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

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.

Rome teleportation experiment analysed in the Wigner representation: the role of the zeropoint fluctuations in complete one-photon polarization-momentum Bell-state analysis

“…The transfer from polarization entanglement to momentum entanglement supposes an increase of the activated ZPF sets of modes, from four to eight. In consequence, the Preparer can store information into the vacuum amplitudes F The ZPF inputs corresponding to the idle channels inside Alice's station contribute to the extraction of the two bits of classical information through the measurement process [29], in which the zeropoint intensity is subtracted at the detectors. These entry point of noise reduce the maximal classical information that can be extracted, in relation to the number of amplified sets of ZPF modes entering the analyser.…”

“…Following the standard particle-like Hilbert space approach, the application of the projection postulate implies that the quantum information that remains in the electromagnetic field after the BSM is stored at Bob's station into a given one-photon polarization state, depending on the result obtained at Alice's station (see Equation (B.8)). In contrast, the autocorrelations of the light field corresponding to F (+) Bob,signal do not give any information about the preparation (see Equations (29) and (31)). Hence, the properties concerning Bob's photon after the BSM are supported by the cross-correlations associated to the field (see Equations (32) to (35)): the behaviour of Bob's photon is inherently linked to Alice's photon, so that there is no disconnection between them mediated through the collapse of the state vector at Alice's station.…”

“…He uses two optical devices C 1 and C 2 , whose action on the Wigner field amplitudes is treated in the same way as for classical optics. For instance, we shall consider the application of a linear transformationM to the beam given in Equation (29). We have:…”

“…In order to quantitatively demonstrate the relationship between the zeropoint inputs at the experimental setup and the optimality of the Bell-state analysis, we shall focus on Equation (36) of reference [29]. This Equation relates the maximun number of distinguishable Bell-state classes of two photons, entangled in n dichotomic degrees of freedom, in experiments in which the two photons are not mixed at the device, with the number of ZPF inputs at the source of entanglement and inside the analyser.…”

“…In contrast, in the Rome teleportation experiment only two entangled photons are used, and the qubit to be teleported is encoded in one of two ZPF inputs in experiments on quantum cryptography [27], partial BSM [28], hyperentanglement and complete BSM [29], and entanglement swapping [30]. Two features of the ZPF constitute the common denominator in these works.…”