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

In this work, we study the Renyi alpha entropies S-alpha((rho) over cap) = ( 1 - alpha)(-1) ln{Tr((rho) over cap (alpha))} 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,... and then, with the help of the analytic continuation, acquire the closed form also for real values of alpha > 0. The quantity S-2((rho) over cap), primarily studied in the literature, will then be a special case of our finding. Subsequently we acquire the Renyi a entropies, with positive integers a, 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 vertical bar n > (pure states) of a single harmonic oscillator with n >= 1, in which theWigner 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 Renyi a entropy dynamics when the current form of a non- Gaussian state is initially prepared.