Partial Bell‐State Analysis with Parametric down Conversion in the Wigner Function Formalism

Abstract

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.

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

“…In both cases the non-null correlations correspond to different polarization components, the only difference being the minus sign that appears in |Ψ − . On the other hand, the case β = ∓π/2 and κ = π (β = −π/2, κ = 0) corresponds to the description of ∓|Φ + (|Φ − ), where the horizontal (vertical) component of a beam is correlated with the horizontal (vertical) component of the conjugated one [16].…”

“…In the Wigner formalism, by putting κ = β = 0 in Eqs. (13) to (16), we obtain the following four beams in order to compactly describe the four states |Ψ + ⊗ |ψ ± and |Ψ + ⊗ |φ ± :…”

“…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.…”

“…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.…”

Wigner’s inequalities in quantum field theory

“…In [19] it is shown that those inequalities may be used for a case of entangled photons for a quantum optical experiment. The contemporary review of this topic and the corresponding references may be found in [20]. However, that class of inequalities is not suitable for testing in HEP experiments.…”