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arXiv:quant-ph/0602216

Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects

M. Ruzzi, Marcelo A. Marchiolli, Eduardo Costa da Silva, D. Galetti

Published Feb 26, 2006• 27 citations
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Abstract

An approach featuring $s$-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.

DOI
10.48550/arxiv.quant-ph/0602216

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An approach featuring sss-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.
Polarization quasiprobability distribution defined in the Stokes space shares many important properties with the Wigner function for position and momentum. Most notably, they both give correct one-dimensional marginal probability distributions and therefore represent the natural choice for the probability distributions in classical hidden-variable models. In this context, negativity of the Wigner function is considered as proof of nonclassicality for a quantum state. On the contrary, the polarization quasiprobability distribution demonstrates negativity for all quantum states. This feature comes from the discrete nature of Stokes variables; however, it was not observed in previous experiments, because they were performed with photon-number averaging detectors. Here we reconstruct the polarization quasiprobability distribution of a coherent state with photon-number resolving detectors, which allows us to directly observe for the first time its negativity. Furthermore we derive a theoretical polarization quasiprobability distribution for any linearly polarized quantum state.
Quasiprobability distributions of the photon number and the momentum-like quadrature appropriate to partial symmetrizations of products of the photon-number and quadrature operators allowing these operators to alternate at most (2r-2) times are good approximations to the Wigner function for these simultaneously considered noncommuting quantities related to the fully symmetric ordering. It is shown that the r-parametrized quasiprobability distributions must take on values at multiples of 1/r. A scheme for simultaneous measurement of the photon number and the momentum-like quadrature is proposed. The r-parametrized quasidistributions can be compared with a joint distribution of the photon number and the momentum-like quadrature measured simultaneously.