We provide a stochastic interpretation of non-commutative Dirichlet forms in the context of quantum filtering. For stochastic processes motivated by quantum optics experiments, we derive an optimal finite time deviation bound expressed in terms of the non-commutative Dirichlet form. Introducing and developing new non-commutative functional inequalities, we deduce concentration inequalities for these processes. Examples satisfying our bounds include tensor products of quantum Markov semigroups as well as Gibbs samplers above a threshold temperature.

中文翻译：

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量子噪声的偏差界限和浓度不等式

我们在量子过滤的背景下提供了对非交换狄利克雷形式的随机解释。对于由量子光学实验激发的随机过程，我们推导出以非交换狄利克雷形式表示的最佳有限时间偏差界。引入和发展新的非交换函数不等式，我们推导出这些过程的集中不等式。满足我们界限的例子包括量子马尔可夫半群的张量积以及高于阈值温度的吉布斯采样器。