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Warianty tytułu
Języki publikacji
Abstrakty
In this work we provide a method for approximating the separable numerical range of a matrix. We also recall the connection between restricted numerical range and entanglement of a quantum state. We show the possibility to establish state separability using computed restricted numerical range. In particular we present a method to obtain separability criteria for arbitrary system partition with use of the separable numerical range.
W pracy przedstawiamy metodę przybliżania separowalnego obrazu numerycznego macierzy. Przybliżamy również związek pomiędzy ograniczonymi obrazami numerycznymi macierzy oraz splataniem stanów kwantowych. Wskazujemy możliwość zbadania separowalności stanu używając wyliczonego ograniczonego obrazu numerycznego. W szczególności przedstawiamy metodę uzyskania kryterium separowalności, dla dowolnego podziału układów kwantowych, używając separowalnego obrazu numerycznego.
Czasopismo
Rocznik
Tom
Strony
149--158
Opis fizyczny
Bibliogr. 27 poz., rys.
Twórcy
autor
- Institute of Theoretical and Applied Informatics, Polish Academy of Sciences Bałtycka 5, 44-100 Gliwice, Poland
autor
- Institute of Theoretical and Applied Informatics, Polish Academy of Sciences Bałtycka 5, 44-100 Gliwice, Poland
Bibliografia
- [1] C. H. Bennett and G. Brassard. Quantum cryptography: Public key distribution and coin tossing. In Proceedings of IEEE International Conference on Computers, Systems and Signal Processing, volume 175. Bangalore, India, 1984.
- [2] R. P. Feynman. Simulating physics with computers. International journal of theoretical physics, 21(6):467–488, 1982.
- [3] D. Deutsch. Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 400(1818):97–117, 1985.
- [4] P. W. Shor. Algorithms for quantum computation: discrete logarithms and factoring. In Foundations of Computer Science, 1994 Proceedings., 35th Annual Symposium on, pages 124–134. IEEE, 1994.
- [5] L. K. Grover. A fast quantum mechanical algorithm for database search. In Proceedings of the twenty-eighth annual ACM symposium on Theory of computing, pages 212–219. ACM, 1996.
- [6] I. Bengtsson and K. ˙ Zyczkowski. Geometry of quantum states: an introduction to quantum entanglement. Cambridge University Press, 2006.
- [7] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki. Quantum entanglement. Reviews of Modern Physics, 81(2):865, 2009.
- [8] R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1994.
- [9] K. E. Gustafson and D. K. M. Rao. Numerical Range: The Field of Values of Linear Operators and Matrices. Springer-Verlag, New York, 1997.
- [10] T. Ando and C. K. Li. Special issue: The numerical range and numerical radius. Linear and Multilinear Algebra, 37(1–3), 1994. Ando, T. and Li, C. K., editors.
- [11] C. K. Li. C-numerical ranges and c-numerical radii. Linear and Multilinear Algebra, 37(1–3):51–82, 1994.
- [12] D. W. Kribs, A. Pasieka, M. Laforest, C. Ryan, and M. P. Silva. Research problems on numerical ranges in quantum computing. Linear and Multilinear Algebra, 57:491–502, 2009.
- [13] M. Marcus. Finite Dimensional Multilinear Algebra. Part I. Marcel Dekker, New York, U.S.A., 1973.
- [14] M. Marcus and B. Wang. Some variations on the numerical range. Linear and Multilinear Algebra, 9:111–120, 1980.
- [15] N. Bebiano, C. K. Li, and J. da Providencia. The numerical range and decomposable numerical range of matrices. Linear and Multilinear Algebra, 29:195–205, 1991.
- [16] C. K. Li and A. Zaharia. Induced operators on symmetry classes of tensors. Trans. Am. Math. Soc., pages 807–836, 2001.
- [17] Z. Puchała, P. Gawron, J.A. Miszczak, Ł. Skowronek, M .D. Choi, and K. ˙ Zyczkowski. Product numerical range in a space with tensor product structure. Linear Algebra and its Applications, 434(1):327–342, 2011.
- [18] P. Gawron, Z. Puchała, J.A. Miszczak, Ł. Skowronek, and K. ˙ Zyczkowski. Restricted numerical range: A versatile tool in the theory of quantum information. Journal of Mathematical Physics, 51:102204, 2010.
- [19] C. F. Dunkl, P. Gawron, J. A. Holbrook, J. A. Miszczak, Z. Puchała, and K. ˙ Zyczkowski. Numerical shadow and geometry of quantum states. Journal of Physics A: Mathematical and Theoretical, 44:335301, 2011.
- [20] C.F. Dunkl, P. Gawron, J.A. Holbrook, Z. Puchala, and K. ˙ Zyczkowski. Numerical shadows: measures and densities on the numerical range. Linear Algebra and its Applications, 2011.
- [21] Z. Puchała, J. A. Miszczak, P. Gawron, C. F. Dunkl, J. A. Holbrook, and K. ˙ Zyczkowski. Restricted numerical shadow and geometry of quantum entanglement. Journal of Physics A: Mathematical and Theoretical, 45:415309, 2012.
- [22] T.-C.Wei and P. M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Physical Review A, 68(4):042307, 2003.
- [23] T. Schulte-Herbrüggen, G. Dirr, U. Helmke, and S. J. Glaser. The significance of the C-numerical range and the local c-numerical range in quantum control and quantum information. Linear and Multilinear Algebra, 56(1-2):3–26, 2008.
- [24] C.-K. Li and Y.-T. Poon. Quantum error correction and generalized numerical ranges. arXiv preprint arXiv:0812.4772, 2008.
- [25] Ł. Rudnicki, Z. Puchała, P. Horodecki, and K. ˙ Zyczkowski. Constructive entanglement test from triangle inequality. Journal of Physics A: Mathematical and Theoretical, 47(42):424035, 2014.
- [26] T. Radtke and S. Fritzsche. Simulation of n-qubit quantum systems. iv. parametrizations of quantum states, matrices and probability distributions. Computer Physics Communications, 179(9):647–664, Nov 2008.
- [27] B. Collins, P. Gawron, A. E. Litvak, and K. ˙ Zyczkowski. Numerical range for random matrices. Journal of Mathematical Analysis and Applications, 418(1):516–533, 2014.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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