发表机构
Leibniz Universität Hannover（汉诺威莱布尼茨大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Bravyi小增量混合猜想的最优常数c=1，利用算子层饼定理的积分表示给出量子熵混合速率的维度无关尖锐界限，并修正相关猜想。
AI 中文摘要
在哈密顿量演化下，量子态系综的冯·诺依曼熵以何种速率变化？Bravyi提出了小增量混合猜想，该猜想将二元系综$\{(1\\!-\\!p,\rho_1),(p,\rho_2)\}$的混合速率控制在$c\\,\\|{H}\\|h_{2}(p)$（其中$h_{2}$为二元熵）。随后，Mariën、Audenaert、Van Acoleyen和Verstraete将Bravyi猜想的证明归结为可分希尔伯特空间上满足$\text{Tr} A\\!=\\!p$和$\text{Tr} B\\!=\\!1$的正迹类算子$A\\!\leq\\! B$的矩阵不等式$\\|[A,\log B]\\|_{1}\\!\leq\\! c\\,h_{2}(p)$，并猜想最优常数为$c\\!=\\!1$。本文利用Cheng和Liu提出的算子层饼定理得到的对易子$[A,\log B]$的精确积分表示，证明了后一猜想。该结果给出了一个与维度无关的尖锐界限，限制了单一组分的幺正演化能改变二元量子系综熵的最大速率。由此可知，混合速率满足具有最优常数的小增量混合，且二分哈密顿量的纠缠速率受$(2\log d\\!+\\!1)\\|H\\|$约束。此外，该结果修正了Lieb和Vershynina关于一般系综混合速率的猜想。
英文摘要
At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble $\{(1\!-\!p,ρ_1),(p,ρ_2)\}$ by $c\,\|{H}\|h_{2}(p)$ (with the binary entropy $h_{2}$). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality $\|[A,\log B]\|_{1}\!\leq\! c\,h_{2}(p)$ for positive trace-class operators $A\!\leq\! B$ with $\text{Tr} A\!=\!p$ and $\text{Tr} B\!=\!1$ on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be $c\!=\!1$. Here, I prove the latter conjecture using an exact integral representation of the commutator $[A,\log B]$ obtained from the operator layer cake theorem, due to Cheng and Liu. The result gives a sharp dimension-independent limit on how rapidly unitary evolution of one component can change the entropy of a binary quantum ensemble. It follows that mixing rates satisfy small incremental mixing with the optimal constant, and entangling rates of bipartite Hamiltonians are bounded by $(2\log d\!+\!1)\|H\|$. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.
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