Deviation bounds and concentration inequalities for quantum noises

Tristan Benoist1, Lisa Hänggli2,3, and Cambyse Rouzé2,3

1Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse Cedex 9, France
2Department of Mathematics, Technische Universität München, 85748 Garching, Germany
3Munich Center for Quantum Science and Technology (MCQST), München, Germany

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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.

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Cited by

[1] Bowen Li and Jianfeng Lu, "Interpolation Between Modified Logarithmic Sobolev and Poincaré Inequalities for Quantum Markovian Dynamics", Journal of Statistical Physics 190 10, 161 (2023).

[2] George Bakewell-Smith, Federico Girotti, Mădălin Guţă, and Juan P. Garrahan, "General Upper Bounds on Fluctuations of Trajectory Observables", Physical Review Letters 131 19, 197101 (2023).

[3] Li Gao and Cambyse Rouzé, "Coarse Ricci curvature of quantum channels", Journal of Functional Analysis 286 8, 110336 (2024).

[4] Federico Girotti, Juan P. Garrahan, and Mădălin Guţă, "Concentration Inequalities for Output Statistics of Quantum Markov Processes", Annales Henri Poincaré 24 8, 2799 (2023).

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