Geodesic
On the geometric probability of entangled mixed states
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 274-296 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

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 measurement-theoretical 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, based on the Peres–Horodecki positive partial transposition criterion, the measurement theoretical aspects of the separability problem are discussed for bipartite systems composed either of two qubits or of qubit-qutrit pairs. The necessary and sufficient conditions for the $2$-qubit state separability are formulated in terms of local $\mathrm{SU(2)\otimes SU(2)}$ invariant polynomials, the determinant of the correlation matrix, and the determinant of the Schlienz–Mahler matrix. Using the projective method of generation of random density matrices distributed according to the Hilbert–Schmidt or Bures measure, the separability (including the absolute separability) probabilities of $2$-qubit and qubit-qutrit pairs have been calculated. 
@article{ZNSL_2015_432_a14,
     author = {A. Khvedelidze and I. Rogojin},
     title = {On the geometric probability of entangled mixed states},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {274--296},
     year = {2015},
     volume = {432},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a14/}
}
TY  - JOUR
AU  - A. Khvedelidze
AU  - I. Rogojin
TI  - On the geometric probability of entangled mixed states
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2015
SP  - 274
EP  - 296
VL  - 432
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a14/
LA  - en
ID  - ZNSL_2015_432_a14
ER  - 
%0 Journal Article
%A A. Khvedelidze
%A I. Rogojin
%T On the geometric probability of entangled mixed states
%J Zapiski Nauchnykh Seminarov POMI
%D 2015
%P 274-296
%V 432
%U http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a14/
%G en
%F ZNSL_2015_432_a14
A. Khvedelidze; I. Rogojin. On the geometric probability of entangled mixed states. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 274-296. http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a14/

[1] E. Schrödinger, “Die gegenwärtige situation in der Quantenmechanik”, Die Naturwissenschaften, 23 (1935), 807–812, 823–828, 844–849 | DOI | DOI | DOI | Zbl

[2] R. F. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model”, Phys. Rev. A, 40 (1989), 4277–4281 | DOI

[3] W. Thirring, R. A. Bertlmann, P. Köhler, H. Narnhofer, “Entanglement or separability; the choice how to factorize the algebra of a density matrix”, Eur. Phys. D, 64 (2011), 181–196 | DOI

[4] L. Gurvits, “Classical deterministic complexity of Edmond's problem and quantum entanglement”, Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, ACM, New York, 2003, 10–19, (electronic) | MR

[5] L. M. Ioannou, “Computational complexity of the quantum separability problem”, Quantum Information Computation, 7:8 (2007), 335–370 ; arXiv: quant-ph/0603199v7 | MR | Zbl

[6] E. A. Morozova, N. N. Chentsov, “Markov invariant geometry on state manifolds”, Itogi Nauki Tehniki. Ser. Sovr. Probl. Matem. Noveish. Dostizh., 36, 1990, 69–102 | MR | Zbl

[7] D. Petz, C. Sudar, “Geometries of quantum states”, J. Math. Phys., 37 (1996), 2662–2673 | DOI | MR | Zbl

[8] K. Zyczkowki, P. Horodecki, A. Sanpera, M. Lewenstein, “Volume of the set of separable states”, Phys. Rev. A, 58 (1998), 883–892 | DOI | MR

[9] K. Zyczkowki, “Volume of the set of separable states. II”, Phys. Rev. A, 60 (1999), 3496–3507 | DOI | MR

[10] J. Schlienz, G. Mahler, “Description of entanglement”, Phys. Rev. A, 52 (1995), 4396–4404 | DOI

[11] J. von Neumann, “Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik”, Göttingen Nachrichten, 1927 (1927), 245–272 | Zbl

[12] L. D. Landau, “Das dämpfungsproblem in der Wellenmechanik”, Zeitschrift für Physik, 45 (1927), 430–441 | DOI | Zbl

[13] J. Dittmann, “On the Riemannian metrics on the space of density matrices”, Rep. Math. Phys., 36 (1995), 309–315 | DOI | MR | Zbl

[14] D. P. Zelobenko, Compact Lie groups and their representations, Translations of Mathematical Monographs, 40, AMS, 1978 | MR

[15] S. M. Deen, P. K. Kabir, G. Karl, “Positivity constraints on density matrices”, Phys. Rev. D, 4 (1971), 1662–1666 | DOI

[16] V. Gerdt, A. Khvedelidze, Yu. Palii, “Constraints on $\mathrm{SU(2)\otimes SU(2)}$ invariant polynomials for entangled qubit pairs”, Yad. Fiz., 74:6 (2001), 919–955

[17] Lin Chen, Dragomir Z. Dokovi, “Dimensions, lengths and separability in finite-dimensional quantum systems”, J. Math. Phys., 54 (2013), 022201, 13 pp. | DOI | MR | Zbl

[18] U. Fano, “Description of states in quantum mechanics by density matrix and operator techniques”, Rev. Mod. Phys., 29 (1957), 74–93 | DOI | MR | Zbl

[19] A. Peres, “Separability criterion for density matrices”, Phys. Rev. Lett., 77 (1996), 1413–1415 | DOI | MR | Zbl

[20] M. Horodecki, P. Horodecki, R. Horodecki, “Separability of mixed states: necessary and sufficient conditions”, Phys. Lett. A, 223 (1996), 1–8 | DOI | MR | Zbl

[21] P. Horodecki, “Separability criterion and inseparable mixed states with positive partial transpose”, Phys. Lett., 232 (1977), 333–339 | DOI | MR

[22] N. Linden, S. Popescu, “On multi-particle entanglement”, Fortschr. Phys., 46 (1998), 567–578 | 3.0.CO;2-H class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[23] M. Grassl, M. Rötteler, T. Beth, “Computing local invariants of qubit systems”, Phys. Rev. A, 58 (1998), 1833–1859 | DOI | MR

[24] R. C. King, T. A. Welsh, P. D. Jarvis, “The mixed two-qubit system and the structure of its ring of local invariants”, J. Phys. A: Math. Theor., 40 (2007), 10083–10108 | DOI | MR | Zbl

[25] V. Gerdt, Yu. Palii, A. Khvedelidze, “On the ring of local invariants for a pair of entangled qubits”, J. Math. Sci., 168:3 (2010), 368–378 | DOI | MR | Zbl

[26] V. Gerdt, A. Khvedelidze, D. Mladenov, Yu. Palii, “$SU(6)$ Casimir invariants and $\mathrm{SU(2)\otimes SU(3)}$ scalars for a mixed qubit-qutrit states”, Zap. Nauchn. Semin. POMI, 387, 2001, 102–121 | MR

[27] C. Quesne, “$SU(2)\otimes SU(2)$ scalars in the enveloping algebra of $SU(4)$”, J. Math. Phys., 17 (1976), 1452–1467 | DOI | MR

[28] M. Kus, K. Źyczkowski, “Geometry of entangled states”, Phys. Rev. A, 63 (2001), 032307, 21 pp. | DOI | MR

[29] F. Verstraete, K. Audenaert, B. D. Moor, “Maximally entangled mixed states of two qubits”, Phys. Rev., 64 (2001), 012316, 6 pp. | DOI

[30] R. Hildebrand, “Positive partial transpose spectra”, Phys. Rev. A, 76 (2007), 052325, 5 pp. | DOI | MR

[31] E. Lubkin, “Entropy of an $n$-system from its correlation with a $k$-reservoir”, J. Math. Phys., 19 (1978), 1028–1031 | DOI | Zbl

[32] S. Lloyd, H. Pagels, “Complexity as thermodynamic depth”, Ann. Phys. NY, 188 (1988), 186–213 | DOI | MR

[33] D. N. Page, “Average entropy of a subsystem”, Phys. Rev. Lett., 71 (1993), 1291–1294 | DOI | MR | Zbl

[34] D. J. C. Bures, “An extension of Kakutani's theorem on infinite product measures to the tensor product of semidefinite $\omega^*$-algebras”, Trans. Am. Mat. Soc., 135 (1969), 199–212 | MR | Zbl

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

[36] H.-J. Sommers, K. Zyczkowski, “Bures volume of the set of mixed quantum states”, J. Phys. A: Math. Theor., 36 (2003), 10083–10100 | DOI | MR | Zbl

[37] A. Uhlmann, “Density operators as an arena for differential geometry”, Rep. Math. Phys., 33 (1993), 253–263 | DOI | MR | Zbl

[38] M. Hübner, “Explicit computation of the Bures distance for density matrices”, Phys. Lett. A, 163 (1992), 239–242 | DOI | MR

[39] M. J. Hall, “Random quantum correlations and density operator distributions”, Phys. Lett. A, 242 (1998), 123–129 | DOI | MR | Zbl

[40] K. Zyczkowski, H.-J. Sommers, “Hilbert–Schmidt volume of the set of mixed quantum states”, J. Phys. A: Math. Theor., 36 (2003), 10115–10130 | DOI | MR | Zbl

[41] S. L. Braunstein, “Geometry of quantum inference”, Phys. Lett. A, 219 (1966), 169–174 | DOI | MR

[42] K. Zyczkowski, H.-J. Sommers, “Induced measures in the space of mixed states”, J. Phys. A: Math. Theor., 34 (2001), 7111–7125 | DOI | MR | Zbl

[43] V. Al. Osipov, H.-J. Sommeres, K. Zyczkowski, “Random Bures mixed states and the distribution of their purity”, J. Phys. A: Math. Theor., 43 (2010), 055302, 22 pp. | DOI | MR | Zbl

[44] J. Ginibre, “Statistical ensembles of complex, quaternion, and real matrices”, J. Math. Phys., 6 (1965), 440–339 | DOI | MR

[45] P. B. Slater, “Dyson indices and Hilbert–Schmidt separability functions and probabilities”, J. Phys. A: Math. Theor., 40 (2007), 14279–14308 | DOI | MR | Zbl

[46] P. B. Slater, “Eigenvalues, separability and absolute separability of two-qubit states”, J. Geom. Phys., 59 (2009), 17–31 | DOI | MR | Zbl

[47] J. Batle, M. Casas, A. Plastino, A. R. Plastino, “Metrics, entanglement, and mixedness in the space of two-qubits”, Phys. Lett. A, 353 (2006), 161–165 | DOI | MR | Zbl