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Physical Review A

Classical statistics inherent in pure quantum states

A. K. Rajagopal

Published Jan 1, 1983• 31 citations
Publisher

Abstract

A new coherent state introduced recently by us is shown to lead to a Wigner distribution function which maximizes the Shannon entropy with given mean, variance, and covariance of position and momentum of a particle. This result is discussed in the light of a recent paper of Senitzky. A more general discussion of entropy based on smoothed Wigner functions is also given.

DOI
10.1103/physreva.27.558

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In a recent paper [PRE 62, 4665 (2000)] (quant-ph/0203102) Manfredi and Feix proposed an alternative definition of quantum entropy based on Wigner phase-space distribution functions and discussed its properties. They proposed also some simple rules to construct positive-definite Wigner functions which are crucial in their discussion. We show, however, that the main results obtained and their discussion are incorrect.

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A new coherent state introduced recently by us is shown to lead to a Wigner distribution function which maximizes the Shannon entropy with given mean, variance, and covariance of position and momentum of a particle. This result is discussed in the light of a recent paper of Senitzky. A more general discussion of entropy based on smoothed Wigner functions is also given.
One of the differences between classical and quantum world is that in the former we can always perform a measurement that gives certain outcomes for all pure states, while such a situation is not possible in the latter one. The degree of randomness of the distribution of the measurement outcomes can be quantified by the Shannon entropy. While it is well known that this entropy, as a function of quantum states, needs to be minimized by some pure states, we would like to address the question how ‘badly’ can we end by choosing initially any pure state, i.e., which pure states produce the maximal amount of uncertainty under given measurement. We find these maximizers for all highly symmetric POVMs in dimension 2, and for all SIC-POVMs in any dimension.
We define the Wigner entropy of a quantum state as the differential Shannon entropy of the Wigner function of the state. This quantity is properly defined only for states that possess a positive Wigner function, which we name Wigner-positive states, but we argue that it is a proper measure of quantum uncertainty in phase space. It is invariant under symplectic transformations (displacements, rotations, and squeezing) and we conjecture that it is lower bounded by $ln\ensuremath{\pi}+1$ within the convex set of Wigner-positive states. It reaches this lower bound for Gaussian pure states, which are natural minimum-uncertainty states. This conjecture bears a resemblance with the Wehrl-Lieb conjecture, and we prove it over the subset of passive states of the harmonic oscillator which are of particular relevance in quantum thermodynamics. Along the way, we present a simple technique to build a broad class of Wigner-positive states exploiting an optical beam splitter and reveal an unexpectedly simple convex decomposition of extremal passive states. The Wigner entropy is anticipated to be a significant physical quantity, for example, in quantum optics where it allows us to establish a Wigner entropy-power inequality. It also opens a way towards stronger entropic uncertainty relations. Finally, we define the Wigner-R\'enyi entropy of Wigner-positive states and conjecture an extended lower bound that is reached for Gaussian pure states.
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.
Shannon entropies of the Wigner function are calculated for ground and excited stationary states of the Particle-In-A-Box and Harmonic Oscillator model systems and examined as a measure of the localization of the phase-space distribution. We show that their behavior is consistent with that of the sum of the position and momentum space entropies as a function of quantum number. Position-momentum correlation is then analyzed in these systems by defining mutual information between position and momentum variables. This mutual information yields non-zero values, in contrast to the quantum covariance, and increases with quantum number.