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

Horssen, Merlijn van and Guţă, Mădălin (2015) Sanov and central limit theorems for output statistics of quantum Markov chains. Journal of Mathematical Physics, 56 (2). 022109/1-022109/14. ISSN 1089-7658

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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.

Item Type: Article
Keywords: Quantum measurement theory, Eigenvalues , Signal generators , Statistical analysis, Markov processes
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: 10.1063/1.4907995
Depositing User: Eprints, Support
Date Deposited: 09 Oct 2017 12:32
Last Modified: 13 Oct 2017 04:43
URI: http://eprints.nottingham.ac.uk/id/eprint/47095

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