Relative entropy representations via tracial joint spectral measures

Abstract:We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}\mu_{A,B}(a,b)$, where $\mu_{A,B}$ denotes Heinävaara's tracial joint spectral measure. This allows us to obtain a unified derivation of several known integral representations of relative entropy from scalar equalities. Scalar inequalities can be used in the same way to derive Pinsker-type bounds. We further show that the testing curve $t\mapsto \mathrm{tr}[A-tB]_+$ is piecewise affine exactly when $A$ and $B$ commute, which is equivalent to their testing region being a polygon. This gives a characterization of noncommutativity through the curvature. As further applications, we derive inequalities for relative entropy variance and the loss of relative entropy under positive trace-preserving maps.

arXiv:2609.12547 [quant-ph]

(or arXiv:2609.12547v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.12547 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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