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∫ alog(a/b) dμA,B(a,b), where μA,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↦ 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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BibTeX
@misc{schmitt2026relative,
title = {{Relative entropy representations via tracial joint spectral measures}},
author = {Lukas Schmitt},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12547},
eprint = {2609.12547},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12547},
}
RIS
TY - UNPB
TI - Relative entropy representations via tracial joint spectral measures
AU - Lukas Schmitt
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12547
UR - https://arxiv.org/abs/2609.12547
AB - We show that for positive semidefinite A and B the relative entropy can be written as D(A|B)=2∫ alog(a/b) dμ_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↦ 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.
ER -