Concentration Inequalities for Output Statistics of Quantum Markov Processes

Girotti, Federico; Garrahan, Juan P.; Guţă, Mădălin

Abstract

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-commutative L2 theory, 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.

Citation

Girotti, F., Garrahan, J. P., & Guţă, M. (2023). Concentration Inequalities for Output Statistics of Quantum Markov Processes. Annales Henri Poincaré, https://doi.org/10.1007/s00023-023-01286-1

• Journal Article Type: Article
• Acceptance Date: Feb 15, 2023
• Online Publication Date: Mar 27, 2023
• Publication Date: Mar 27, 2023
• Deposit Date: May 3, 2023
• Publicly Available Date: May 3, 2023
• Journal: Annales Henri Poincaré
• Print ISSN: 1424-0637
• Electronic ISSN: 1424-0661
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.1007/s00023-023-01286-1
• Keywords: Mathematical Physics; Nuclear and High Energy Physics; Statistical and Nonlinear Physics
• Public URL: https://nottingham-repository.worktribe.com/output/19453824
• Publisher URL: https://link.springer.com/article/10.1007/s00023-023-01286-1
• Additional Information: Received: 10 November 2022; Accepted: 15 February 2023; First Online: 27 March 2023; : ; : Data sharing was not applicable to this article as no datasets were generated or analysed during the current study.