Quantum Information Processing

, Volume 10, Issue 5, pp 633–651 | Cite as

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

  • D. Salgado
  • J. L. Sánchez-Gómez
  • M. Ferrero


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.


Entanglement Separability Cone Bipartite Multipartite 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • D. Salgado
    • 1
    • 2
  • J. L. Sánchez-Gómez
    • 3
  • M. Ferrero
    • 4
  1. 1.D.G. Metodología, Calidad y Tecnologías de la Información y las ComunicacionesInsto. Nacional de EstadísticaMadridSpain
  2. 2.Dpto. Ingeniería InformáticaUniversidad Antonio de NebrijaMadridSpain
  3. 3.Dpto. Física TeóricaUniversidad Autónoma de MadridMadridSpain
  4. 4.Dpto. FísicaUniversidad de OviedoOviedoSpain

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