量子马尔可夫半群的广义皮卡德不等式
Generalized Poincaré inequality for quantum Markov semigroups
AI总结：
本文提出了一种针对量子马尔可夫半群的广义皮卡德不等式，通过非交换导数和链规则估计，推广了半交换结果，并应用于非交换辛钦和亚指数集中不等式的恢复。
摘要：
我们为在迹von Neumann代数上作用的迹对称量子马尔可夫半群证明了一个非交换的(p,p)-皮卡德不等式，仅假设存在谱间隙。扩展Huang和Tropp的半交换结果，我们的论证利用马尔可夫扩张来获得狄利克雷形式的链规则估计，并运用融合自由积来定义适当的非交换导数。我们进一步将该论证推广到非迹σ-有限von Neumann代数，在更弱的GNS-detailed balance假设下进行。作为应用，我们恢复了非交换辛钦和亚指数集中不等式。
英文摘要：
We prove a noncommutative $p$-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial $σ$-finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial $σ$-finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.
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