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

## 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.

- Authors:

- School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD (United Kingdom)
- School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 22405048

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FUNCTIONS; MARKOV PROCESS; PHASE TRANSFORMATIONS; QUANTUM OPERATORS; STATISTICS

### Citation Formats

```
Horssen, Merlijn van, E-mail: merlijn.vanhorssen@nottingham.ac.uk, and Guţă, Mădălin, E-mail: madalin.guta@nottingham.ac.uk.
```*Sanov and central limit theorems for output statistics of quantum Markov chains*. United States: N. p., 2015.
Web. doi:10.1063/1.4907995.

```
Horssen, Merlijn van, E-mail: merlijn.vanhorssen@nottingham.ac.uk, & Guţă, Mădălin, E-mail: madalin.guta@nottingham.ac.uk.
```*Sanov and central limit theorems for output statistics of quantum Markov chains*. United States. doi:10.1063/1.4907995.

```
Horssen, Merlijn van, E-mail: merlijn.vanhorssen@nottingham.ac.uk, and Guţă, Mădălin, E-mail: madalin.guta@nottingham.ac.uk. Sun .
"Sanov and central limit theorems for output statistics of quantum Markov chains". United States.
doi:10.1063/1.4907995.
```

```
@article{osti_22405048,
```

title = {Sanov and central limit theorems for output statistics of quantum Markov chains},

author = {Horssen, Merlijn van, E-mail: merlijn.vanhorssen@nottingham.ac.uk and Guţă, Mădălin, E-mail: madalin.guta@nottingham.ac.uk},

abstractNote = {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.},

doi = {10.1063/1.4907995},

journal = {Journal of Mathematical Physics},

number = 2,

volume = 56,

place = {United States},

year = {Sun Feb 15 00:00:00 EST 2015},

month = {Sun Feb 15 00:00:00 EST 2015}

}