通过迹联合谱测度的相对熵表示
Relative entropy representations via tracial joint spectral measures
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中文总结 AI 辅助
本文通过迹联合谱测度给出相对熵的统一积分表示，并利用标量等式与不等式推导Pinsker型界，同时以测试曲线曲率刻画非交换性，进而推导相对熵方差与映射下损失的界。
中文摘要 AI 辅助
我们证明对于正半定矩阵$A$和$B$，相对熵可以写成$D(A\\|B)=2\int a\log(a/b)\\,\mathrm{d}\mu_{A,B}(a,b)$，其中$\mu_{A,B}$表示Heinävaara的迹联合谱测度。这使我们能够从标量等式统一推导出相对熵的多个已知积分表示。标量不等式可以以相同方式用于推导Pinsker型界。我们进一步证明，测试曲线$t\mapsto \mathrm{tr}[A-tB]_+$恰好当$A$和$B$可交换时是分段仿射的，这等价于它们的测试区域是多边形。这给出了通过曲率对非交换性的刻画。作为进一步的应用，我们推导了相对熵方差以及在正迹保持映射下相对熵损失的若干不等式。
英文摘要
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.
发表机构
- Institute for Theoretical Physics, ETH Zurich（苏黎世联邦理工学院理论物理研究所）
- IBM Quantum, IBM Research Europe – Zurich（IBM欧洲苏黎世研究中心）
机构由 AI 辅助整理，请以论文原文为准。
