# An analytic approach to the problem of separability of quantum states based upon the theory of cones

An analytic approach to the problem of separability of quantum states based upon the theory of cones Exploiting the cone structure of the set of unnormalized mixed quantum states, we offer an approach to detect separability independently of the dimensions of the subsystems. We show that any mixed quantum state can be decomposed as ρ = (1−λ)C ρ  + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice $${C_{\rho}=M_{1}\otimes M_{2}}$$ we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ equals 1. We give a first extension of this result to detect genuine entanglement in multipartite states and show a natural connection between the multipartite separability problem and the classification of pure states under stochastic local operations and classical communication. We argue that this approach is not exhausted with the first simple choices included herein. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# An analytic approach to the problem of separability of quantum states based upon the theory of cones

Quantum Information Processing, Volume 10 (5) – Dec 25, 2010
19 pages

/lp/springer_journal/an-analytic-approach-to-the-problem-of-separability-of-quantum-states-T8iz5DUSJG
Publisher
Springer Journals
Subject
Physics; Mathematics, general; Quantum Physics; Physics, general; Theoretical, Mathematical and Computational Physics; Computer Science, general
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-010-0223-9
Publisher site
See Article on Publisher Site

### Abstract

Exploiting the cone structure of the set of unnormalized mixed quantum states, we offer an approach to detect separability independently of the dimensions of the subsystems. We show that any mixed quantum state can be decomposed as ρ = (1−λ)C ρ  + λE ρ , where C ρ is a separable matrix whose rank equals that of ρ and the rank of E ρ is strictly lower than that of ρ. With the simple choice $${C_{\rho}=M_{1}\otimes M_{2}}$$ we have a necessary condition of separability in terms of λ, which is also sufficient if the rank of E ρ equals 1. We give a first extension of this result to detect genuine entanglement in multipartite states and show a natural connection between the multipartite separability problem and the classification of pure states under stochastic local operations and classical communication. We argue that this approach is not exhausted with the first simple choices included herein.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Dec 25, 2010

### References

• Quantum information with Gaussian states
Wang, X.-B.; Hiroshima, T.; Tomita, A.; Hayashi, M.

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