No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

Dimensions, lengths, and separability in finite-dimensional quantum systems

Rent:

Rent this article for

USD

10.1063/1.4790405

### Abstract

Many important sets of normalized states in a multipartite quantum system of finite dimension *d*, such as the set of all separable states, are real semialgebraic sets. We compute dimensions of many such sets in several low-dimensional systems. By using dimension arguments, we show that there exist separable states which are not convex combinations of *d* or less pure product states. For instance, such states exist in bipartite *M*⊗*N* systems when (*M* − 2)(*N* − 2) > 1. This solves an open problem proposed by DiVincenzo, Terhal and Thapliyal about 12 years ago. We prove that there exist a separable state ρ and a pure product state, whose mixture has smaller length than that of ρ. We show that any real , which is invariant under all partial transpose operations, is a convex sum of real pure product states. In the case of the 2⊗*N* system, the number *r* of product states can be taken to be . We also show that the general multipartite separability problem can be reduced to the case of real states. Regarding the separability problem, we propose two conjectures describing as a semialgebraic set, which may eventually lead to an analytic solution in some low-dimensional systems such as 2⊗4, 3⊗3, and 2⊗2⊗2.

© 2013 American Institute of Physics

Received 26 October 2012
Accepted 15 January 2013
Published online 06 February 2013

Acknowledgments: We thank the referee for his valuable comments. The first author was mainly supported by MITACS and NSERC. The CQT is funded by the Singapore MoE and the NRF as part of the Research Centres of Excellence programme. The second author was supported in part by an NSERC Discovery Grant.

Article outline:

I. INTRODUCTION

II. SOME SEMIALGEBRAIC SETS AND SEPARABILITY CONJECTURES

III. DIMENSION COMPUTATIONS FOR SOME QUANTUM SYSTEMS

IV. REAL SEPARABLE STATES

V. CONCLUSIONS

/content/aip/journal/jmp/54/2/10.1063/1.4790405

http://aip.metastore.ingenta.com/content/aip/journal/jmp/54/2/10.1063/1.4790405

Article metrics loading...

/content/aip/journal/jmp/54/2/10.1063/1.4790405

2013-02-06

2014-04-20

Full text loading...

### Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content