# Evenly distributed unitaries: On the structure of unitary designs

## Abstract

We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.

- Authors:

- Institute for Mathematical Sciences, Imperial College London, Princes Gate, London SW7 2PE (United Kingdom) and QOLS, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BW (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 20929700

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 5; Other Information: DOI: 10.1063/1.2716992; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; GROUP THEORY; INFORMATION THEORY; MATRICES; POLYNOMIALS; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS

### Citation Formats

```
Gross, D., Audenaert, K., and Eisert, J.
```*Evenly distributed unitaries: On the structure of unitary designs*. United States: N. p., 2007.
Web. doi:10.1063/1.2716992.

```
Gross, D., Audenaert, K., & Eisert, J.
```*Evenly distributed unitaries: On the structure of unitary designs*. United States. doi:10.1063/1.2716992.

```
Gross, D., Audenaert, K., and Eisert, J. Tue .
"Evenly distributed unitaries: On the structure of unitary designs". United States.
doi:10.1063/1.2716992.
```

```
@article{osti_20929700,
```

title = {Evenly distributed unitaries: On the structure of unitary designs},

author = {Gross, D. and Audenaert, K. and Eisert, J.},

abstractNote = {We clarify the mathematical structure underlying unitary t-designs. These are sets of unitary matrices, evenly distributed in the sense that the average of any tth order polynomial over the design equals the average over the entire unitary group. We present a simple necessary and sufficient criterion for deciding if a set of matrices constitutes a design. Lower bounds for the number of elements of 2-designs are derived. We show how to turn mutually unbiased bases into approximate 2-designs whose cardinality is optimal in leading order. Designs of higher order are discussed and an example of a unitary 5-design is presented. We comment on the relation between unitary and spherical designs and outline methods for finding designs numerically or by searching character tables of finite groups. Further, we sketch connections to problems in linear optics and questions regarding typical entanglement.},

doi = {10.1063/1.2716992},

journal = {Journal of Mathematical Physics},

number = 5,

volume = 48,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}