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.


81P05 General and philosophical questions in quantum theory
82B10 Quantum equilibrium statistical mechanics (general)
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[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
[3] S. Popescu, Phys. Rev. Lett. 72 pp 797– (1994) · Zbl 0973.81506
[4] N. D. Mermin, in: Quantum Mechanics without Observer, (1996)
[5] N. Gisin, Phys. Lett. A 210 pp 151– (1996) · Zbl 1073.81512
[6] R. Horodecki, Phys. Lett. A 210 pp 377– (1996) · Zbl 1073.81517
[7] J. Blank, Acta Univ. Carolinae, Math. Phys. 18 pp 3– (1977)
[8] C. H. Bennett, Phys. Rev. Lett. 76 pp 722– (1996)
[9] R. Horodecki, Phys. Lett. A 200 pp 340– (1995) · Zbl 1020.81533
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