Concentration Inequalities for Output Statistics of Quantum Markov Processes
Concentration Inequalities for Output Statistics of Quantum Markov Processes
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DOI：
10.1007/s00023-023-01286-1
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发表时间：
2022-06
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影响因子：
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通讯作者：
Federico Girotti;J. P. Garrahan;M. Guţă
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作者：
Federico Girotti;J. P. Garrahan;M. Guţă
我们推导出量子马尔可夫过程中测量结果的时间平均值的新的浓度界。这推广了经典马尔可夫链的众所周知的界限，它提供了对时间加性量在其平均值周围的有限时间波动的约束。我们采用谱，扰动和鞅技术，连同非交换理论，推导：（i）量子马尔可夫链测量结果的时间平均的Bernstein型浓度界，（ii）同一过程的Hoeffding型浓度界，（iii）连续时间量子马尔可夫过程计数过程的Bernstein型浓度界的推广，（iv）经典马尔可夫链的经验通量的新的浓度边界，其拓宽了相应的经典边界的适用范围，超出了经验平均值。我们还建议潜在的应用，我们的结果参数估计，并考虑扩展到可还原的量子通道，多时间统计和时间依赖的测量，并评论连接到所谓的热力学不确定性关系。
We derive new concentration bounds for time averages of measurement outcomes in quantum Markov processes. This generalizes well-known bounds for classical Markov chains, which provide constraints on finite-time fluctuations of time-additive quantities around their averages. We employ spectral, perturbation and martingale techniques, together with non-commutativetheory, to derive: (i) a Bernstein-type concentration bound for time averages of the measurement outcomes of a quantum Markov chain, (ii) a Hoeffding-type concentration bound for the same process, (iii) a generalization of the Bernstein-type concentration bound for counting processes of continuous-time quantum Markov processes, (iv) new concentration bounds for empirical fluxes of classical Markov chains which broaden the range of applicability of the corresponding classical bounds beyond empirical averages. We also suggest potential application of our results to parameter estimation and consider extensions to reducible quantum channels, multi-time statistics and time-dependent measurements, and comment on the connection to so-called thermodynamic uncertainty relations.