Geometric conditions for saturating the data processing inequality

TL;DR

This paper introduces geometric techniques to derive operator equations from the saturation of the data processing inequality for various quantum distinguishability measures, extending known results and providing new explicit formulas.

Contribution

It develops a geometric approach to connect DPI saturation with operator equations for broad classes of quantum distinguishability measures, including new results for $ extalpha$-$z$ Rényi divergences.

Findings

Derived a formula linking DPI saturation to operator equations for all distinguishability measures studied.

Established that for many measures, the operator equation is both necessary and sufficient for DPI saturation.

Explicitly computed new results for $ extalpha$-$z$ Rényi divergences and quantum $f$-divergences.

Abstract

The data processing inequality (DPI) is a scalar inequality satisfied by distinguishability measures on density matrices. For some distinguishability measures, saturation of the scalar DPI implies an operator equation relating the arguments of the measure. These results are typically derived using functional analytic techniques. In a complementary approach, we use geometric techniques to derive a formula that gives an operator equation from DPI saturation for any distinguishability measure; moreover, for a broad class of distinguishability measures, the derived operator equation is sufficient to imply saturation as well. Our operator equation coincides with known results for the sandwiched R\'{e}nyi relative entropies, and gives new results for $\alpha$-$z$ R\'{e}nyi relative entropies and a family of of quantum $f$-divergences, which we compute explicitly.