Strong subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states

Recently, there has been a surge of interest in using Rényi 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 Rényi entropies for bosonic and fermionic Gaussian states. We show that for bosons the Rényi 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 Rényi index $\alpha$ for which subadditivity and strong subadditivity are valid in each case.

Figure 1

Violation of strong subadditivity (SSA) of bosonic Rényi entropies for three-mode Gaussian states. The plots were created by randomly generating $\sim 200k$ bona fide correlation matrices and computing for different $\alpha$ the Rényi conditional mutual information ${\mathcal{I}}_{\alpha }(A:C|B)$, where $A$, $B$, and $C$ correspond to three single-mode subsystems.

Figure 2

Violation of strong subadditivity (SSA) of fermions for three-mode Gaussian states. The plots were created by randomly generating $\sim 300k$ bona fide correlation matrices and computing for different $\alpha$ the Rényi conditional mutual information ${\mathcal{I}}_{\alpha }(A:C|B)$ for single-mode subsystems $A$, $B$, and $C$.