On the separability problem for quantum composite systems
Matematičeskoe modelirovanie, Tome 26 (2014) no. 11, pp. 65-70.

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The present article addresses the so-called “quantum separability problem”, the mathematical issue that lies in foundations of quantum information and communication theory. The separability problem consist in elaboration of efficient computational algorithms for determination of whether a given state of a composite quantum system admits representation in a product form, with factors corresponding to each subsystem. The measurement theoretical aspects of this problem are discussed and the geometric probability of the mixed separable/entangled states in quantum systems composed from 2-qubits and qubit-qutrit pairs are computed.
Keywords: statistical measures.
Mots-clés : quantum information, entanglement, random matrices
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A. M. Khvedelidze; I. A. Rogojin. On the separability problem for quantum composite systems. Matematičeskoe modelirovanie, Tome 26 (2014) no. 11, pp. 65-70. http://geodesic.mathdoc.fr/item/MM_2014_26_11_a9/

[1] Schrodinger E., “Die gegenwartige situation in der quantenmechanik”, Naturwiss, 23 (1935), 807–812 | DOI

[2] Morozova E. A., Chentsov N. N., “Markov invariant geometry on manifolds of states”, Journal of Soviet Mathematics, 56:5 (1991), 2648–2669 | DOI | Zbl

[3] Dittmann J., “On the riemannian metric on the space of density matrices”, Reports on Mathematical Physics, 36:2 (1995), 309–315 | DOI | MR | Zbl

[4] Petz D., Sudar C., “Geometries of quantum states”, Journal of Mathematical Physics, 37 (1996), 2662–2673 | DOI | MR | Zbl

[5] Zyczkowski K., Horodecki P., Sanpera A., Lewenstein M., “On the volume of the set of mixed entangled states”, Phys. Rev., A58 (1998), 883 | DOI | MR

[6] Pittenger A. O., Rubin M. H., “Convexity and the separability problem of quantum mechanical density matrices”, Linear Algebra and its Applications, 346 (2002), 47–71 | DOI | MR | Zbl

[7] Gurvits L., “Classical deterministic complexity of Edmonds' problem and quantum entanglement”, STOC'03, ACM, New York, NY, USA, 2003, 10–19 | MR | Zbl

[8] Ioannou L. M., “Computational complexity of the quantum separability problem”, Quantum Info. Comput., 7 (2007), 335–370 | MR | Zbl

[9] Braunstein S. L., “Geometry of quantum inference”, Phys. Lett., A219 (1996), 189 | MR

[10] Physics A: Mathematical and General, 34 (2001), 7111–7125 | DOI | MR | Zbl

[11] Ginibre J., “Statistical ensembles of complex, quaternion, and real matrices”, Journal of Mathematical Physics, 6:3 (1965), 440–449 | DOI | MR | Zbl

[12] Bures D., “An extension of Kakutani's theorem on infinite product measures to the tensor product of semidefinite algebras”, Trans. Am. Mat. Soc., 135 (1969), 199 | MR | Zbl

[13] Braunstein S. L., Caves C. M., “Statistical distance and the geometry of quantum states”, Phys. Rev. Lett., 72 (1994), 3439–3443 | DOI | MR | Zbl

[14] Peres A., “Separability criterion for density matrices”, Phys. Rev. Lett., 77 (1996), 1413 | DOI | MR | Zbl

[15] Horodecki P., “Separability criterion and inseparable mixed states with positive partial transposition”, Phys. Lett., 232 (1977), 333 | DOI | MR

[16] Gerdt V., Khvedelidze A., Palii Y., “Constraints on $\mathrm{SU(2)}\otimes \mathrm{SU(2)}$ invariant polynomials for a pair of entangled qubits”, Physics of Atomic Nuclei, 74 (2011), 893–900 | DOI

[17] Gerdt V., Khvedelidze A., Mladenov D., Palii Y., “$\mathrm{SU(6)}$ casimir invariants and $\mathrm{SU(2)}\otimes \mathrm{SU(3)}$ scalars for a mixed qubit-qutrit state”, Journal of Mathematical Sciences, 179 (2011), 690–701 | DOI | MR

[18] Zyczkowski K., Sommers H. J., “Hilbert–Schmidt volume of the set of mixed quantum states”, Journal of Physics A: Mathematical and General, 36:39 (2003), 10115 | DOI | MR | Zbl

[19] Zyczkowski K., Sommers H. J., “Bures volume of the set of mixed quantum states”, Journal of Physics A: Mathematical and General, 36 (2003), 10083 | DOI | MR | Zbl

[20] Slater P. B., “Dyson indices and Hilbert–Schmidt separability functions and probabilities”, Journal of Physics A: Mathematical and General, 40 (2007), 14279 | DOI | MR | Zbl