# Behavior of the maximum likelihood in quantum state tomography

## Abstract

Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d. They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good. Here, we consider the loglikelihood ratio. Because of the positivity constraint ρ ≥ 0, quantum state space does not generally satisfy local asymptotic normality (LAN), meaning the classical null theory for the loglikelihood ratio (the Wilks theorem) should not be used. Thus, understanding and quantifying how positivity affects the null behavior of this test statistic is necessary for its use in model selection for state tomography. We define a new generalization of LAN, metric-projected LAN, show that quantum state space satisfies it, and derive a replacement for the Wilks theorem. In addition to enabling reliable model selection, our results shed more light on the qualitative effects of the positivity constraint on state tomography.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of New Mexico, Albuquerque, NM (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1335564

- Report Number(s):
- SAND-2016-8677J

Journal ID: ISSN 1367-2630; 647145

- DOE Contract Number:
- NA0003525

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: New Journal of Physics; Journal Volume: 20

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Blume-Kohout, Robin J, and Scholten, Travis L.
```*Behavior of the maximum likelihood in quantum state tomography*. United States: N. p., 2018.
Web. doi:10.1088/1367-2630/aaa7e2.

```
Blume-Kohout, Robin J, & Scholten, Travis L.
```*Behavior of the maximum likelihood in quantum state tomography*. United States. doi:10.1088/1367-2630/aaa7e2.

```
Blume-Kohout, Robin J, and Scholten, Travis L. Thu .
"Behavior of the maximum likelihood in quantum state tomography". United States.
doi:10.1088/1367-2630/aaa7e2. https://www.osti.gov/servlets/purl/1335564.
```

```
@article{osti_1335564,
```

title = {Behavior of the maximum likelihood in quantum state tomography},

author = {Blume-Kohout, Robin J and Scholten, Travis L.},

abstractNote = {Quantum state tomography on a d-dimensional system demands resources that grow rapidly with d. They may be reduced by using model selection to tailor the number of parameters in the model (i.e., the size of the density matrix). Most model selection methods typically rely on a test statistic and a null theory that describes its behavior when two models are equally good. Here, we consider the loglikelihood ratio. Because of the positivity constraint ρ ≥ 0, quantum state space does not generally satisfy local asymptotic normality (LAN), meaning the classical null theory for the loglikelihood ratio (the Wilks theorem) should not be used. Thus, understanding and quantifying how positivity affects the null behavior of this test statistic is necessary for its use in model selection for state tomography. We define a new generalization of LAN, metric-projected LAN, show that quantum state space satisfies it, and derive a replacement for the Wilks theorem. In addition to enabling reliable model selection, our results shed more light on the qualitative effects of the positivity constraint on state tomography.},

doi = {10.1088/1367-2630/aaa7e2},

journal = {New Journal of Physics},

number = ,

volume = 20,

place = {United States},

year = {Thu Feb 22 00:00:00 EST 2018},

month = {Thu Feb 22 00:00:00 EST 2018}

}