# 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:
- 1432340

- Report Number(s):
- SAND2018-0673J

Journal ID: ISSN 1367-2630; 660116; TRN: US1802658

- Grant/Contract Number:
- NA0003525

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- New Journal of Physics

- Additional Journal Information:
- Journal Volume: 20; Journal Issue: 2; Journal ID: ISSN 1367-2630

- Publisher:
- IOP Publishing

- Country of Publication:
- United States

- Language:
- English

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

### Citation Formats

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

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

```
Scholten, Travis L., and Blume-Kohout, Robin. Thu .
"Behavior of the maximum likelihood in quantum state tomography". United States. doi:10.1088/1367-2630/aaa7e2. https://www.osti.gov/servlets/purl/1432340.
```

```
@article{osti_1432340,
```

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

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

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},

issn = {1367-2630},

number = 2,

volume = 20,

place = {United States},

year = {2018},

month = {2}

}