In this paper, the theory of parametric down-conversion in the Wigner representation is applied to Ekert's quantum cryptography protocol. We analyse the relation between two-photon entanglement and (non-secure) quantum key distribution within the Wigner framework in the Heisenberg picture. Experiments using two-qubit polarization entanglement generated in nonlinear crystals are analysed in this formalism, along with the effects of eavesdropping attacks in the case of projective measurements.

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.

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 wavelike 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 Bellstate analysis.

TheWigner formalism in the Heisenberg picture constitutes a bridge that connects QuantumOptics to Stochastic Optics. The vacuum field appears explicitly in the formalism, and the wavelikeaspects of light are emphasised. In addition, the zeropoint intensity as a threshold for detection is acommon denominator in both theories. In this paper, after summarising the basic rules of the Wignerapproach and its application to parametric down-conversion, some new results are presented thatdelve into the physical meaning of the zeropoint field in optical quantum communication. Specifically,the relationship between Bell-state distinguishability and the number of sets of zeropoint modesthat take part in the experiment is analysed in terms of the coupling between the phases of thedifferent fields involved and the subtraction of the zeropoint intensity at the detectors. Additionally,the connection between the compatibility theorem in quantum cryptography and zeropoint fieldis stressed.

AcknowledgementsThe authors would like to thank Prof. E. Santos for revising the manuscript, and for helpful suggestions and comments on the work. We are grateful for the insights gained in conversations with R. Risco-Delgado. AbstractThe Wigner representation of parametric down conversion in the Heisenberg picture is applied to the study of the Rome teleportation experiment. We investigate the physical meaning of the zeropoint inputs at the different areas of the experimental setup. In particular, we establish a quantitative relationship between the zeropoint sets of modes that are needed for the preparation of the quantum state to be teleported, the idle channels inside the one-photon polarization-momentum Bell-state analyser, and the possibility of performing teleportation of a polarization state whith certainty. arXiv:1707.09624v1 [quant-ph] 30 Jul 2017 degrees of freedom (polarization or momentum) of Alice's photon [15]. The other photon of the down converted pair is sent to Bob. The advantage of this teleportation scheme is that a complete BSM of one-photon polarizationmomentum Bell-states is possible. Nevertheless, the input state cannot be supplied by an external system, and this presents some limitations, such as the inability to teleport entangled or mixed states [2]. The result obtained by Alice is communicated to Bob, who uses this information to apply one out of four unitary transformations, in order to reproduce the original state. This experiment was proposed by Popescu [20] and performed in Rome [15] by using two-photon momentum entanglement, and polarization coding for preparing the qubit to be teleported. The Innsbruck and Rome experiments omitted the final stage of teleportation, the unitary transformations applied by Bob after the classical communication in order to reconstruct the unknown state. This was accomplished using nuclear magnetic resonance [21], and later the first long-distance optical quantum teleportation experiment with active feed-forward in real time was accomplished [22].The Wigner representation of quantum optics in the Heisenberg picture (WRHP) has been applied in recent years to the study of experiments on quantum communication using photons generated via parametric down conversion. The WRHP formalism of PDC resembles nonlinear classical optics, by taking into consideration the vacuum inputs at the nonlinear crystal and the different linear optical devices placed between the source of down converted photons and the detectors. The zeropoint field (ZPF) appears as a stochastic field that couples with the laser beam into the crystal, giving rise to the down converted beams [23,24]. In this way, the signals emitted by the crystal constitute needles of radiation generated by the amplification of vacuum fluctuations [25]. These signals propagate according to the classical Maxwell equations, so that the WRHP formalism of PDC emphasizes the wave-like aspects in the generation and propagation of light [26]. The Wigner function of PDC is positive, and corresponds to the Gaussian Wi...