# Compatibility of subsystem states and convex geometry

## Abstract

The subsystem compatibility problem, which concerns the question of whether a set of subsystem states are compatible with a state of the entire system, has received much study. Here we attack the problem from a new angle, utilizing the ideas of convexity that have been successfully employed against the separability problem. Analogously to an entanglement witness, we introduce the idea of a compatibility witness, and prove a number of properties about these objects. We show that the subsystem compatibility problem can be solved numerically and efficiently using semidefinite programming, and that the numerical results from this solution can be used to extract exact analytic results, an idea which we use to disprove a conjecture about the subsystem problem made by Butterley et al. [Found. Phys. 36, 83 (2006)]. Finally, we consider how the ideas can be used to tackle some important variants of the compatibility problem; in particular, the case of identical particles (known as N-representability in the case of fermions) is considered.

- Authors:

- Department of Mathematics, University of York, Heslington, York YO10 5DD (United Kingdom)

- Publication Date:

- OSTI Identifier:
- 20982233

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 75; Journal Issue: 3; Other Information: DOI: 10.1103/PhysRevA.75.032102; (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; COMPATIBILITY; FERMIONS; MATHEMATICAL SOLUTIONS; PROGRAMMING; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS

### Citation Formats

```
Hall, William.
```*Compatibility of subsystem states and convex geometry*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVA.75.032102.

```
Hall, William.
```*Compatibility of subsystem states and convex geometry*. United States. doi:10.1103/PHYSREVA.75.032102.

```
Hall, William. Thu .
"Compatibility of subsystem states and convex geometry". United States.
doi:10.1103/PHYSREVA.75.032102.
```

```
@article{osti_20982233,
```

title = {Compatibility of subsystem states and convex geometry},

author = {Hall, William},

abstractNote = {The subsystem compatibility problem, which concerns the question of whether a set of subsystem states are compatible with a state of the entire system, has received much study. Here we attack the problem from a new angle, utilizing the ideas of convexity that have been successfully employed against the separability problem. Analogously to an entanglement witness, we introduce the idea of a compatibility witness, and prove a number of properties about these objects. We show that the subsystem compatibility problem can be solved numerically and efficiently using semidefinite programming, and that the numerical results from this solution can be used to extract exact analytic results, an idea which we use to disprove a conjecture about the subsystem problem made by Butterley et al. [Found. Phys. 36, 83 (2006)]. Finally, we consider how the ideas can be used to tackle some important variants of the compatibility problem; in particular, the case of identical particles (known as N-representability in the case of fermions) is considered.},

doi = {10.1103/PHYSREVA.75.032102},

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}

}