In this work, we study the Rényi $\alpha$ entropies $S_{\alpha }\left(\widehat{\rho }\right)={(1-\alpha )}^{-1}ln\left\{\text{Tr}\left({\widehat{\rho }}^{\alpha }\right)\right\}$ 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 $\alpha =2,3,4,\cdots$ and then, with the help of the analytic continuation, acquire the closed form also for real values of $\alpha >0$. The quantity $S_2\left(\widehat{\rho }\right)$, primarily studied in the literature, will then be a special case of our finding. Subsequently we acquire the Rényi $\alpha$ entropies, with positive integers $\alpha$, 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 $|n\rangle$ (pure states) of a single harmonic oscillator with $n\ge 1$, 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 $\alpha$ entropy dynamics when the current form of a non-Gaussian state is initially prepared.

• Received 16 April 2018
• Revised 24 May 2018

DOI:https://doi.org/10.1103/PhysRevE.97.062141