Geometric conditions for saturating the data processing inequality

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 Rényi relative entropies, and gives new results for $\alpha$-$z$ Rényi relative entropies and a family of of quantum $f$-divergences, which we compute explicitly.

arXiv comment13 pages + 2 pages appendices; v2 contains some small corrections, additional pedagogical explanations, and is published in J. Phys. A: Math. Theor (2022)

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Cited by in corpus (2)

• Approximate Petz recovery from the geometry of density operators
• Riemannian-geometric generalizations of quantum fidelities and Bures-Wasserstein distance