The state space of a composite quantum system, the set of density matrices \( \mathfrak{P} \) +, is decomposable into the space of separable states \( \mathfrak{S} \) + and its complement, the space of entangled states. An explicit construction of such a decomposition constitutes the so-called separability problem. If the space \( \mathfrak{P} \) + is endowed with a certain Riemannian metric, then the separability problem admits a measuretheoretic formulation. In particular, one can define the “geometric probability of separability” as the relative volume of the space of separable states \( \mathfrak{S} \) + with respect to the volume of all states. In the present note, using the Peres–Horodecki positive partial transposition criterion, we discuss the measure-theoretic aspects of the separability problem for bipartite systems composed either of two qubits or of a qubit-qutrit pair. Necessary and sufficient conditions for the separability of a two-qubit state are formulated in terms of local SU(2) ⊗ SU(2) invariant polynomials, the determinant of the correlation matrix, and the determinant of the Schlienz–Mahler matrix. Using the projective method of generating random density matrices distributed according to the Hilbert–Schmidt or Bures measure, we calculate the probability of separability (including that of absolute separability) of a two-qubit and qubit-qutrit pair. Bibliograhpy: 47 titles.
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References
E. Schrödinger, “Die gegenw¨artige Situation in der Quantenmechanik,” Die Naturwissenschaften, 23, 807–812, 823–828, 844–849 (1935).
R. F. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Phys. Rev. A, 40, 4277–4281 (1989).
W. Thirring, R. A. Bertlmann, P. Köhler, and H. Narnhofer, “Entanglement or separability: the choice of how to factorize the algebra of a density matrix,” Eur. Phys. D, 64, 181–196 (2011).
L. Gurvits, “Classical deterministic complexity of Edmond’s problem and quantum entanglement,” in: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 1019 (electronic), ACM, New York (2003).
L. M. Ioannou, “Computational complexity of the quantum separability problem,” Quantum Inf. Comput., 7, No. 8 (2007), 335–370; arXiv:quant-ph/0603199v7.
E. A. Morozova and N. N. Chentsov, “Markov invariant geometry on state manifolds,” Itogi Nauki Tekhniki, 36, 69–102 (1990).
D. Petz and C. Sudar, “Geometries of quantum states,” J. Math. Phys., 37, 2662–2673 (1996).
K. Zyczkowki, P. Horodecki, A. Sanpera, and M. Lewenstein, “Volume of the set of separable states,” Phys. Rev. A, 58, 883–892 (1998).
K. Zyczkowki, “Volume of the set of separable states. II,” Phys. Rev. A, 60, 3496–3507 (1999).
J. Schlienz and G. Mahler, “Description of entanglement,” Phys. Rev. A, 52, 4396–4404 (1995).
J. von Neumann, “Warscheinlichtkeitstheoretischer Aufbau der Quantemechanik,” Nachrichten Göttingen, 1927, 245–272 (1927).
L. D. Landau, “Das D¨ampfungsproblem in der Wellenmechanik,” Z. Physik, 45, 430–441 (1927).
J. Dittmann, “On the Riemannian metrics on the space of density matrices,” Rep. Math. Phys., 36, 309–315 (1995).
D. P. Zelobenko, Compact Lie Groups and Their Representations, Amer. Math. Soc., Providence, Rhode Island (1978).
S. M. Deen, P. K. Kabir, and G. Karl, “Positivity constraints on density matrices,” Phys. Rev. D, 4, 1662–1666 (1971).
V. Gerdt, A. Khvedelidze, and Yu. Palii, “Constraints on SU(2) ⊗ SU(2) invariant polynomials for entangled qubit pairs,” Yad. Fiz., 74, No. 6, 919–955 (2001).
L. Chen and D. Z. Dokovic, “Dimensions, lengths and separability in finite-dimensional quantum systems,” J. Math. Phys., 54, 022201 (2013).
U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques,” Rev. Mod. Phys., 29, 74–93 (1957).
A. Peres, “Separability criterion for density matrices,” Phys. Rev. Lett., 77, 1413–1415 (1996).
M. Horodecki, P. Horodecki, and R. Horodecki, “Separability of mixed states: necessary and sufficient conditions,” Phys. Lett. A, 223, 1–8 (1996).
P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transpose,” Phys. Lett., 232, 333–339 (1977).
N. Linden and S. Popescu, “On multi-particle entanglement,” Fortschr. Phys., 46, 567–578 (1998).
M. Grassl, M. Rötteler, and T. Beth, “Computing local invariants of qubit systems,” Phys. Rev. A, 58, 1833–1859 (1998).
R. C. King, T. A. Welsh, and P. D. Jarvis, “The mixed two-qubit system and the structure of its ring of local invariants,” J. Phys. A, 40, 10083–10108 (2007).
V. Gerdt, Yu. Palii, and A. Khvedelidze, “On the ring of local invariants for a pair of entangled qubits,” J. Math. Sci., 168, 368–378 (2010).
V. Gerdt, A. Khvedelidze, D. Mladenov, and Yu. Palii, “SU(6) Casimir invariants and SU(2) ⊗ SU(3) scalars for a mixed qubit-qutrit states,” Zap. Nauchn. Semin. POMI, 387, 102–121 (2001).
C. Quesne, “SU(2)⊗ SU(2) scalars in the enveloping algebra of SU(4),” J. Math. Phys., 17, 1452–1467 (1976).
M. Kuś and K. Zyczkowski, “Geometry of entangled states,” Phys. Rev. A, 63, 032307 (2001).
F. Verstraete, K. Audenaert, and B. D. Moor, “Maximally entangled mixed states of two qubits,” Phys. Rev. A, 64, 012316 (2001).
R. Hildebrand, “Positive partial transpose spectra,” Phys. Rev. A, 76, 052325 (2007).
E. Lubkin, “Entropy of an n-system from its correlation with a k-reservoir,” J. Math. Phys., 19, 1028–1031 (1978).
S. Lloyd and H. Pagels, “Complexity as thermodynamic depth,” Ann. Phys., 188, 186–213 (1998).
D. N. Page, “Average entropy of a subsystem,” Phys. Rev. Lett., 71, 1291–1294 (1993).
D. J. C. Bures, “An extension of Kakutani’s theorem on infinite product measures to the tensor product of semidefinite ω*-algebras,” Trans. Amer. Math. Soc., 135, 199–212 (1969).
S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett., 72, 3439–3443 (1994). 1003
H.-J. Sommers and K. Zyczkowski, “Bures volume of the set of mixed quantum states,” J. Phys. A, 36, 10083–10100 (2003).
A. Uhlmann, “Density operators as an arena for differential geometry,” Rep. Math. Phys., 33, 253–263 (1993).
M. Hübner, “Explicit computation of the Bures distance for density matrices,” Phys. Lett. A, 163, 239–242 (1992).
M. J. Hall, “Random quantum correlations and density operator distributions,” Phys. Lett. A, 242, 123–129 (1998).
K. Zyczkowski and H.-J. Sommers, “Hilbert–Schmidt volume of the set of mixed quantum states,” J. Phys. A, 36, 10115–10130 (2003).
S. L. Braunstein, “Geometry of quantum inference,” Phys. Lett. A, 219, 169–174 (1966).
K. Zyczkowski and H.-J. Sommers, “Induced measures in the space of mixed states,” J. Phys. A, 34, 7111–7125 (2001).
V. A. Osipov, H.-J. Sommeres, and K. Zyczkowski, “Random Bures mixed states and the distribution of their purity,” J. Phys. A, 43, 055302 (2010).
J. Ginibre, “Statistical ensembles of complex, quaternion, and real matrices,” J. Math. Phys., 6, 440–339 (1965).
P. B. Slater, “Dyson indices and Hilbert–Schmidt separability functions and probabilities,” J. Phys. A, 40, 14279–14308 (2007).
P. B. Slater, “Eigenvalues, separability and absolute separability of two-qubit states,” J. Geom. Phys., 59, 17–31 (2009).
J. Batle, M. Casas, A. Plastino, and A. R. Plastino, “Metrics, entanglement, and mixedness in the space of two-qubits,” Phys. Lett. A, 353, 161–165 (2006).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 432, 2015, pp. 274–296.
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Khvedelidze, A., Rogojin, I. On the Geometric Probability of Entangled Mixed States. J Math Sci 209, 988–1004 (2015). https://doi.org/10.1007/s10958-015-2542-y
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DOI: https://doi.org/10.1007/s10958-015-2542-y