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

Physical Review. E
Ilki Kim

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.

References

Apr 12, 2006·Physical Review Letters·Michael M WolfJ Ignacio Cirac
Apr 7, 2010·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Ilki Kim, Günter Mahler
Dec 12, 2012·Physical Review Letters·Gerardo AdessoAlessio Serafini
Feb 13, 2015·Proceedings of the National Academy of Sciences of the United States of America·Fernando BrandãoStephanie Wehner
Nov 14, 2015·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Avijit MisraA K Rajagopal
Sep 15, 2016·Physical Review. E·Sumiyoshi Abe
Jun 18, 2017·Physical Review Letters·Jader P SantosMauro Paternostro

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