# Probabilistic interpretation of the reduction criterion for entanglement

## Abstract

Inspired by the idea of conditional probabilities, we introduce a variant of conditional density operators. But unlike the conditional probabilities which are bounded by 1, the conditional density operators may have eigenvalues exceeding 1 for entangled states. This has the consequence that although any bivariate classical probability distribution has a natural separable decomposition in terms of conditional probabilities, we do not have a quantum analogue of this separable decomposition in general. The 'nonclassical' eigenvalues of conditional density operators are indications of entanglement. The resulting separability criterion turns out to be equivalent to the reduction criterion introduced by Horodecki [Phys. Rev. A 59, 4206 (1999)] and Cerf et al. [Phys. Rev. A 60, 898 (1999)]. This supplies an intuitive probabilistic interpretation for the reduction criterion. The conditional density operators are also used to define a form of quantum conditional entropy which provides an alternative mechanism to reveal quantum discord.

- Authors:

- School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6 (Canada)
- (China)

- Publication Date:

- OSTI Identifier:
- 20982258

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.75.032312; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DECOMPOSITION; DENSITY; DISTRIBUTION; EIGENFUNCTIONS; EIGENVALUES; ENTROPY; PROBABILISTIC ESTIMATION; QUANTUM ENTANGLEMENT; QUANTUM OPERATORS; REDUCTION

### Citation Formats

```
Zhang, Zhengmin, Luo, Shunlong, and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080 Beijing.
```*Probabilistic interpretation of the reduction criterion for entanglement*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.032312.

```
Zhang, Zhengmin, Luo, Shunlong, & Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080 Beijing.
```*Probabilistic interpretation of the reduction criterion for entanglement*. United States. doi:10.1103/PHYSREVA.75.032312.

```
Zhang, Zhengmin, Luo, Shunlong, and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080 Beijing. Thu .
"Probabilistic interpretation of the reduction criterion for entanglement". United States.
doi:10.1103/PHYSREVA.75.032312.
```

```
@article{osti_20982258,
```

title = {Probabilistic interpretation of the reduction criterion for entanglement},

author = {Zhang, Zhengmin and Luo, Shunlong and Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100080 Beijing},

abstractNote = {Inspired by the idea of conditional probabilities, we introduce a variant of conditional density operators. But unlike the conditional probabilities which are bounded by 1, the conditional density operators may have eigenvalues exceeding 1 for entangled states. This has the consequence that although any bivariate classical probability distribution has a natural separable decomposition in terms of conditional probabilities, we do not have a quantum analogue of this separable decomposition in general. The 'nonclassical' eigenvalues of conditional density operators are indications of entanglement. The resulting separability criterion turns out to be equivalent to the reduction criterion introduced by Horodecki [Phys. Rev. A 59, 4206 (1999)] and Cerf et al. [Phys. Rev. A 60, 898 (1999)]. This supplies an intuitive probabilistic interpretation for the reduction criterion. The conditional density operators are also used to define a form of quantum conditional entropy which provides an alternative mechanism to reveal quantum discord.},

doi = {10.1103/PHYSREVA.75.032312},

journal = {Physical Review. A},

number = 3,

volume = 75,

place = {United States},

year = {Thu Mar 15 00:00:00 EDT 2007},

month = {Thu Mar 15 00:00:00 EDT 2007}

}