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.
Similar content being viewed by others
References
Schrödinger E.: Discussion of probability relations between separated systems. Proc. Cambridge Phil. Soc. 31, 555 (1935)
Gisin N.: Quantum nonlocality: how does nature do it?. Science 326, 1357 (2009)
Wang X.-B., Hiroshima T., Tomita A., Hayashi M.: Quantum information with Gaussian states. Phys. Rep. 448, 1 (2007)
Nielsen M., Chuang I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)
Woronowicz S.: Positive maps of low dimensional matrix algebras. Rep. Math. Phys. 10, 165 (1976)
Peres A.: Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)
Horodecki M., Horodecki P., Horodecki R.: Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 8 (1996)
Horodecki P.: Separability criterion and inseparable mixed states with positive partial transposition. Phys. Lett. A 232, 333 (1997)
Rudolph O.: Further results on the cross norm criterion for separability. Quant. Inf. Process 4, 219 (2005)
Chen K., Wu L.-A.: A matrix realignment method for recognizing entanglement. Quant. Inf. Comp. 3, 193 (2003)
Lewenstein M., Bruss D., Cirac J., Kraus B., Samsonowicz J., Sanpera A., Tarrach R.: Separability and distillability in composite quantum systems-a primer. J. Mod. Opt. 47, 2841 (2000a)
Plenio M., Virmani S.: An introduction to entanglement measures. Quant. Inf. Comp. 7, 1 (2007)
Horodecki R., Horodecki P., Horodecki M., Horodecki K.: Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)
Gühne O., Tóth G.: Entanglement detection. Phys. Rep. 474, 1 (2009)
Doherty A., Parrilo P., Spedalieri M.: Distinguishing separable and entangled states. Phys. Rev. Lett. 88, 187904 (2002)
Doherty A., Parrilo P., Spedalieri F.: A complete family of separability criteria. Phys. Rev. A 69, 022308 (2004)
Pérez-García D.: Deciding separability with a fixed error. Phys. Lett. A 330, 149 (2004)
Terhal B.: Bell inequalities and the separability criterion. Phys. Lett. A 271, 319 (2000)
Lewenstein M., Kraus B., Cirac J.I., Horodecki P.: Optimization of entanglement witnesses. Phys. Rev. A 62, 052310 (2000)
Bruss D., Cirac J.I., Horodecki P., Hulpke F., Kraus B., Lewenstein M., Sanpera A.: Reflections upon separability and distillability. J. Mod. Opt. 49, 1399 (2002)
Amico L., Fazio R., Osterloh A., Vedral V.: Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)
Hill R., Waters S.: On the cone of positive semidefinite matrices. Lin. Alg. Appl. 90, 81 (1987)
Lewenstein M., Sanpera A.: Separability and entanglement of composite quantum systems. Phys. Rev. Lett. 80, 2261 (1998)
Karnas S., Lewenstein M.: Separable approximations of density matrices of composite quantum systems. J. Phys. A Math. Gen. 34, 6919 (2001)
Dür W., Vidal G., Cirac J.I.: Three qubits can be entangled in two inequivalent ways. Phys. Rev. A 62, 062314 (2000)
Lamata L., León J., Salgado D., Solano E.: Inductive classification of multipartite entanglement under SLOCC. Phys. Rev. A 74, 052336 (2006)
Werner R.: Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. Phys. Rev. A 40, 4277 (1989)
Barker G.: Theory of cones. Lin. Alg. Appl. 39, 263 (1981)
Gühne O., Lütkenhaus N.: Nonlinear entanglement witnesses, covariance matrices and the geometry of separable states. J. Phys. C Conf. Ser. 67, 012004 (2007)
Gurvits, L.: Classical deterministic complexity of Edmonds’ problem and quantum entanglement. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pp. 10–19. See also quant-ph/0303055 (2003)
Horn R., Johnson C.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)
Pittenger A., Rubin M.: Note on separability of the Werner states in arbitrary dimensions. Opt. Comm. 179, 447 (2000)
Deng D.-L., Chen J.-L.: Sufficient and necessary condition of separability for generalized Werner states. Ann. Phys. 324, 408 (2008)
Eisert, J., Gross, D.: Multiparticle entanglement. In: Bruss, D., Leuchs, G. Lectures on Quantum Information., Wiley-VCH, Weinheim (2006)
Greenberger D., Horne M., Shimony A., Zeilinger A.: Bell’s theorem without inequalities. Am. J. Phys. 58, 1131 (1990)
Bastin T., Krins S., Mathonet P., Godefroid M., Lamata L., Solano E.: Operational families of entanglement classes for symmetric N-qubit states. Phys. Rev. Lett. 103, 070503 (2009)
van Loan C.F.: Generalizing the singular value decomposition. SIAM J. Numer. Anal. 13, 76 (1976)
Churchill R., Brown J.: Complex Variables and Its Applications. McGraw Hill, New York (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Salgado, D., Sánchez-Gómez, J.L. & Ferrero, M. An analytic approach to the problem of separability of quantum states based upon the theory of cones. Quantum Inf Process 10, 633–651 (2011). https://doi.org/10.1007/s11128-010-0223-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11128-010-0223-9