Relative entropy representations via tracial joint spectral measures

• URL: http://arxiv.org/abs/2609.12547v1
• Date: Fri, 11 Sep 2026 07:56:19 GMT
• Title: Relative entropy representations via tracial joint spectral measures
• Abstract summary: We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A|B)=2int alog(a/b),mathrmd_A,B(a,b)$, where $_A,B$ denotes Heinvaara's tracial joint spectral measure.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that for positive semidefinite $A$ and $B$ the relative entropy can be written as $D(A\|B)=2\int a\log(a/b)\,\mathrm{d}μ_{A,B}(a,b)$, where $μ_{A,B}$ 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.

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