Lipschitz continuity of quantum-classical conditional entropies with respect to angular distance and related properties

TL;DR

This paper establishes a Lipschitz continuity bound for quantum-classical conditional entropies based on angular distance, which is independent of the conditioning system's dimension, and explores related properties and limitations for quantum entropies.

Contribution

It introduces a new Lipschitz continuity bound for quantum-classical conditional entropies with respect to angular distance, improving upon previous bounds and analyzing its limitations for quantum entropies.

Findings

Lipschitz bound is independent of the conditioning system's dimension.

The bound is sharper than previous bounds in some cases.

Characterization of states saturating Fuchs--van de Graaf inequality.

Abstract

We derive a Lipschitz continuity bound for quantum-classical conditional entropies with respect to angular distance, with a Lipschitz constant that is independent of the dimension of the conditioning system. This bound is sharper in some situations than previous continuity bounds, which were either based on trace distance (where Lipschitz continuity is not possible), or based on angular distance but did not include a conditioning system. However, we find that the bound does not directly generalize to fully quantum conditional entropies. To investigate possible counterexamples in that setting, we study the characterization of states which saturate the Fuchs--van de Graaf inequality and thus have angular distance approximately equal to trace distance. We give an exact characterization of such states in the invertible case. For the noninvertible case, we show that the situation appears to be significantly more elaborate, and seems to be strongly connected to the question of characterizing the set of fidelity-preserving measurements.