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Abstrakty
We analyze the facial structures of the convex set consisting of all two-qubit separable states. One of the faces is a four-dimensional convex body generated by the trigonometric moment curve arising from polyhedral combinatorics. Another one is an eight-dimensional convex body, which is the convex hull of a homeomorphic image of the two-dimensional sphere. Extreme points consist of points on the surface, and any two of tchem determine an edge. We also reconstruct the trigonometric moment curve in any even-dimensional affine space using the qubit-qudit systems, and characterize the facial structures of the convex hull.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Strony
385--400
Opis fizyczny
Bibliogr. 24 poz., rys.
Twórcy
autor
- Department of Mathematics, and Institute of Mathematics, Seoul National University, Seoul 151-742, Korea
Bibliografia
- [1] E. Alfsen and F. Shultz, Unique decompositions, faces, and automorphisms of separable states, J. Math. Phys. 51 (2010), 052201.
- [2] E. Alfsen and F. Shultz, Finding decompositions of a class of separable states, Linear Algebra Appl. 437 (2012), pp. 2613-2629.
- [3] A. I. Barvinok and I. Novik, A centrally symmetric version of the cyclic polytope, Discrete Comput. Geom. 39 (2008), pp. 76-99.
- [4] E.-S. Byeon and S.-H. Kye, Facial structures for positive linear maps in the two dimensional matrix algebra, Positivity 6 (2002), pp. 369-380.
- [5] L. Chen and D. Ž. Djoković, Qubit-qudit states with positive partial transpose, Phys. Rev. A 86 (2012), 062332.
- [6] L. Chen and D. Ž. Djoković, Dimensions, lengths and separability in finite-dimensional quantum systems, preprint. arXiv:1206.3775.
- [7] H.-S. Choi and S.-H. Kye, Facial structures for separable states, J. Korean Math. Soc. 49 (2012), pp. 623-639.
- [8] M.-D. Choi, Positive linear maps, in: Operator Algebras and Applications (Kingston, 1980), Proc. Sympos. Pure Math., Vol. 38. Part 2, Amer. Math. Soc., 1982, pp. 583-590.
- [9] D. Gale, Neighborly and cyclic polytopes, in: Covexity (Seattle, 1961), Proc. Sympos. Pure Math., Vol. 7, Amer. Math. Soc., 1963, pp. 225-232.
- [10] K.-C. Ha and S.-H. Kye, Construction of entangled states with positive partial transposes based on indecomposable positive linear maps, Phys. Lett. A 325 (2004), pp. 315-323.
- [11] K.-C. Ha and S.-H. Kye, Separable states with unique decompositions, preprint. arXiv:1210.1088.
- [12] K.-C. Ha and S.-H. Kye, Exposedness of Choi type entanglement witnesses and applications to lengths of separable states, preprint. arXiv:1211.5675.
- [13] M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett. A 223 (1996), pp. 1-8.
- [14] S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Pure Appl. Math., Vol. 15, Interscience Publishers, 1966.
- [15] Y.-H. Kiem, S.-H. Kye, and J. Lee, Existence of product vectors and their partial conjugates in a pair of spaces, J. Math. Phys. 52 (2011), 122201.
- [16] K. A. Kirkpatrick, Uniqueness of a convex sum of products of projectors, J. Math. Phys. 43 (2002), pp. 684-686.
- [17] B. Kraus, J. I. Cirac, S. Karnas, and M. Lewenstein, Separability in 2 ×N composite quantum systems, Phys. Rev. A 61 (2000), 062302.
- [18] S.-H. Kye, Facial structures for positive linear maps between matrix algebras, Canad. Math. Bull. 39 (1996), pp. 74-82.
- [19] A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77 (1996), pp. 1413-1415.
- [20] R. Puente, Cyclic convex bodies and optimization moment problems, Linear Algebra Appl. 426 (2007), pp. 596-609.
- [21] R. Sanyal, F. Sottile, and B. Sturmfels, Orbitopes, Mathematika 57 (2011), pp. 275-314.
- [22] Z. Smilansky, Convex hulls of generalized moment curves, Israel J. Math. 52 (1985), pp. 115-128.
- [23] C. Vinzant, Edges of the Barvinok-Novik orbitope, Discrete Comput. Geom. 46 (2011), pp. 479-487.
- [24] S. L. Woronowicz, Positive maps of low dimensional matrix algebras, Rep. Math. Phys. 10 (1976), pp. 165-183.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
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bwmeta1.element.baztech-2dd43fdd-edb7-4e2b-9f0b-2cf8fa6ae041