Rényi α entropies of quantum states in closed form: Gaussian states and a class of non-Gaussian states.

Abstract

In this work, we study the Rényi α entropies S_{α}(ρ[over ̂])=(1-α)^{-1}ln{Tr(ρ[over ̂]^{α})} of quantum states for N bosons in the phase-space representation. With the help of the Bopp rule, we derive the entropies of Gaussian states in closed form for positive integers α=2,3,4,⋯ and then, with the help of the analytic continuation, acquire the closed form also for real values of α>0. The quantity S_{2}(ρ[over ̂]), primarily studied in the literature, will then be a special case of our finding. Subsequently we acquire the Rényi α entropies, with positive integers α, in closed form also for a specific class of the non-Gaussian states (mixed states) for N bosons, which may be regarded as a generalization of the eigenstates |n〉 (pure states) of a single harmonic oscillator with n≥1, in which the Wigner functions have negative values indeed. Due to the fact that the dynamics of a system consisting of N oscillators is Gaussian, our result will contribute to a systematic study of the Rényi α entropy dynamics when the current form of a non-Gaussian state is initially prepared.