Linear entropy in quantum phase space

TL;DR

This paper introduces a method to calculate quantum Renyi entropy in phase space for fermions and bosons, enabling analysis of purity, fidelity, and entanglement using sampled distributions, with applications to thermal states and non-Markovian reservoirs.

Contribution

It presents a novel approach to compute quantum entropy from phase space samples, especially using Gaussian methods for complex quantum states.

Findings

Entropy can be computed from sampled phase space distributions for certain states.

Classical phase space methods face divergences and are unsuitable for entropy calculation.

Gaussian phase space methods effectively handle non-Markovian dynamics and entanglement.

Abstract

We calculate the quantum Renyi entropy in a phase space representation for either fermions or bosons. This can also be used to calculate purity and fidelity, or the entanglement between two systems. We show that it is possible to calculate the entropy from sampled phase space distributions in normally ordered representations, although this is not possible for all quantum states. We give an example of the use of this method in an exactly soluble thermal case. The quantum entropy cannot be calculated at all using sampling methods in classical symmetric (Wigner) or antinormally ordered (Husimi) phase spaces, due to inner product divergences. The preferred method is to use generalized Gaussian phase space methods, which utilize a distribution over stochastic Green's functions. We illustrate this approach by calculating the reduced entropy and entanglement of bosonic or fermionic modes coupled to a time-evolving, non-Markovian reservoir.