Wigner representation for polarization-momentum hyperentanglement generated in parametric down-conversion, and its application to complete Bell-state measurement

Depósito de Investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an article published by Springer Nature The European Physical Journal D, Vol. 68 on November 2014, available at: https://doi.org/10.1140/epjd/e2014-50368-y © 2014 EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag Wigner representation for polarization-momentum hyperentanglement generated in parametric down-conversion, and its application to complete Bell-state measurement A. Casado1,S.Guerra 2,andJ.Pl´acido3. 1Departamento de F´ısica Aplicada III, Escuela T´ecnica Superior de Ingenier´ıa, Universidad de Sevilla, 41092 Sevilla, Spain. Electronic address: [email protected]s 2Centro Asociado de la Universidad Nacional de Educaci´on a Distancia de Las Palmas de Gran Canaria, 35004 Las Palmas de Gran Canaria, Spain. 3Grupo de Ingenier´ıa T´ermica e Instrumentaci´on, Universidad de Las Palmas de Gran Canaria, 35017 Las Palmas de Gran Canaria, Spain. PACS: 42.50.-p, 03.67.-a, 03.65.Sq, 03.67.Dd Abstract We apply the Wigner function formalism to the study of two-photon polarization-momentum hyperentanglement generated in parametric downconversion. It is shown that the consideration of a higher number of degrees of freedom is directly related to the extraction of additional uncorrelated sets of zeropoint modes at the source. We present a general expression for the description of the quantum correlations corresponding to the sixteen Bell base states, in terms of four beams whose amplitudes are correlated through the stochastic properties of the zeropoint field. A detailed analysis of the two experiments on complete Bell-state measurement included in [Walborn et al., Phys. Rev. A 68, 042313 (2003)] is made, emphasizing the role of the zeropoint field. Finally, we investigate the relationship between the zeropoint inputs at the source and the analysers, and the limits on optimal Bell-state measurement. Keywords: Entanglement, Bell-state analysis, parametric down-conversion, Wigner representation, zeropoint field. 1 1INTRODUCTION In the last two decades, parametric down-conversion (PDC) assumed an important role for the practical implementation of the quantum theory of information, such as quantum cryptography [1], quantum dense coding [2] and teleportation [3]. The use of PDC as a source of entanglement involves the necessity of performing a complete Bell-state measurement (BSM), which is required in many quantum communication schemes. In this context, entangled photon pairs produced in PDC have been used for experiments on a partial Bell-state measurement [4], in which entanglement involves only one degree of freedom, and a complete Bell-state measurement, in which hyperentanglement (entanglement between two or more degrees of freedom) takes part [5]. More recently, these states have been used as an essential part of cluster states [6] which are of great value in the field of quantum computing [7]. The use of enlarged Hilbert spaces opened the door for a complete BSM using linear optics and single photon detectors, first by considering that one of the degrees of freedom was in a fixed quantum state [8], and encoding the information in the other. Further studies about the actual limits for BSM in these enlarged spaces have shown that, for two photons hyperentangled in ndegrees of freedom, the number of mutually distinguishable sets of Bell states is bounded above by 2n+1 [9]. More recently, it has been shown that at most 2n+1 −1 classes out of 4nhyper-Bell states can be distinguished with one copy of the input state, and that complete distinguishability is possible with two copies, within the class of devices obeying linear evolution and local measurement (LELM) [10]. In case the two photons are not brought together at the LELM apparatus the maximun number of distinguishable classes is 2n. The Wigner representation of quantum optics provides an alternative to the standard Hilbert space formalism for the study of quantum information and for its practical implementation with PDC. In the Wigner representation within the Heisenberg picture (WRHP) the generation and propagation of PDC light is treated as in classical optics by taking into account the zeropoint field (ZPF) entering the crystal and the different optical devices placed between the source and the detectors. Finally, the vacuum fluctuations of the electromagnetic field are subtracted at the detectors [11, 12]. Hence, the peculiarities of the quantum world with respect to the image that classical physics offers are represented, in this context, by (i) the existence of 2 a stochastic zeropoint field whose amplitudes are distributed according to a positive Wigner function, and (ii) the way in which the signal is separated from the zeropoint background in the detection process. These two features give rise to the typical counterintuitive results within the quantum domain. In essence, manipulating entanglement is a common denominator in the different manifestations of quantum communication. In the WRHP, twophoton polarization entanglement is represented by two stochastic light beams, whose correlation properties arise from the coupling between two zeropoint beams (each containing two sets of uncorrelated zeropoint modes) and the laser beam at the crystal [12], following the classical Maxwell equations [13, 14]. On the other hand, entanglement manipulation involves the change in the correlation properties of the light beams when they are going through the different optical devices, and this is related to the way in which the vacuum modes are redistributed at the field amplitudes. The WRHP description of the four polarization Bell base states was made in [15] along with the application of the formalism to experiments on quantum cryptography. In [16] partial Bell-state analysis was studied, and the fermionic behaviour of two photons described by the singlet state, when they reach a balanced beam-splitter, was explained using purely wave-mechanical arguments based on the Wigner representation. More recently, the WRHP formalism has been applied to the description of entanglement swapping using PDC light, and it has been shown that the generation of mode entanglement between two initially non interacting photons is related to the quadruple correlation properties of the electromagnetic field, through the stochastic properties of the vacuum [17]. These works emphasised the role of the zeropoint field in the generation and propagation of light in experimental implementations of quantum communication, including the existence of a relevant ZPF noise entering the idle channels of the analysers. Concretely, according to [15] the effects of eavesdropping attacks in the case of projective measurements are directly related to the inclusion of some fundamental noise, that also turns out to be fundamental to reproduce the quantum results. In this way, we can state that the zeropoint field carries the quantum information which is extracted at the source, and also introduces some fundamental noise at the idle channels of the analyzers. These two features of the zeropoint field are a common denominator in optical experiments on quantum information. The standard Hilbert-space formulation of quantum optics considers vacuum fluctuations in an implicit way through the Heisenberg principle and 3 the use of normal ordering for the calculation of photodetection probabilities. In contrast, the Wigner function offers the possibility of stating specifically what the role of vacuum fluctuations is in the generation and measurement of quantum information in quantum optical information processing using PDC. In this way, the motivation for this paper and further works comes from the following questions: What is the relationship between enlarging the Hilbert space and the zeropoint field activated at the source? What is the role of the zeropoint at the different stages of a BSM experiment? What is the relationship between the zeropoint modes entering the source, the ones activated at the idle channels of the analyzers, and the maximal information that can be generated in each experiment? Thus, there is considerable motivation for the application of the WRHP approach to the description of hyperentanglement and its application to complete Bell-state analysis, in order to investigate the role of the zeropoint field in this area. The paper is organised as follows: In Section 2 we shall describe twophoton polarization-momentum hyperentanglement generated in PDC, by means of four correlated light beams. This description involves the consideration of eight uncorrelated sets of vacuum modes, distributed in four ZPF entering beams at the source. We shall obtain a compact expression for the description of the sixteen Bell base states in the WRHP, in terms of four two-by-two correlated beams. In Section 3 we shall apply this formalism to study the experiments proposed in reference [8] in which one of the degrees of freedom is in a fixed quantum state. We shall put the emphasis on the role of the zeropoint field at the different steps of each experiment, in order to make clear its relevant contribution to the signal fields arriving at the detectors. This analysis is important, not only for the theoretical aspects concerning optical experiments on quantum information, but also for its relation with optical tests of Bell’s inequalities, in which the vacuum field entering the idle channels of the analysers gives rise to enhancement [18, 19]. In Section 4 we shall demonstrate that the number of independent sets of zeropoint modes entering the source represents an upper bound to the maximum number of Bell states that can be distinguished in hyperentanglement-assisted Bell-state analysis. Also, we shall establish the relationship between the ZPF inputs at the source and the analysers, and the maximum number of distinguishable Bell-state classes, in LELM apparatus in which the left and right input channels are not brought together. Finally, in Section 5 we shall present the main conclusions of this work, and sketch further steps for future research. In order to a better understanding of the WRHP approach in this paper, we 4 have included some fundamental ideas in Appendix A. 2 POLARIZATION-MOMENTUM HYPERENTANGLEMENT IN THE WRHP Let us start by considering the following situation: a type-I two-crystal source is pumped by a laser beam. The first (second) crystal emits pairs of horizontal (vertical) polarized photons in superimposed emission cones. Because the photons are emitted on opposite sides of the cone, two sets of conjugated beams, (a1,b 2)and(a2,b 1), which are represented by wave vectors kai,kbi (i=1,2), can be selected [20]. If the coherence volume of the laser contains the two-crystal interaction region, the quantum state corresponding to a photon pair is usually expressed, in a particle-like description, as: |Φ+⊗|ψ+=1 √2[|H1|H2+|V1|V2]⊗1 √2[|a1|b2+|b1|a2].(1) The state given by (1) is one of the sixteen base states corresponding to the two-photon hyperentanglement on polarization and momentum degrees of freedom, of the form |Π⊗|η,where|Π(|η) is the four-dimensional vector representing one of the polarization (momentum) Bell base states [8]: Ψ±=1 √2[|H1|V2±|V1|H2];Φ±=1 √2[|H1|H2±|V1|V2], (2) ψ±=1 √2[|a1|b2±|b1|a2];φ±=1 √2[|a1|a2±|b1|b2].(3) The study of polarization-momentum hyperentanglement in the WRHP is based on the same ideas that were developed in references [11] and [12]. The key point in this case is that the selection of two sets of correlated beams, (a1,b 2)and(a2,b 1), implies the consideration of eight sets of vacuum modes which are “activated” at the crystal via the coupling with the laser beam (see fig.1). The set of representative modes, corresponding to the entering zeropoint beam of wave vector kxi, is represented by the vacuum amplitudes: 5 {αxi,λ}≡αk,λ;k∈[k]xi;x=a, b ;λ=H, V ;i=1,2.(4) Laser FOUR CORRELATED BEAM S Fa2(+) Fa1(+) FOUR ZPF BEA MS TWO-CRYSTAL SOURCE (TYPE-I PDC) Fb1(+) Fb2 (+) Figure 1: Sets of vacuum modes (on the left), which are “activated” and coupled with the laser inside the crystal. The correlation properties of the four beams (on the right) are related to the way in which the vacuum amplitudes are distributed in the field amplitudes. Hence, each of the sixteen Bell base states is characterized, in the WRHP approach, by a “particular” set of correlations. In order to focus on the main points we shall first describe the generation of the beams concerning one of the sixteen states, the one corresponding to Eq. (1). The Hamiltonian corresponding to the electromagnetic field can be expressed in the following way: H=Hfree +Hint = λ=H,V  k ¯hωk,λα∗ k,λαk,λ + ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ i¯hgV 2 2  i, j =1 i=j  λ=H,V  k∈[k]ai k∈[k]bj f(k,k)exp(−iωpt)α∗ k,λα∗ k,λ +c.c. ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (5) 6 where the crucial difference between (5) and Eq. (1) of [12] is that we have made the change V→V/2, in order to consider that the energy of the classical wave corresponding to the laser (with frequency ωpand momentum kp), which is proportional to the squared amplitude, must be divided into four beams. On the other hand, f(k,k)isafunctionwhichisdifferentfrom zero only when the momentum matching condition is fulfilled, and gis a constant related to the coupling parameter. The evolution equation for αk,λ is given by the Hamilton (canonical) equations, taking √¯hαk,λ as coordinates and √¯hα∗ k,λ as canonical momenta. We have: ˙αk,λ =−iωk,λαk,λ +gV 2 k f(k,k)exp(−iωpt)α∗ k,λ,(6) where {k,λ}represents any mode belonging to (4). The integration is performed to second order in the coupling constant (g=gΔt), from t=−Δt to t=0,Δtbeing the interaction time inside the two-crystal source. For t>0 there is a free evolution. By substituting αk,H (t)andαk,V (t) in Eq. (A.1) we obtain the following four correlated beams leaving the crystal (for more details see [11, 12]): F(+) a1(r,t)=F(+) p(r,t;{αa1,H ;α∗ b2,H})ia1+F(+) s(r,t;{αa1,V ;α∗ b2,V })ja1,(7) F(+) b2(r,t)=F(+) q(r,t;{αb2,H ;α∗ a1,H})ib2+F(+) r(r,t;{αb2,V ;α∗ a1,V })jb2,(8) F(+) b1(r,t)=F(+) p(r,t;{αb1,H ;α∗ a2,H})ib1+F(+) s(r,t;{αb1,V ;α∗ a2,V })jb1,(9) F(+) a2(r,t)=F(+) q(r,t;{αa2,H;α∗ b1,H})ia2+F(+) r(r,t;{αa2,V ;α∗ b1,V })ja2,(10) where each polarization component is a linear transformation of the ZPF entering the nonlinear medium. We have included the sets of relevant zeropoint modes, for a better understanding of the correlation properties. For instance, 7 F(+) pis only correlated to F(+) qbecause {αa1,H}({α∗ b2,H}) is correlated to {α∗ a1,H}({αb2,H}), as it can be seen from Eq. (A.4). The same reasoning applies to the correlations F(+) r↔F(+) s,F(+) p↔F(+) q,andF(+) r↔F(+) s. Hence, the beam F(+) a1(F(+) b1) is correlated to F(+) b2(F(+) a2), but there is no correlation between the two beams corresponding to each photon. Taking r=0at the center of the source, we have the cross-correlation: F(+) p(0,t)F(+) q(0,t )=gV 2ν(t−t),(11) where ν(t−t) vanishes when |t−t|is greater than the correlation time between the beams F(+) a1and F(+) b2[13]. Similar expressions hold for F(+) r(0,t)F(+) s(0,t ), F(+) r(0,t)F(+) s(0,t ),andF(+) p(0,t)F(+) q(0,t ). On the other hand, taking for instance the polarization amplitude F(+) p at a point rand times tand t, we have the autocorrelation: F(+) p(r,t)F(−) p(r,t )−[F(+) ZPF,a1(r,t)·ia1][F(−) ZPF,a1(r,t )·ia1] =g2|V|2 4μ(t−t),(12) where F(+) ZPF,a1is the zeropoint beam corresponding to mode a1,andμ(t−t) is a correlation function which goes to zero when |t−t|is greater than the coherence time of PDC light. Similar expressions hold for F(+) s,F(+) q,F(+) r, and for the corresponding primed amplitudes. 2.1 The sixteen hyper-Bell states in the WRHP The four beams given in Eqs. (7) to (10) are correlated through the ZPF entering the two-crystal source, which is “amplified” via the activation of the eight sets of vacuum modes {kxi,λ (x=a, b;i=1,2; λ=H, V )}.The beams F(+) a1and F(+) b1can be locally manipulated, allowing for the possibility of distributing the vacuum amplitudes in sixteen different ways, each corresponding to the generation of a concrete Bell base state. Hence, the possibility of performing superdense coding is explained in the WRHP framework through the change of the correlation properties of the light beams (represented by the four nonvanishing correlations p↔qand r↔s) when the 8 1 2 BS BS HWP Mx My ZPFX1 X1H A2H X1V A2V Y1V B2V Y1H B2H ZPFY1 ZPFA2 ZPFB2 a2 x1 y1 b2 Figure 3: Hyperentangled-Bell-state analyzer using a fixed entangled state in polarization. We have represented two optical devices, Mxand My,in order to account for the two possibilities, depending on the value of xand y. The zeropoint field at the idle channels of the analyzers is represented by four beams. 15 F(+) x1=[F(+) six1+F(+) pjx1]eiϕ1,(24) F(+) y1=[F(+) siy1+F(+) pjy1]eiϕ2,(25) F(+) a2=F(+) qia2+F(+) rja2(26) F(+) b2=F(+) qib2+F(+) rjb2.(27) Now, we define the matrices ˆ Mxand ˆ My:inthecasex=a,y=b,ˆ Mx=ˆ I (identity matrix) and ˆ My=ˆ MHWP; on the other hand, if x=b,y=a,then ˆ Mx=ˆ MHWP and ˆ My=ˆ I. In this way, the beams entering the BSs are F(+) x1=ˆ MxF(+) x1;F(+) y1=ˆ MyF(+) y1,(28) F(+) a2=F(+) a2;F(+) b2=ˆ MHWP F(+) b2=F(+) rib2+F(+) qjb2.(29) From Eqs. (28) and (29) it can be easily seen that, for x=a,y=b(x=b, y=a), the only non-null cross-correlations are those concerning the same polarization (orthogonal polarizations). For this reason, the CNOT operation marks the polarization state. This operation does not introduce additional zeropoint fluctuations, because the HWPs do not activate zeropoint modes. Finally, the BSs transform the beams given in Eqs. (28) and (29) without the consideration of additional zeropoint modes, because there is no idle channel at the BSs. Finally, if we consider the zeropoint amplitudes at the idle channels of the PBSs (see Fig. 3), the field amplitudes at the detectors X1H,X1V,Y1H and Y1Vcan be expressed by the following compact expression: F(+) Z1λ=inλ √2im1(Z)[ˆ MxF(+) x1]+im2(Z)[ˆ MyF(+) y1]·uλ+F(+) vac,Z1λ,(30) 16 where: Z=X,Y ,λ=H, V ;uH≡i,uV≡j;nH=1,nV=0;m1(X)=1, m1(Y)=0;m2(X)=0,m2(Y)=1. In the same way, we have the following expression for the field amplitudes at the detectors A2H,A2V,B2Hand B2V: F(+) V2λ=inλ √2im1(V)F(+) a2+im2(V)[ˆ MHWP F(+) b2]·uλ+F(+) vac,V2λ,(31) where: V=A, B,λ=H, V ;m1(A)=1,m1(B)=0;m2(A)=0,m2(B)= 1. Now, using Eqs. (11) and (A.7), and taking into account that the ZPF inputs at the PBSs are uncorrelated with the signals and with each other, we shall consider the following cases: •Case I (x=a,y=b) corresponds to the states |Ψ+⊗|ψ±.After some easy algebra, we obtain: PA1H,A2H kA1HkA2H =PB1H,B2H kB1HkB2H =PA1V,A2V kA1VkA2V =PB1V,B2V kB1VkB2V =g2|V|2|ν(0)|2 16 |eiϕ1+eiϕ2|2, (32) and PA1H,B2H kA1HkB2H =PB1H,A2H kB1HkA2H =PA1V,B2V kA1VkB2V =PB1V,A2V kB1VkA2V =g2|V|2|ν(0)|2 16 |eiϕ1−eiϕ2|2. (33) From Eqs. (32) and (33) it can be seen that, in the case ϕ1=ϕ2=0 (ϕ1=0,ϕ2=π), which corresponds to the momentum state |ψ+ (|ψ−), only the four probabilities in Eq. (32) (Eq. (33)) are nonzero. •Case II (x=b,y=a) corresponds to the states |Ψ+⊗|φ±.Inthis case, we obtain: PA1H,A2V kA1HkA2V =PA1V,A2H kA1VkA2H =PB1H,B2V kB1HkB2V =PB1V,B2H kB1VkB2H =g2|V|2|ν(0)|2 16 |eiϕ1−eiϕ2|2. (34) 17 The above probabilities are non-null only in the case ϕ1=π,ϕ2=0 (or viceversa), i.e. the WRHP description of the momentum state |φ−. On the other hand: PA1H,B2V kA1HkB2V =PA1V,B2H kA1VkB2H =PB1H,A2V kB1HkA2V =PB1V,A2H kB1VkA2H =g2|V|2|ν(0)|2 16 |eiϕ1+eiϕ2|2. (35) From Eq. (35), the corresponding probabilities are non-null only in the case ϕ1=ϕ2= 0, i.e. the situation corresponding to the momentum state |φ+. There is a discrepancy with Walborn’s Table II in ref. [8], with respect to the states |φ−and |φ+, which is due to the consideration of the complex factor ifor the reflected amplitudes at the BSs [21]. Then, the corresponding joint probabilities for these two states are exchanged with respect to the work of Walborn et al. 4 ZPF AND THE LIMITS ON OPTIMAL BSM Let us consider the following situation: two photons entangled in ndichotomic degrees of freedom enter a LELM apparatus via separate spatial channels, designated Land R. Each photon contains 2nmodes, and a unitary matrix transforms the 2n+1 input modes to 2n+1 output modes to the detectors. In [10] it is demonstrated that: (i) A single detector click cannot discriminate between any of the Bell states, so that the 2n+1 possibilities for the second detector event form a simple upper bound on distinguishable Bell-state classes using LELM devices; (ii) There can be at most 2n+1 −1 distinguishable classes of hyper-Bell states for two bosons. This number is reduced to 2nin the case that the left and right channels are not brought together in the apparatus. The experiments described in Sec. 3 are just an example of this situation in the case of photons, for n= 2. This section is divided in two parts: (1) First, we shall investigate the relationship between the statement (i) and the zeropoint field inputs when optical experiments using parametric down conversion are considered; (2) We shall study the 18 relationship between the number of vacuum inputs at the source and the analysers, and the maximum number of distinguishable classes, 2n,inthe kind of experiments as the ones described in Sec. 3. 1. Given an optical n-qubit state, the maximum number of mutually distinguishable sets of Bell states is bounded above by 2n+1 [9, 10]. The demonstration of this point in the Hilbert space is based on a particlelike description, which contrasts to the image in the WRHP formalism. For instance, in [16] it was stressed that two-photon entanglement in only one degree of freedom implies the consideration of four independent sets of zeropoint modes at the source, which are “activated” through a coupling with the laser inside the crystal. In this paper, the generation of polarization-momentum hyperentanglement is represented via the consideration of eight sets of independent vacuum modes, which are amplified at the two-crystal source. Hence, hyperentanglement, i.e. entanglement in Hilbert spaces of higher dimensions is closely related to the inclusion of more sets of vacuum modes entering the source. With an increasing number of vacuum inputs, the possibility of extracting more information from the zeropoint field also increases. As we shall demonstrate below, for a given n, the maximal distinguishability in a Bell-like experiment is bounded by the number of independent vacuum sets of modes which are extracted at the source, being this number equal to 2n+1. In order to prove this statement, we shall consider the following lemmas: Lemma I: For a two-photon n-qubit state generated via PDC, the number of independent sets of zeropoint modes which are necessary for the generation of entanglement, is just 2n+1. Proof: By considering the situation described in this paper, in which n= 2, the interaction Hamiltonian given in Eq. (5) gives rise to linear evolution equations for the amplitudes αk,λ [see Eq. (6)]. Hence, the 2n+1 sets of field amplitudes outgoing the crystal are generated from an identical number of independent sets of ZPF amplitudes entering the nonlinear source, which are “amplified” via the coupling with the laser beam. This result is also true for n= 1, as it has been demonstrated elsewhere [12, 16]. For n>2, the interaction Hamiltonian is also quadratic, which implies that the evolution equations for the vacuum amplitudes are linear. Hence, this result is valid for any n≥1. 19 Lemma II: The propagation of the 2n+1 sets of field amplitudes outgoing the crystal through a LELM device gives rise to 2n+1 output amplitudes to the detectors. Each of them will include, at least, the 2n+1 sets of input ZPF amplitudes at the source. Proof: Given the fact that linear mode transformations lead to Bogoliubov transformations of the mode operators, which are generated via quadratic Hamiltonians [22], the total set of unitary transformations, including the generation of PDC light at the crystal and the action of linear optical devices between the nonlinear source and the detectors, are represented by quadratic Hamiltonians in mode operators, which give rise to linear equations (see Fig. 4). When passing to the Wigner representation, in which the destruction (creation) operator ˆak,λ (ˆa† k,λ) is substituted by a complex amplitude αk,λ (α∗ k,λ), each of the 2n+1 output field amplitudes at the detectors will include, generally, the 2n+1 sets of field amplitudes outgoing the crystal, and so the 2n+1 input ZPF amplitudes at the nonlinear source (see Lemma I). Lemma III: The 2n+1 possibilities for the second detection event, which constitutes a simple upper bound on distinguishable Bell-state classes from LELM devices, is just the number of the input ZPF sets of modes at the source. Proof: This result follows from lemmas Iand II. Hence, the upper bound of 2n+1, which limits the optimality of any Bell-state analysis, coincides with the number of relevant ZPF modes at the source. This key point represents the relationship between the information that can be effectively measured in an experiment and the ZPF inputs at the source. In other words, the number of relevant sets of zeropoint modes at the source represents a limit on optimal Bell-state analysis in the enlarged Hilbert space. From [10], it is well known that a single detector event cannot discriminate any of the Bell base states. Let us consider that a PBS is placed at each of the beams (13) to (16) in order to measure the single and joint detection probabilities, following a setup similar to Fig. 1 of Ref. [15], and taking into account the vacuum inputs at the idle channels of the PBSs. Using the autocorrelation properties of the light field given in Eq. (12), the expression (A.5) for the single detection probability, and the field amplitudes at the detectors, it can be easily demonstrated 20 that the single detection probabilities are identical, and independent of the parameters β,κ,(x, y)and(ϕ1,ϕ 2). The same reasoning holds for the experiments developed in Sec.3, by using the field amplitudes at the detectors given in Eqs. (21) and (22) [(30) and (31)] for the first (second) experiment. SOURCE (PDC) ZPF INPUT MODES OUTGOING MODES TOWARDS THE DETECTORS ZPF NOISE AT THE DETECTORS DETECTION AMPLIFIED MODES LINEAR EVOLUTION LINEAR EVOLUTION OF VACUUM MODES LASER BEAM Figure 4: The ZPF inputs at the non linear source are amplified and propagate through linear devices. The total set of transformations are represented by a quadratic Hamiltonian which gives rise to linear evolution equations. The total number of modes outgoing towards the detectors is equal to the entering ZPF modes at the crystal, which are amplified via the coupling with the laser beam. We have also represented the ZPF inputs at the detectors. 2. The noise entering the idle channels of the analyzers limits the optimality of the Bell-state analysis, and this idea is worthy of consideration. In Section 3 we have applied the WRHP formalism to complete BSM, in the case where one of the degrees of freedom is in a fixed (ancillary) state, and the information is encoded in the other degree of freedom. These experiments correspond to a general class of setups in which the two photons are not mixed in the apparatus, so that the maximun number of distinguishable classes of Bell states, is just 2n[10]. From the point of view of the WRHP approach, this is exactly the number of non-vanishing cross-correlations between the field amplitude at each detector on the left (right) side, and the whole set of amplitudes at the detectors on the right (left) side, in the optimal situation of maximal 21 distinguishability. On the other hand, each cross-correlation property of the field of the kind of Eq. (11) is related to the probability of a joint detection, in which the subtraction of the zeropoint intensity at each detector is relevant (A.6). In order to be able to measure such a correlation, the beam has to be divided, being necessary a zeropoint contribution through the idle channel of the PBS, in order to preserve the commutation relations. This zeropoint beam introduces two sets of vacuum modes, one of them corresponding to vertical polarization and the other to the horizontal one, which are uncorrelated with the signal entering the other channel. For instance, if n=1(n= 2), i.e. the photon pair is described by two (four) correlated beams, and two (four) correlations are generated by the source between the “amplified” zeropoint fluctuations, almost two (four) entry points of noise are necessary for measuring such correlations, so that four (eight) sets of zeropoint amplitudes enter the analyser. In the general case of ndegrees of freedom, the total number of entry points of noise at the analysers is just 2n,sothat2 n+1 sets of vacuum amplitudes must be taken into account at the Bell state analyser. The net effect of the vacuum inputs at the BSM station is to decrease the optimality of the Bell-state analysis. Following [10], given a detection at, for instance, the left side, the 2npossible joint detections at the right side, in the case in which the photons are not mixed at the LELM device, gives the maximaum number of sets of Bell states that can be distinghished in these experiments. However, following the WRHP formalism, the upper bound, 2n+1,isgivenbythenumberof ZPF sets of modes which are amplified at the source. Which quantity must be subtracted from 2n+1 in order to obtain 2n? The answer is just 2n, which constitutes the number of zeropoint sets of vacuum modes on the right side (the area corresponding to the second detection), and also the total number of entry points of noise at the analyser. For instance, in Walborn’s experiments we can determine the number of Bell states classes that can be distinguished by subtracting the number of channels with noise from the number of ZPF entry modes, so that in this case we have 8 −4 = 4. Due of the fact that one of the degrees of freedom is in a fixed state, this number will coincide with the number of the Bell base states corresponding to the other degree of freedom. 22 Hence, for a given n,ifNZPF,S =2 n+1 is the number of sets of zeropoint modes at the source, NZPF,x =2 n,x=L, R,isthenumberofsets of vacuum modes entering the idle channels of the analysers at the left or right area, and Nic =2 nis the number of idle channels (entry points of noise) at the analyser, the maximum number of mutually distinguishable classes of Bell states, Nmax,class, will be given by: Nmax,class =NZPF,S −NZPF,x =NZPF,S −Nic.(36) 5CONCLUSIONS The zeropoint field at the optical experiments on quantum information is not merely a mathematical tool which gives rise, after being subtracted, to a broad class of theoretical results. From our point of view, the vacuum field has a “visible” presence in these experiments, and this is what we are trying to demonstrate in this paper. Using the WRHP approach we have analysed polarization-momentum hyperentanglement in detail, showing the close relationship between enlarging Hilbert spaces and ZPF inputs at the source, so that the possibility for extracting more information, when an enlarged Hilbert space is used, is due to the consideration of a greater number of ZPF inputs which are amplified at the source. Eqs. (13) to (16) give the WRHP description of the sixteen Bell-like states in terms of four two-by-two correlated beams. Each of the states is described by giving the value of four parameters: two of them (βand κ) indicate the polarization Bell-state, and the other two, the dichotomic couples (x,y)and(ϕ1,ϕ2), are related to the momentum. Let us emphasize that these parameters can be locally controlled because all of them appear in Eqs. (13) and (14), corresponding to photon 1. Hence dense coding [8] and superdense coding [9] are justified, in the context of WRHP, by the possibility of changing the correlation properties of the four beams through the action of a linear optical device which operates in the same way as in classical optics. This contrasts with the usual description in the Hilbert space, in which local operations are represented by unitary operators. In the Wigner framework, the effect of a linear optical device on a beam accounts for a change in the distribution of the zeropoint amplitudes inside the field components, so that there is a change in the correlation properties. Given that these operations 23 do not introduce additional zeropoint noise, the information encoded in the zeropoint amplitudes entering the source can be manipulated in order to succesfully complete a quantum dense coding protocol. We have analysed two experimental setups for complete BSM, each using a fixed state in one of the two degrees of freedom, so that the information is encoded in the other, as it appears in reference [8]. As we have already pointed out, once within the Wigner framework, the typical quantum results appear precisely as a consequence of the role of the zeropoint field in the production, propagation and detection of light. Quantum correlations can then be explained solely in terms of the propagation of those vacuum amplitudes through the experimental setup, and their subsequent subtraction at the detectors. Hence, the Wigner formalism allows for an interpretation of these experiments in terms of waves, where photons are just wave-packets carrying the zeropoint amplitudes through the experimental setup, and finally detected. We have explained how the zeropoint inputs contribute to distinguishability in a Bell-state analysis in which both down-converted photons are not brought together at the LELM device. In this situation, the difference between the amplified zeropoint modes at the source (2n+1) and the ZPF inputs at the Bell-state analyser (2n), gives the maximal distinguishability of Bellstate classes (2n). The influence of the zeropoint inputs in a general strategy of Bell-state measurement, in which both photons are brought together at the apparatus [23], will be the aim of further research. The main conclusion of this work is that the WRHP approach allows for the possibility of obtaining additional information to the one provided by the standard Hilbert space formalism, just by considering the role of the zeropoint field at the different steps of an optical quantum communication experiment using PDC. Likewise, there is a close relationship between the zeropoint extracted at the source, the corresponding zeropoint field entering the vacuum channels of the analyzers, and the maximal information that can be extracted in a concrete experiment, as we have discussed in Sec.4. This idea will be developed in further works. 6 ACKNOWLEDGEMENTS The authors would like to thank Prof. E. Santos for revising the manuscript, and for helpful suggestions and comments on the work. A. Casado acknowl- 24