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Separability criterion for density matrices. (English) Zbl 0947.81003
Summary: A quantum system consisting of two subsystems is separable if its density matrix can be written as $$\rho=\sum_A w_A\rho'_A\otimes\rho"_A$$, where $$\rho'_A$$ and $$\rho"_A$$ are density matrices for the two subsystems, and the positive weights $$w_A$$ satisfy $$\sum w_A=1$$. In this letter, we prove that a necessary condition for separability is that a matrix, obtained by partial transposition of $$\rho$$, has only non-negative eigenvalues. Some examples show that this criterion is more sensitive than Bell’s inequality for detecting quantum inseparability.

##### MSC:
 81P05 General and philosophical questions in quantum theory 82B10 Quantum equilibrium statistical mechanics (general)
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##### References:
 [1] A. Peres, in: Quantum Theory: Concepts and Methods (1993) · Zbl 0820.00011 [2] R. F. Werner, Phys. Rev. A 40 pp 4277– (1989) · Zbl 1371.81145 · doi:10.1103/PhysRevA.40.4277 [3] S. Popescu, Phys. Rev. Lett. 72 pp 797– (1994) · Zbl 0973.81506 · doi:10.1103/PhysRevLett.72.797 [4] N. D. Mermin, in: Quantum Mechanics without Observer, (1996) [5] N. Gisin, Phys. Lett. A 210 pp 151– (1996) · Zbl 1073.81512 · doi:10.1016/S0375-9601(96)80001-6 [6] R. Horodecki, Phys. Lett. A 210 pp 377– (1996) · Zbl 1073.81517 · doi:10.1016/0375-9601(95)00930-2 [7] J. Blank, Acta Univ. Carolinae, Math. Phys. 18 pp 3– (1977) [8] C. H. Bennett, Phys. Rev. Lett. 76 pp 722– (1996) · doi:10.1103/PhysRevLett.76.722 [9] R. Horodecki, Phys. Lett. A 200 pp 340– (1995) · Zbl 1020.81533 · doi:10.1016/0375-9601(95)00214-N
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