On the Strong Subadditivity of the R\'enyi entropies for bosonic and fermionic Gaussian states

TL;DR

This paper investigates the conditions under which R\'enyi entropies satisfy subadditivity and strong subadditivity for bosonic and fermionic Gaussian states, revealing that bosons always satisfy subadditivity while fermions generally do not.

Contribution

It provides a detailed analysis of the subadditivity properties of R\'enyi entropies for Gaussian states, including explicit intervals of the R\'enyi index where these properties hold.

Findings

Bosonic R\'enyi entropies always satisfy subadditivity.

Fermionic R\'enyi entropies generally do not satisfy subadditivity or strong subadditivity.

Explicit R\'enyi index intervals where subadditivity and strong subadditivity hold for both bosons and fermions.

Abstract

Recently, there has been a surge of interest in using R\'enyi entropies as quantifiers of correlations in many-body quantum systems. However, it is well known that in general these entropies do not satisfy the strong subadditivity inequality, which is a central property ensuring the positivity of correlation measures. In fact, in many cases they do not even satisfy the weaker condition of subadditivity. In the present paper we shed light on this subject by providing a detailed survey of R\'enyi entropies for bosonic and fermionic Gaussian states. We show that for bosons the R\'enyi entropies always satisfy subadditivity, but not necessarily strong subadditivity. Conversely, for fermions both do not hold in general. We provide the precise intervals of the R\'enyi index $\alpha$ for which subadditivity and strong subadditivity are valid in each case.