Abstract
We describe a computable analytical criterion for separability of bipartite mixed states in arbitrary dimension. The criterion stipulates that a certain norm on the state space (the computable cross-norm) is bounded by 1 for separable states. The criterion is shown to be independent of the well-known positive partial transpose (PPT) criterion. In other words, the criterion detects some bound entangled states but fails for some free entangled states.
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References
Bruβ, D.: Characterizing entanglement, J.Math.Phys. 43 (2002), 4237–4251.
Chen, K., Wu, L.-A. and Yang, L.: A matrix realignment method for recognizing entanglement, quant-ph/0205017v1; published as: Chen, K. and Wu, L.-A.: A matrix realignment method for recognizing entanglement, Quantum.Inf.Comp. 3 (2003), 193, quant-ph/0205017v5.
Donald, M. J., Horodecki, M. and Rudolph, O.: The uniqueness theorem for entanglement measures, J.Math.Phys. 43 (2002), 4252–4272.
Ekert, A. and Knight, P. L.: Entangled quantum systems and the Schmidt decomposition, Amer.J.Phys. 63 (1995), 415–423.
Horodecki, M., Horodecki, P. and Horodecki, R.: Separability of mixed states: Necessary and sufficient conditions, Phys.Lett.A 223 (1996), 1–8.
Horodecki, M., Horodecki, P. and Horodecki, R.: Separability of mixed quantum states: Linear contractions approach, quant-ph/0206008.
Horodecki, P.: Direct estimation of elements of quantum states algebra and entanglement detection via linear contractions, Phys.Lett.A 319 (2003), 1.
Horodecki, P., Horodecki, M. and Horodecki, R.: Mixed-state entanglement and distillation: Is there a bound entanglement in nature? Phys.Rev.Lett. 80 (1998), 5239.
Horodecki, P., Horodecki, M. and Horodecki, R.: Bound entanglement can be activated, Phys.Rev.Lett. 82 (1998), 1056.
Kadison, R. V. and Ringrose, J. R.: Fundamentals of the Theory of Operator Algebras, I & II, Academic Press, Orlando, 1983, 1986.
Peres, A.: Separability criterion for density matrices, Phys.Rev.Lett. 77 (1996), 1413–1415.
Rudolph, O.: A separability criterion for density operators, J.Phys.A Math.Gen. 33 (2000), 3951–3955.
Rudolph, O.: Further results on the cross-norm criterion for separability, quant-ph/0202121.
Rudolph, O.: Some properties of the computable cross-norm criterion for separability, Phys.Rev.A 67 (2003), 032312.
Schatten, R.: Norm Ideals of Completely Continuous Operators, 2nd edn, Springer, Berlin, 1970.
Werner, R. F.: Quantum States with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model, Phys.Rev.A 40 (1989), 4277–4281.
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Rudolph, O. Computable Cross-norm Criterion for Separability. Letters in Mathematical Physics 70, 57–64 (2004). https://doi.org/10.1007/s11005-004-0767-7
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DOI: https://doi.org/10.1007/s11005-004-0767-7