Phase-space quantum distributions and information theory

Abstract

The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.