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We apply the Wigner function formalism to partial Bell-state analysis using polarization entanglement produced in parametric down conversion. Two-photon statistics at a beam-splitter are reproduced by a wave-like description with zeropoint fluctuations of the electromagnetic field. In particular, the fermionic behaviour of two photons in the singlet state is explained from the invariance on the correlation properties of two light beams going through a balanced beam-splitter. Moreover, we show that a Bell-state measurement introduces some fundamental noise at the idle channels of the analyzers. As a consequence, the consideration of more independent sets of vacuum modes entering the crystal appears as a need for a complete Bell-state analysis.

The theory of parametric down conversion (PDC) in the Wigner formalism, along with the theory of detection, was treated in a series of papers [

The development of quantum information in the recent years, alongside with the important role of parametric down conversion for experimental schemes, has motivated the application of the Wigner approach to some relevant contexts, up to now almost exclusively linked to the Hilbert domain. Examples of these are quantum cryptography [

Recently, the Wigner formalism has been applied to experiments on quantum cryptography based on the Ekert’s protocol [

Bell-state measurements constitute another key aspect in the field of quantum information, posing a relevant problem in quantum dense coding and teleportation schemes. In this context, entangled photon pairs produced in parametric down conversion have also been used in the last decades for experiments on partial Bell-state measurement [

The paper is organized as follows. In Section

We will start this section by reviewing the basic concepts in the Hilbert framework [

Let us now go to the Wigner formalism. The Wigner transformation stablishes a correspondence between a field operator acting on a vector in the Hilbert space and a (complex) amplitude of the field. In the case of zeropoint field, these amplitudes follow a particular stochastic distribution, given by the Wigner function of the vacuum.

Quantum predictions corresponding to the state

As said before, amplitudes

In expressions (

The four Bell-states can be generated by manipulating only one beam, and this is related to the possibility of sending two bits of classical information via the manipulation of only one particle [

Let us now focus on the experimental setup in Figure

Setup for quantum dense coding. The polarization rotator and the wave retarder used by Bob allow a transmission to Alice of any of the Bell-states. Alice’s station consists on a balanced beam-splitter, two polaryzing beam-splitters, and detectors DH1, DV1, DH2, and DV2.

Now, let the wave retarder introduce a phase shift

The combination

Finally, the case

This description of the four Bell-states is equivalent to the one in [

The general expressions (

We will now study the action of a balanced beam-splitter on the correlation properties of the light beams. For the sake of clarity, we suppose an identical distance separating the source from the BS’s, so the contribution of the phase shift in [

Because there is one beam at each input port, it is not necessary to consider the vacuum field at the beam-splitter [

For the light beams at the outgoing channels we have

We now calculate the cross-correlations between the components of

Let us again consider the situation in Figure

The two beams are recombined at Alice’s Bell-state analyser by means of a balanced beam-splitter (BS), and the horizontal and vertical polarization components of each outgoing beams are separated at the polarizing beam-splitters PBS1 and PBS2. Finally, all coincidence detection probabilities can be measured with the detectors DH1, DH2, DV1, and DV2.

We focus now on the fields at the detectors. We will suppose that there is the same distance between the BS and any of the detectors, and again the phase factor corresponding to the propagation of the different amplitudes is irrelevant. Owing to the fact that each polaryzing beam-splitter reflects (transmits) the horizontal (vertical) polarization, the electric field at the detector (DH1, DV1, DH2, DV2) is the superposition of (

To calculate joint detection probabilities we use (see [

For instance, let us show the calculation of

The rest of the probabilities can be obtained similarly. By using (

From (

recording a coincidence of DH1 and DH2 (DV1 and DV2) is not possible, for whatever

recording a coincidence of DH1 and DV2 (DV1 and DH2) is possible only in the case

recording a coincidence of DH1 and DV1 (DH2 and DV2) is possible only for

when

We have applied the Wigner approach to study two-photon statistics at a balanced beam splitter. We have also treated an experimental setup for partial Bell-state analysis. The Wigner formalism allows for an interpretation of these experiments in terms of waves, but, however, the whole formalism lies inside the quantum domain, the zeropoint field being an alternative to the role of vacuum fluctuations in the Hilbert space.

As we already pointed out, once in the Wigner framework, the typical quantum results appear precisely as a consequence of the introduction of the zeropoint field. This vacuum field enters in the crystal and also in the rest of optical devices. Finally, it is substracted in the detection process. Quantum correlations can be then explained solely in terms of the propagation of those vacuum amplitudes through the experimental setup, and their subsequent substraction at the detectors.

At the beginning of Section

For the Wigner representation, the corpuscular aspect of light appears as just an interplay of (Maxwell) waves, including a zeropoint vacuum field. The action of the beam-splitter must be treated as in the classical framework: a part of each entering beam is transmitted, and the other part is reflected, without any change in their polarization properties. From (

At this point it is worth to stop again at the question: how can we explain a typical particle behaviour (the bosonic nature of photons in the Hilbert space description), from a wave-like description, just with the inclusion of vacumm fluctuations of the electromagnetic field? By inspection of (

The Bell-state measurement performed at Alice’s station only identifies the states

Recently, the problem of performing complete Bell-state measurements has been solved by considering a higher number of degrees of freedom (hyperentanglement). Hyperentanglement is also a convenient resource for some other recent and important applications in the field of quantum computing. An example of these is one-way quantum computation using clusters states [

The authors thank Professor E. Santos for helpful suggestions and comments on the work. They also thank R. Risco and D. Rodríguez for their comments and careful reading of the manuscript. A. Casado acknowledge the support from the Spanish MCI Project no. FIS2008-05596.