量子熵收缩与超压缩性导致的因子分解
Quantum Entropy Contraction and Factorization from Hypercontractivity
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中文总结 AI 辅助
本文证明超压缩性蕴含量子通道的熵收缩，无需详细平衡条件，并给出KMS对称半群的修正对数Sobolev界，进而应用于相对熵的近似张量化。
中文摘要 AI 辅助
我们证明了超压缩性意味着单个量子通道的熵收缩，无需详细平衡条件。对于关于忠实不变态σ为KMS对称的原始量子马尔可夫半群，我们得到了修正的对数Sobolev界α₁≥λ/(2+log‖σ⁻¹‖∞)，其中λ为谱隙。这去除了通过对数Sobolev常数进行比较时对L_p正则性的假设。作为应用，我们证明了两个条件期望平均的超压缩性意味着相对熵的近似张量化。
英文摘要
We prove that hypercontractivity implies entropy contraction for a single quantum channel, without a detailed balance condition. For primitive quantum Markov semigroups that are KMS-symmetric with respect to a faithful invariant state \(σ\), we obtain the modified Log-Sobolev bound $α_1\geq \fracλ{(2+\log\|σ^{-1}\|_\infty)}$ where $λ$ is the spectral gap. This removes the assumption of \(L_p\)-regularity for the comparison through the log-Sobolev constant. As an application, we show that the hypercontractivity of an average of two conditional expectations implies the approximate tensorization of relative entropy.
