Sanov and central limit theorems for output statistics of quantum Markov chains
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 proces...
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American Institute of Physics
2015
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| Online Access: | https://eprints.nottingham.ac.uk/47095/ |
| _version_ | 1848797466743275520 |
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| author | Horssen, Merlijn van Guţă, Mădălin |
| author_facet | Horssen, Merlijn van Guţă, Mădălin |
| author_sort | Horssen, Merlijn van |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T20:04:20Z |
| format | Article |
| id | nottingham-47095 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T20:04:20Z |
| publishDate | 2015 |
| publisher | American Institute of Physics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-470952020-05-04T17:02:30Z https://eprints.nottingham.ac.uk/47095/ Sanov and central limit theorems for output statistics of quantum Markov chains Horssen, Merlijn van Guţă, Mădălin 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. American Institute of Physics 2015-02-18 Article PeerReviewed 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 Quantum measurement theory Eigenvalues Signal generators Statistical analysis Markov processes https://doi.org/10.1063/1.4907995 doi:10.1063/1.4907995 doi:10.1063/1.4907995 |
| spellingShingle | Quantum measurement theory Eigenvalues Signal generators Statistical analysis Markov processes Horssen, Merlijn van Guţă, Mădălin Sanov and central limit theorems for output statistics of quantum Markov chains |
| title | Sanov and central limit theorems for output statistics of quantum Markov chains |
| title_full | Sanov and central limit theorems for output statistics of quantum Markov chains |
| title_fullStr | Sanov and central limit theorems for output statistics of quantum Markov chains |
| title_full_unstemmed | Sanov and central limit theorems for output statistics of quantum Markov chains |
| title_short | Sanov and central limit theorems for output statistics of quantum Markov chains |
| title_sort | sanov and central limit theorems for output statistics of quantum markov chains |
| topic | Quantum measurement theory Eigenvalues Signal generators Statistical analysis Markov processes |
| url | https://eprints.nottingham.ac.uk/47095/ https://eprints.nottingham.ac.uk/47095/ https://eprints.nottingham.ac.uk/47095/ |