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.

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

References in corpus (7)

• Renyi generalizations of the conditional quantum mutual information
• Recoverability for optimized quantum $f$-divergences
• Equality conditions of Data Processing Inequality for $\alpha$-$z$ Rényi relative entropies
• Reverse test and quantum analogue of classical fidelity and generalized fidelity
• Approximate Petz recovery from the geometry of density operators
• A fidelity measure for quantum states based on the matrix geometric mean
• Revisiting the equality conditions of the data processing inequality for the sandwiched Rényi divergence

Cited by in corpus (2)

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