# Comparison of mixed quantum states

## Abstract

In this article, we study the problem of comparing mixed quantum states: given n unknown mixed quantum states, can one determine whether they are identical or not with an unambiguous quantum measurement? We first study the universal comparison of mixed quantum states, and prove that this task is generally impossible to accomplish. Then, we focus on the unambiguous comparison of n mixed quantum states arbitrarily chosen from a set of k mixed quantum states. The condition for the existence of an unambiguous measurement operator which can produce a conclusive result when the unknown states are actually the same and the condition for the existence of an unambiguous measurement operator when the unknown states are actually different are studied independently. We derive a necessary and sufficient condition for the existence of the first measurement operator, and a necessary condition and two sufficient conditions for the second. Furthermore, we find that the sufficiency of the necessary condition for the second measurement operator has a simple and interesting dependence on n and k. At the end, a unified condition is obtained for the simultaneous existence of these two unambiguous measurement operators.

- Authors:

- Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, 230026 Anhui (China)

- Publication Date:

- OSTI Identifier:
- 22051287

- Resource Type:
- Journal Article

- Journal Name:
- Physical Review. A

- Additional Journal Information:
- Journal Volume: 84; Journal Issue: 1; Other Information: (c) 2011 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1050-2947

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; COMPARATIVE EVALUATIONS; MIXED STATES; QUANTUM INFORMATION; QUANTUM OPERATORS

### Citation Formats

```
Pang Shengshi, and Wu Shengjun.
```*Comparison of mixed quantum states*. United States: N. p., 2011.
Web. doi:10.1103/PHYSREVA.84.012336.

```
Pang Shengshi, & Wu Shengjun.
```*Comparison of mixed quantum states*. United States. doi:10.1103/PHYSREVA.84.012336.

```
Pang Shengshi, and Wu Shengjun. Fri .
"Comparison of mixed quantum states". United States. doi:10.1103/PHYSREVA.84.012336.
```

```
@article{osti_22051287,
```

title = {Comparison of mixed quantum states},

author = {Pang Shengshi and Wu Shengjun},

abstractNote = {In this article, we study the problem of comparing mixed quantum states: given n unknown mixed quantum states, can one determine whether they are identical or not with an unambiguous quantum measurement? We first study the universal comparison of mixed quantum states, and prove that this task is generally impossible to accomplish. Then, we focus on the unambiguous comparison of n mixed quantum states arbitrarily chosen from a set of k mixed quantum states. The condition for the existence of an unambiguous measurement operator which can produce a conclusive result when the unknown states are actually the same and the condition for the existence of an unambiguous measurement operator when the unknown states are actually different are studied independently. We derive a necessary and sufficient condition for the existence of the first measurement operator, and a necessary condition and two sufficient conditions for the second. Furthermore, we find that the sufficiency of the necessary condition for the second measurement operator has a simple and interesting dependence on n and k. At the end, a unified condition is obtained for the simultaneous existence of these two unambiguous measurement operators.},

doi = {10.1103/PHYSREVA.84.012336},

journal = {Physical Review. A},

issn = {1050-2947},

number = 1,

volume = 84,

place = {United States},

year = {2011},

month = {7}

}