Sanov and central limit theorems for output statistics of quantum Markov chains

Horssen, Merlijn van; Gu??, M?d?lin

Abstract

In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not.

Citation

Horssen, M. V., & Guţă, M. (2015). Sanov and central limit theorems for output statistics of quantum Markov chains. Journal of Mathematical Physics, 56(2), Article 022109. https://doi.org/10.1063/1.4907995

• Journal Article Type: Article
• Acceptance Date: Feb 2, 2015
• Publication Date: Feb 18, 2015
• Deposit Date: Oct 9, 2017
• Publicly Available Date: Oct 9, 2017
• Journal: Journal of Mathematical Physics
• Print ISSN: 0022-2488
• Electronic ISSN: 1089-7658
• Publisher: American Institute of Physics
• Peer Reviewed: Peer Reviewed
• Volume: 56
• Issue: 2
• Article Number: 022109
• DOI: https://doi.org/10.1063/1.4907995
• Keywords: Quantum measurement theory, Eigenvalues , Signal generators , Statistical analysis, Markov processes
• Public URL: https://nottingham-repository.worktribe.com/output/744827
• Publisher URL: https://doi.org/10.1063/1.4907995
• Contract Date: Oct 9, 2017