Entropy and Wigner Distribution Functions Revisited

Laila A. Al‐Essa, Wafa F. Alfwzan, F. M. Aldosari +2 more

The Wigner quasi-probability distribution function is an important tool for estimating the quantumness of quantum states. The Wigner distribution function (WDF) is related monotonically to the coherent field state density operator and their probability distributions. In this paper, we analyze the time-dependent resonator coherent field states produced by a flux qubit’s interaction with a resonator through a two-photon coupling. The WDF dynamics of the coherent field states is investigated under the effects of the qubit-resonator detuning and the intrinsic decoherence. The WDF non-classicality associated with the mixedness resonator entropy dynamics can be controlled by varying the qubit-resonator interaction, detuning as well as the decoherence. The intrinsic decoherence clearly affects the WDF non-classicality, identified by the negative values of the WDF. Finally, by fixing one or both of the complex phase space components, we are able to explore the dynamics of the Wigner-distribution functions.

Joachim J. Włodarz

Élliott H. Lieb

An upper bound is proved for the Lp norm of Woodward’s ambiguity function in radar signal analysis and of the Wigner distribution in quantum mechanics when p>2. A lower bound is proved for 1≤p<2. In addition, a lower bound is proved for the entropy. These bounds set limits to the sharpness of the peaking of the ambiguity function or Wigner distribution. The bounds are best possible and equality is achieved in the Lp bounds if and only if the functions f and g that enter the definition are both Gaussians.

Amalia Torre

Nuno Costa Dias, João Nuno Prata

Abstract We address a recent conjecture stated by Z. Van Herstraeten and N. J. Cerf. They claim that the Shannon entropy for positive Wigner functions is bounded below by a positive constant, which can be attained only by Gaussian pure states. We introduce an alternative definition of entropy for all absolutely integrable Wigner functions, which is the Shannon entropy for positive Wigner functions. Moreover, we are able to prove, in arbitrary dimension, that this entropy is indeed bounded below by a positive constant, which is not very distant from the constant suggested by Van Herstraeten and Cerf. We also prove an analogous result for another conjecture stated by the same authors for the Rényi entropy of positive Wigner functions. As a by-product we prove a new inequality for the radar-ambiguity function (and for the Wigner distribution) which is reminiscent of Lieb’s inequalities.