ ) := e Σ + K + +Σ K +Σ − K − , where the c-number parameters Σ ± and Σ satisfy thefollowing mathematical relations: A + + A C + = (Σ + /β ) sinh( β )cosh( β ) − (Σ / β ) sinh( β ) ,C − = (Σ − /β ) sinh( β )cosh( β ) − (Σ / β ) sinh( β ) ,A C = [cosh( β ) − (Σ / β ) sinh( β )] − , with β = [(Σ / − Σ + Σ − ] / . Consequently, substi-tuting the definitions of C ± and C in these relations,we obtain a coupled set of nonlinear equations that per-mits us not only to establish a link between { Σ ± , Σ } and { Ω ± , Ω , Λ ± , Λ } , but also to verify the general composi-tion formula (2). (cid:4) An interesting consequence from this mathematicalstatement is associated with the construction process of ageneral family of unitary transformations where the abs-tract operator (1) has a central role. To carry out thistask let us first establish a corollary which is directly re-lated to the Lemma II.1.
Corollary II.1
For Ω + = ξ , Ω − = − ξ ∗ , and Ω = i ω ,with ξ ∈ C and ω ∈ R , the abstract operator (1) repre-sents a generator of unitary transformations associatedwith the su (1 , Lie algebra.
Proof II.2
Basically, the idea is obtaining a specific sub-set of arbitrary c-number parameters such that T (Ω ± , Ω ) [ T (Ω ± , Ω )] † = [ T (Ω ± , Ω )] † T (Ω ± , Ω ) = . Hence, let us initially investigate under what circumstan-ces the mathematical relation T (Ω ± , Ω ) [ T (Ω ± , Ω )] † = is verified. In fact, this condition can be promptly derivedfrom the general composition formula (2) for Λ ± = Ω ∗∓ , Λ = Ω ∗ , and Σ ± = Σ = 0 , with the additional res-trictions Λ ± = − Ω ± and Λ = − Ω . Consequently, theequalities Ω ± = − Ω ∗∓ and Ω = − Ω ∗ are satisfied onlyfor Ω + = ξ , Ω − = − ξ ∗ , and Ω = i ω , with ξ ∈ C and ω ∈ R . This same particular subset of arbitrary c-numberparameters can also be obtained from the analysis of [ T (Ω ± , Ω )] † T (Ω ± , Ω ) = , which implies that T ( ξ, ω ) gives a family of unitary trans-formations for a general class of representations associ-ated with the su (1 , Lie algebra. (cid:4)
To illustrate our results, let us consider as a firstexample the single-mode bosonic representation of theHeisenberg-Weyl algebra where the generators K ± and K are expressed as K + = (1 / a † , K − = (1 / a , and K = (1 / a † a +1 / a and a † being, respectively,the boson annihilation and creation operators satisfyingthe well-known commutation relation [ a , a † ] = . In thiscase, the unitary operator T ( ξ, ω ) assumes the form T ( ξ, ω ) = e [ ξ a † +i ω ( a † a +1 / ) − ξ ∗ a ] , (8)which coincides with the squeezing operator S ( ξ ) := e ( ξ a † − ξ ∗ a ) when ω = 0 . It is worth mentioning that, in particular, equation (8)can be promptly used to derive the quantum analogousof the Fresnel transform in classical optics [15].Another interesting example encompasses the two-mode bosonic representation in which the generatorsare specifically given by K + = a † b † , K − = ab , and K = (1 / a † a + b † b + 1), where [ a , a † ] = [ b , b † ] = .In this context, the unitary operator T ( ξ, ω ) = e ξ a † b † +i( ω/ a † a + b † b +1) − ξ ∗ ab (9)recovers the two-mode squeezing operator S ( ξ ) = exp( ξ a † b † − ξ ∗ ab ) for ω = 0 . Moreover, the action of T ( ξ, ω ) in the annihilation ope-rators for each mode of the electromagnetic field impliesin the following results: T † ( ξ, ω ) aT ( ξ, ω ) = [cosh( φ ) + i( ω/ φ ) sinh( φ )] a + ( ξ/φ ) sinh( φ ) b † , (10) T † ( ξ, ω ) bT ( ξ, ω ) = [cosh( φ ) + i( ω/ φ ) sinh( φ )] b + ( ξ/φ ) sinh( φ ) a † , (11)being φ = [ | ξ | − ( ω/ ] / . This set of unitary transfor-mations for ω = 0 has its counterpart in the quantum des-cription of physical processes involving parametric ampli-fication [10, 16]. Recently, Pielawa et al . [17] have pro-posed a new method for generating two-mode squeezingin high-Q resonators using a beam of atoms (which actsas a reservoir for the field) with random arrival times. Inparticular, the authors have used the unitary two-modesqueezing operator S ( ξ ) to bring an effective Hamilto-nian – which describes resonant single-photon processes– to the well-known Jaynes-Cummings form, where thenew bosonic operators are connected to the old ones bytwo-mode squeezing transformations. Another interes-ting application is based on the analogy between phononsin an axially time-dependent ion trap and quantum fieldsin an expanding/contracting universe, where the multi-mode squeezing operator represents the basic mechanismfor cosmological particle creation [18].It is worth noticing that there are some textbooks onquantum optics [3] which discuss particular cases (if onecompares with those exposed here) of one- and two-modesqueezing operators in different contexts in physics andtheir connections with nonclassical states of the electro-magnetic field. However, the dynamical origin of the realparameter ω present in equation (9), for example, hasnot been investigated up until now in literature, and thisfact will be our object of study in the next section. Forthis intent, we obtain the exact solutions of the Heisen-berg equations for a specific nonresonant system like theparametric amplifier, and show the link between time-evolution operator and T ( ξ, ω ). III. PARAMETRIC AMPLIFIER
The parametric amplifier model proposed by Louisell et al . [10] consists basically of two coupled modes of theelectromagnetic field, which play a symmetrical role inthe amplification process [16]. Such dynamical elementsare usually described by the Hamiltonian ( ~ = 1) H ( t ) = ω a a † a + ω b b † b + κ ab e i ηt + κ ∗ a † b † e − i ηt , (12)where a ( b ) is the annihilation operator for the signal(idler) mode, η provides the frequency of the pump field(which has been assumed strong enough to be expressedin classical terms), and κ represents a complex couplingconstant being proportional to the second-order suscep-tibility of the nonlinear medium and to the amplitude ofthe pump. Moreover, let us introduce a small deviation δ in the usual definition of η such that η = ω a + ω b + δ with δ/ ( ω a + ω b ) ≪ δ comes from a non-perfect match between the fre-quencies η and ω a + ω b ). In this case, the solutions of theHeisenberg equations are given by a ( t ) = e − i( ω a + δ/ t n [cosh( ϕt ) + i( δ/ ϕ ) sinh( ϕt )] a (0) − i( κ ∗ /ϕ ) sinh( ϕt ) b † (0) o , b ( t ) = e − i( ω b + δ/ t n [cosh( ϕt ) + i( δ/ ϕ ) sinh( ϕt )] b (0) − i( κ ∗ /ϕ ) sinh( ϕt ) a † (0) o , plus their Hermitian conjugates for ϕ = [ | κ | − ( δ/ ] / fixed. These solutions lead us to verify that [ a ( t ) , a † ( t )] =[ b ( t ) , b † ( t )] = , which implies in the unitariety of thetime-evolution operator U ( t ). This fact leads us to showthat n a ( t ) − n b ( t ) = n a (0) − n b (0) (conservation law) for n c ( t ) := c † ( t ) c ( t ) is easily seen to hold; therefore, theintensity correlation function h n a ( t ) n b ( t ) i can be writtenas h n a ( t ) n b ( t ) i = h n a ( t ) i + h n a ( t )[ n b (0) − n a (0)] i . Forinstance, when the initial state coincides with the num-ber states {| n a , n b i} n a ,n b ∈ N , the second term on the rhs ofthis equation is reduced to ( n b − n a ) h n a , n b | n a ( t ) | n a , n b i for n a = n b . So, if one considers n a = n b , there is no con-tribution from this term and consequently, h n a ( t ) n b ( t ) i corresponds to the maximum violation of the Cauchy-Schwarz inequality h a † ab † b i ≤ h a † a i for the parame-tric amplifier [3].Next, we show how the time-evolution operator U ( t )can be connected with the unitary operator T ( ξ, ω ). Tocarry out this task, let us initially mention that the time-dependent global phase factors present in the solutions ofthe Heisenberg equations are obtained through the actionof the rotation operator R ( ω a , ω b , δ ; t ) = e − i t [ ( ω a + δ/ a † a +( ω b + δ/ b † b ] (13)on the annihilation operators a (0) and b (0). The nextstep then consists in noticing that for φ = ϕt , ω = δt ,and ξ = − i κ ∗ t , the operator T ( κ, δ ; t ) = e − i t [ κ ∗ a † b † − ( δ/ a † a + b † b +1)+ κ ab ] (14)is responsible for generating the terms between braces inthe solutions a ( t ) and b ( t ) – see equations (10) and (11).After these considerations, it is easy to show that U ( t ) := R ( ω a , ω b , δ ; t ) T ( κ, δ ; t ) (15)provides the solutions obtained above by means of themathematical operation sketched in the identities a ( t ) = U † ( t ) a (0) U ( t ) and b ( t ) = U † ( t ) b (0) U ( t ). Note that theparticular nonresonant system here studied represents afirst dynamical application of the results obtained in Sec-tion II for the two-mode bosonic representation, wherethe connection between the time-evolution operator (15)and the unitary operator (9) is promptly established. Itis worth emphasizing that different physical systems canalso be used for explaining the dynamical origin of thereal parameter ω present in T ( ξ, ω ) (e.g., see Ref. [17]),but this fact still deserves be carefully investigated. IV. AN INITIAL STUDY ON ENTANGLEMENTVIA WIGNER FUNCTION
Let us initially consider a specific subset of bipartitephysical systems described by continuous variables suchthat the parametric amplifier model here studied (or thesimple model for parametric frequency conversion pro-posed by Louisell et al . [10], and subsequently studied indetail by Tucker and Walls [19]) constitutes a particularelement. Furthermore, let us also assume that ρ ( t ) = U ( t ) ρ (0) U † ( t ) describes the dynamics of any bipartitesystem belonging to this subset whose unitary time-evolution operator U ( t ) is related to the Hamiltonianoperator H ( t ); by hypothesis, the initial density operator ρ (0) represents the system prepared at the initial instant t = 0 in any disentangled (entangled) state. The symme-tric characteristic function for this class of bipartite phy-sical systems is given by C ( G ; t ) := Tr[ D ( G ) ρ ( t )], where D ( G ) := exp( − G † EO ) defines a displacement operatorwritten in terms of the matrices G † = (cid:0) ξ ∗ a ξ a ξ ∗ b ξ b (cid:1) , E = I ⊗ S with I = diag(1 ,
1) and S = diag(1 , −
1) beingthe respective 2 × O † = (cid:0) a † a b † b (cid:1) . Note that due to the cyclic pro-perty of the trace operation, the symmetric characteristicfunction can also be evaluated through the mathematicalstatement Tr[ U † ( t ) D ( G ) U ( t ) ρ (0)], where the time evo-lution of the displacement operator plays an importantrole.Next, let us suppose that the action of the unitarytime-evolution operator on the matrix O transforms theannihilation (creation) operators a ( a † ) and b ( b † ) fol-lowing the general rule O H ( t ) = U † ( t ) O U ( t ) = T ( t ) O ,where T ( t ) = µ a ( t ) ν a ( t ) χ a ( t ) η a ( t ) ν ∗ a ( t ) µ ∗ a ( t ) η ∗ a ( t ) χ ∗ a ( t ) µ b ( t ) ν b ( t ) χ b ( t ) η b ( t ) ν ∗ b ( t ) µ ∗ b ( t ) η ∗ b ( t ) χ ∗ b ( t ) represents a 4 × O H ( t ) with specific initial conditions that preserve thequantum mechanics (the subscript H indicates that ope-rators are in the Heisenberg picture). For instance, when t = 0 the c-number functions should necessarily imply inthe mathematical identity T (0) = diag(1 , , , t ≥ (cid:2) a H ( t ) , a † H ( t ) (cid:3) = , (cid:2) b H ( t ) , b † H ( t ) (cid:3) = , [ a H ( t ) , b H ( t )] = 0 , (cid:2) a H ( t ) , b † H ( t ) (cid:3) = 0 , provide extra relations for the c-number functions whichpermit us to solve completely the Heisenberg equations,namely | µ a ( t ) | − | ν a ( t ) | + | χ a ( t ) | − | η a ( t ) | = 1 , | µ b ( t ) | − | ν b ( t ) | + | χ b ( t ) | − | η b ( t ) | = 1 ,µ a ( t ) ν b ( t ) − ν a ( t ) µ b ( t ) + χ a ( t ) η b ( t ) − η a ( t ) χ b ( t ) = 0 ,µ a ( t ) µ ∗ b ( t ) − ν a ( t ) ν ∗ b ( t ) + χ a ( t ) χ ∗ b ( t ) − η a ( t ) η ∗ b ( t ) = 0 . The first immediate consequence of these results is that U † ( t ) D ( G ) U ( t ) will produce a new displacement opera-tor D ( Y ) = exp( − Y † EO ) with Y † = (cid:0) β ∗ a β a β ∗ b β b (cid:1) ,where the new elements are connected with the old onesthrough the equalities β a = µ ∗ a ( t ) ξ a − ν a ( t ) ξ ∗ a + µ ∗ b ( t ) ξ b − ν b ( t ) ξ ∗ b ,β b = χ ∗ a ( t ) ξ a − η a ( t ) ξ ∗ a + χ ∗ b ( t ) ξ b − η b ( t ) ξ ∗ b , plus their respective complex conjugates. For conve-nience in our calculations, β a ( b ) means a short nota-tion for β a ( b ) = β a ( b ) ( ξ a , ξ b ; t ). It is worth noticing that C ( G ; t ) may now be written as C ( Y ; 0) = Tr[ D ( Y ) ρ (0)](i.e., the symmetric characteristic function C ( G ; t ) is thusspecified in terms of the form it takes at t = 0, which cor-roborates the results obtained by Mollow and Glauber[16] for the parametric amplifier model), and this factwill bring some insights into the study of entanglementfor bipartite systems via Wigner function.The Wigner function for this particular subset of bi-partite physical systems may then be defined as the four-dimensional Fourier transform of the symmetric charac-teristic function C ( G ; t ), that is, W ( X ; t ) = Z d ξ a d ξ b π exp (cid:0) G † EX (cid:1) C ( G ; t ) , (16)with X † := (cid:0) α ∗ a α a α ∗ b α b (cid:1) . Note that the variables ofintegration ξ a and ξ b can be changed to β a and β b in thisequation, once the Jacobian matrix of the transformationhas a determinant whose absolute value is equal to one– this fact implies in the identity d ξ a d ξ b = d β a d β b .Hence, the integration can now be conveniently carriedout through the mathematical relation W ( X ; t ) = Z d β a d β b π exp (cid:0) Y † EZ (cid:1) C ( Y ; 0)= W ( Z ; 0) , (17)where Z † := (cid:0) γ ∗ a γ a γ ∗ b γ b (cid:1) defines a new matrix withelements given by[30] γ a = µ ∗ a ( t ) α a − ν a ( t ) α ∗ a + µ ∗ b ( t ) α b − ν b ( t ) α ∗ b ,γ b = χ ∗ a ( t ) α a − η a ( t ) α ∗ a + χ ∗ b ( t ) α b − η b ( t ) α ∗ b , and their respective complex conjugates. Thus, equa-tion (17) asserts that W ( X ; t ) can also be expressed interms of the form it takes at the initial time t = 0, sincethe new variables γ a ( α a , α b ; t ) and γ b ( α a , α b ; t ) carry theinformation of the dynamical entanglement between thevariables α a and α b . Furthermore, this result permits usto show that for a given bipartite system initially pre-pared in any entangled state, it will remain entangled forall t ≥
0; otherwise, if the density operator is describedat t = 0 as ρ (0) = ρ a (0) ⊗ ρ b (0) (disentangled state),the appearance of entanglement in the Wigner functionwill depend exclusively on the dynamics provided by theHamiltonian operator H ( t )[31] – here associated with thetime-dependent matrix Z . Following, let us apply theresults obtained until now to the parametric amplifiermodel, where different initial states of the electromag-netic field will be considered.Now we focus our efforts in evaluating W ( Z ; 0) fora well-known family of two-mode electromagnetic fieldswhere, in particular, both modes are initially prepared inthe coherent states, number states, and thermal states,respectively. For instance, the symmetric characteristicfunctions in these situations are given by C coh ( Y ; 0) = e − ( | β a | + | β b | ) +2i[Im( β a ζ ∗ a )+Im( β b ζ ∗ b )] , C n ( Y ; 0) = e − ( | β a | + | β b | ) L n a ( | β a | ) L n b ( | β b | ) , C th ( Y ; 0) = e − [ (1+2¯ n a ) | β a | +(1+2¯ n b ) | β b | ] , while their respective Wigner functions can be expressedas W coh ( Z ; 0) = 4 e − ( | γ a − ζ a | + | γ b − ζ b | ) , (18) W n ( Z ; 0) = 4( − n a + n b e − ( | γ a | + | γ b | ) L n a (4 | γ a | ) × L n b (4 | γ b | ) , (19) W th ( Z ; 0) = 4 [(1 + 2¯ n a )(1 + 2¯ n b )] − × e − [ (1+2¯ n a ) − | γ a | +(1+2¯ n b ) − | γ b | ] , (20)with L n ( z ) denoting a Laguerre polynomial and ¯ n a ( b ) being the mean number of photons for a chaotic lightfield. Note that if we consider the solutions obtainedfrom the Heisenberg equations for the parametric am-plifier model (see previous section), it is easy to showthat µ a ( t ), η a ( t ), ν b ( t ), and χ b ( t ) (once the further time-dependent coefficients do no exist) determine completelythe c-numbers β a ( b ) ( ξ a , ξ b ; t ) and γ a ( b ) ( α a , α b ; t ). Indeed,this last step leads us to characterize precisely C ( Y ; 0)and W ( Z ; 0). Moreover, when δ = 0, κ ∈ R + , and( ω a + ω b ) t = 3 π/
2, Eq. (18) coincides with W EPR ( γ a , γ b )for ζ a ( b ) = 0 (two-mode vacuum state), this function be-ing that used by Braunstein and Kimble [22] in the theo-retical description of teleportation involving continuousquantum variables.[32]Finally, let us say some few words about the resultsobtained in this section. It is important to emphasizethat equation (17) permits us to describe qualitativelythe entanglement of a specific subset of bipartite sys-tems, where now we can promptly identify two distinctorigins of this quantum effect: the first associated withthe entanglement in the initial conditions (here labelledby kinematical entanglement), while the second is res-ponsible for the dynamical entanglement via Hamiltonianoperator. In addition, the normally ordered moments h a † p ( t ) a q ( t ) b † r ( t ) b s ( t ) i = Γ ( p,q,r,s ) ξ a ,ξ ∗ a ,ξ b ,ξ ∗ b e ( | ξ a | + | ξ b | ) × C ( ξ a , ξ ∗ a , ξ b , ξ ∗ b ; t ) (cid:12)(cid:12)(cid:12)(cid:12) ξ a ,ξ ∗ a ,ξ b ,ξ ∗ b =0 , (21)with Γ ( p,q,r,s ) ξ a ,ξ ∗ a ,ξ b ,ξ ∗ b := ( − q + s ∂ p + q + r + s ∂ ξ p a ∂ ξ ∗ q a ∂ ξ r b ∂ ξ ∗ s b and { p, q, r, s } ∈ N , can also be used to investigate somerecent proposals of inseparability criteria for continu-ous bipartite quantum states [21, 23]. In the next sec-tion, we will establish a reasonable measure of entangle-ment which is based on the results obtained by Pi¸atekand Leo´nski [11] for the intermode correlations in phasespace. V. ENTROPY FUNCTIONALS FORCONTINUOUS PHASE-SPACEREPRESENTATIONS
In order to establish a quantitative estimate of entan-glement for bipartite systems, let us introduce some basicdefinitions of entropy functionals in continuous phase-space representations. The first definition is based onthe joint entropy [11]E[ H ; t ] := − Z d α a d α b π H ( α a , α b ; t ) ln [ H ( α a , α b ; t )] , (22)where H ( α a , α b ; t ) := h α a , α b | ρ ( t ) | α a , α b i denotes theHusimi function in the continuous coherent-state repre-sentations for a bipartite system described by the densityoperator ρ ( t ). Note that E[ H ; t ] presents certain proper-ties inherent to its definition which deserve be mentioned:(i) the probability distribution function H ( α a , α b ; t ) isstrictly positive and limited to the interval [0 , H ; t ] can be considered asa natural extension of that definition employed by Wehrl[12] for information entropy.[33]The second definition consists of functionals related tothe partial entropiesE[ H ( A ) ; t ] := − Z d α a π H ( A ) ( α a ; t ) ln h H ( A ) ( α a ; t ) i (23)andE[ H ( B ) ; t ] := − Z d α b π H ( B ) ( α b ; t ) ln h H ( B ) ( α b ; t ) i , (24)which depend basically on the marginal Husimi functions H ( A ) ( α a ; t ) = Z d α b π H ( α a , α b ; t ) , H ( B ) ( α b ; t ) = Z d α a π H ( α a , α b ; t ) . Such marginal Husimi functions carry information of theentanglement between the subsystems ‘A’ and ‘B’, sincethe partial trace over the continuous variables α a ( b ) ofa particular subsystem ‘A’ (‘B’) allows the introduction,via time-evolution operator U ( t ) and/or initial densityoperator ρ (0), of important correlations in the bipartitestates.Let us derive now some mathematical relations amongthese entropy functionals from the Araki-Lieb inequality[25], i.e., (cid:12)(cid:12)(cid:12) E[ H ( A ) ; t ] − E[ H ( B ) ; t ] (cid:12)(cid:12)(cid:12) ≤ E[ H ; t ] ≤ E[ H ( A ) ; t ]+ E[ H ( B ) ; t ] . (25)For instance, the rhs of this inequality corresponds to thesubadditivity property for the Wehrl’s entropy functio-nals, while the equal sign reflects the complete disentan-glement between the subsystems ‘A’ and ‘B’ of the jointsystem. In addition, the conditional entropies [11, 12]E[ H / H ( A ) ; t ] = E[ H ; t ] − E[ H ( A ) ; t ] (26)and E[ H / H ( B ) ; t ] = E[ H ; t ] − E[ H ( B ) ; t ] (27)lead us not only to establish the balance equationE[ H / H ( A ) ; t ]+E[ H ( A ) ; t ] = E[ H / H ( B ) ; t ]+E[ H ( B ) ; t ] , (28)but also to determine the further inequalitiesE[ H / H ( A ) ; t ] ≤ E[ H ( B ) ; t ]and E[ H / H ( B ) ; t ] ≤ E[ H ( A ) ; t ]from the subadditivity property. It is worth noticingthat the equal signs hold in both situations only whenthe c-numbers α a and α b are functionally uncorrelated,namely, the bipartite Husimi function H ( α a , α b ; t ) fac-torizes in the product of the marginal Husimi functions H ( A ) ( α a ; t ) H ( B ) ( α b ; t ).In what concerns the class of entropic functionals listeduntil now, it is convenient to define a new functional li-mited to the closed interval [0 ,
1] for any t ≥
0, whichpermits us to avoid any ambiguity in the significance ofthe subadditivity property. Thus, let us introduce thecorrelation functionalC[ H ; t ] := 2 (cid:18) − E[ H ; t ]E[ H ( A ) ; t ] + E[ H ( B ) ; t ] (cid:19) , (29)which can be used to measure the functional correlation(that is, the ‘degree of entanglement’) between the parts‘A’ and ‘B’ of the joint system, having as reference a fac-torizable bipartite Husimi function H ( α a , α b ; t ). In thisdefinition, the inferior limit of the closed interval [0 , A. Coherent states
In this first example, we consider the bipartite purestates ρ (0) = | ζ a , ζ b ih ζ a , ζ b | as being the initial state at t = 0 of the model governed by the Hamiltonian (12).Here, | ζ a , ζ b i = | ζ a i ⊗ | ζ b i where | ζ a ( b ) i characterizes thecoherent states related to the signal (idler) mode of theelectromagnetic field. Thus, after some nontrivial alge-bra, the bipartite Husimi function assumes the closed-form H coh ( α a , α b ; t ) = | A | e − ( | γ a − ζ a | + | γ b − ζ b | ) × e [ A ∗ + ( γ a − ζ a )( γ b − ζ b ) ] (30)such that γ a ( b ) = e i( ω a ( b ) + δ/ t [cosh( ϕt ) − i( δ/ ϕ ) sinh( ϕt )] α a ( b ) + i( κ ∗ /ϕ ) e − i( ω b ( a ) + δ/ t sinh( ϕt ) α ∗ b ( a ) ,A + = i( κ ∗ /ϕ ) sinh( ϕt )cosh( ϕt ) + i( δ/ ϕ ) sinh( ϕt ) ,A = [cosh( ϕt ) + i( δ/ ϕ ) sinh( ϕt )] − . Furthermore, the marginal Husimi functions H ( A )coh ( α a ; t ) = | A | e −| A || α a − ǫ a | (31)and H ( B )coh ( α b ; t ) = | A | e −| A || α b − ǫ b | , (32)with ǫ a ( b ) = n [cosh( ϕt ) + i( δ/ ϕ ) sinh( ϕt )] ζ a ( b ) − i( κ ∗ /ϕ ) sinh( ϕt ) ζ ∗ b ( a ) o e − i( ω a ( b ) + δ/ t , permit us to verify how the dynamical entanglement in-troduces correlations between the continuous variables ζ a ( b ) and ζ ∗ b ( a ) through the label ǫ a ( b ) . Indeed, for t = 0 itis possible to demonstrate that H coh ( α a , α b ; 0) factorizesin the product H ( A )coh ( α a ; 0) H ( B )coh ( α b ; 0). However, if oneconsiders t >
0, the dynamics governed by the processof parametric amplification introduces significant corre-lations between the signal and idler modes. Following, letus quantify these correlations with the help of Eq. (29).The Gaussian structures present in the bipartite andmarginal Husimi functions allow us to obtain closed formexpressions for the respective joint and partial entropies,namely,E[ H coh ; t ] = 2 + ln (cid:2) | κ | /ϕ ) sinh ( ϕt ) (cid:3) (33)andE[ H ( A , B )coh ; t ] = 1 + ln (cid:2) | κ | /ϕ ) sinh ( ϕt ) (cid:3) . (34)Consequently, the correlation functional can be promptlyevaluated as follows:C[ H coh ; t ] = ln (cid:2) | κ | /ϕ ) sinh ( ϕt ) (cid:3) (cid:2) | κ | /ϕ ) sinh ( ϕt ) (cid:3) . (35)Besides, if one considers the limit ϕt ≫ ϕt = 0 we obtain C[ H coh ; 0] = 0, which corroboratesthe factorization process of the bipartite Husimi function H coh ( α a , α b ; 0). B. Thermal states
Now, let us suppose that both signal and idler modeswere initially prepared in the thermal states. In this case,the initial density operator admits the expansion ρ th (0) = 1¯ n a ¯ n b Z d ζ a d ζ b π e − ( ¯ n − a | ζ a | +¯ n − b | ζ b | ) P ( ζ a , ζ b ) , where ¯ n a ( b ) represents the average occupation number ofeach mode, and P ( ζ a , ζ b ) = | ζ a , ζ b ih ζ a , ζ b | is the projectorof the coherent states. This typical example of bipartitemixed states constitutes an important starting point inthe investigation process of the dynamical entanglementdue to the parametric amplifier model. So, after somecalculations, the bipartite Husimi function can be writtenas H th ( α a , α b ; t ) = [ G ¯ n a , ¯ n b (1 , t )] − e − ( ℓ b | α a | + ℓ a | α b | ) × e ℓ ab α a α b ) (36)with ℓ a = G ¯ n a , ¯ n b (1 , t ) G ¯ n a , ¯ n b (1 , t ) , ℓ b = G ¯ n a , ¯ n b (0 , t ) G ¯ n a , ¯ n b (1 , t ) ,ℓ ab = i( κ/ϕ ) sinh( ϕt ) [cosh( ϕt ) − i( δ/ ϕ ) sinh( ϕt )] × (¯ n a + ¯ n b + 1) [ G ¯ n a , ¯ n b (1 , t )] − e i( ω a + ω b + δ ) t , and G ¯ n a , ¯ n b ( x, y ; t ) = (¯ n a x + 1)(¯ n b y + 1)+ (¯ n a + ¯ n b + 1)( | κ | /ϕ ) sinh ( ϕt );while its joint entropy assumes the closed-formE[ H th ; t ] = 2 + ln [ G ¯ n a , ¯ n b (1 , t )] . (37)In addition, the marginal Husimi functions H ( A )th ( α a ; t ) = [ G ¯ n a , ¯ n b (1 , t )] − e − [ G ¯ na, ¯ nb (1 , t ) ] − | α a | (38)and H ( B )th ( α b ; t ) = [ G ¯ n a , ¯ n b (0 , t )] − e − [ G ¯ na, ¯ nb (0 , t ) ] − | α b | (39)permit also to determine explicitly the partial entropiesE[ H ( A )th ; t ] = 1 + ln [ G ¯ n a , ¯ n b (1 , t )] , (40)E[ H ( B )th ; t ] = 1 + ln [ G ¯ n a , ¯ n b (0 , t )] . (41) In this moment becomes important noticing that the sub-additivity property is not violated for any t ≥
0, the equa-lity E[ H th ; 0] = E[ H ( A )th ; 0] + E[ H ( B )th ; 0] being consistentwith the factorization of H th ( α a , α b ; 0) in the product H ( A )th ( α a ; 0) H ( B )th ( α b ; 0).Following, let us investigate the correlation functionalC[ H th ; t ] for t >
0, since C[ H th ; 0] = 0 reflects the disen-tanglement between the bipartite mixed states. In fact,for t = 0 this functional assumes any values into the openinterval (0 , et al . [21] throughthe evaluation of the inverse negativity coefficient (thismeasure also leads us to estimate quantitatively the ‘de-gree of entanglement’ of a particular class of bipartitestates such as those studied in this section); furthermore,(ii) the dynamics here characterized by the physical sys-tem under investigation produces a maximum estimateof entanglement for any initially disentangled states, thisvalue (namely, 1) being considered as a specific quantumsignature of the parametric amplifier model. VI. CONCLUSIONS
We have established a set of interesting formal resultswithin the scope of quantum optics and quantum infor-mation theory that allows us, among other things, • to define a new class of unitary squeezing transfor-mations related to su (1 ,
1) Lie algebra which gene-ralizes, for certain particular representations of itsgenerators, the one- and two-mode squeezing ope-rators [3]; • to discuss, from the physical point of view, howthe generalized two-mode squeezing operator canbe generated through a slightly modified version ofthe parametric amplifier model [10]; • to obtain a general integral representation for thebipartite Wigner function whose integrand is ex-pressed as a product of two terms which are res-ponsible for the dynamical and kinematical entan-glement; and finally, • to estimate quantitatively the ‘degree of entangle-ment’ related to an ideal bipartite system (we arediscarding the unvoidable coupling with the envi-ronment in this context) by means of a theoreticalframework [11] which is based on the Wehrl’s ap-proach [12] for the entropy functionals.In fact, equation (29) and its respective properties men-tioned properly in the body of the text permit us to es-timate, through an entropic approach, the entanglementeffects for a wide class of electromagnetic field states, in-cluding Gaussian and non-Gaussian states. As a con-cluding remark, it is worth mentioning that these re-sults have also potential applications in modern researchon quantum teleportation [22], quantum tomography[4, 26], and quantum computation [27] (once continuous-variable entanglement can also be efficiently producedusing squeezed light and linear optics [28]), as well as onthe foundations of quantum mechanics through its exten- sions to the su (2) Lie algebra [29]. Acknowledgements
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Phys. Rev. A Phys. Rev. A Phys.Rev. A J. Phys. A: Math.Theor. T − ( t ) andconsequently, to establish the identities Y = T − ( t ) G and Z = T − ( t ) X .[31] According to Mollow and Glauber [16]: ‘The fact thatthe Wigner function has this property is a consequenceof the form taken by the Hamiltonian (12). It may beshown that whenever the Hamiltonian of a system of os-cillators is given by a quadratic form in the creation andannihilation operators, the Wigner function is constantalong classical trajectories. This property does not ex-tend to systems with arbitrary Hamiltonians, as it doesin the case of the classical phase-space distribution.’ Be-yond these fundamental features, it is worth mentioningthat recent studies on the characterization and quantifi-cation of entanglement for symmetric and asymmetricbipartite Gaussian states have contributed considerablyto our comprehension of this important nonclassical ef-fect in quantum optics and quantum information theory[20, 21]. In this sense, we believe that the formalism herepresented for the Wigner function can help to extendsuch results in order to include some important dyna- mical effects that allow us to better understand certainpeculiarities on the entanglement process for a specificsubset of bipartite physical systems described by conti-nuous variables [2].[32] It is worth noticing that the cases here studied repre-sent a specific set of disentangled bipartite states whosecharacteristic and Wigner functions are written as a pro-duct of two functions associated with each mode sepa-rately, i.e., since ρ (0) = ρ a (0) ⊗ ρ b (0) we promptlyobtain the relations C ( Y ; 0) = C a ( β a ; 0) C b ( β b ; 0) and W ( Z ; 0) = W a ( γ a ; 0) W b ( γ b ; 0). In this situation, the dy-namical entanglement originated from the Hamiltonian H ( t ) allows us to correlate the complex variables α a and α b (or ξ a and ξ b ) present in γ a ( b ) ( β a ( b ) ).[33] Recently, Mintert and ˙Zyczkowski [24] evaluated explici-tly the Wehrl and generalized R´enyi-Wehrl entropy func-tionals for any pure states describing N × N bipartitequantum systems. For this intent, they properly definedthe Husimi function for the su ( N ) × su ( N ) coherent-staterepresentations, and showed that: (i) the Wehrl entropyfunctional is minimal iff the pure states above mentionedare separable, (ii) the excess of this quantity is equal tothe subentropy of the mixed states obtained by the par-tial trace of the bipartite pure states; and finally, (iii)these functionals can be considered as alternative mea-sures of entanglement. Here, our intention is to establishan additional measure of entanglement (limited to the in-terval [0 , su (1 , × su (1 ,,