Preservation of a quantum Renyi relative entropy implies existence of a recovery map

TL;DR

This paper demonstrates that equality in the data processing inequality for quantum sandwiched Rényi relative entropy with alpha greater than one implies the existence of a recovery map, extending known results for quantum relative entropy.

Contribution

It establishes that preservation of quantum sandwiched Rényi relative entropy under quantum channels implies the existence of a recovery map, generalizing the equality condition in DPI.

Findings

Equality in DPI for sandwiched Rényi entropy implies a recovery map.

Uses an interpolating family of Lp-norms for the proof.

Extends known results from quantum relative entropy to Rényi entropy.

Abstract

It is known that a necessary and sufficient condition for equality in the data processing inequality (DPI) for the quantum relative entropy is the existence of a recovery map. We show that equality in DPI for a sandwiched R\'enyi relative $\alpha$-entropy with $\alpha>1$ is also equivalent to this property. For the proof, we use an interpolating family of $L_p$-norms with respect to a state.