Numerical Shadow

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Numerical range

General

1. [1] 2. [2] 3. [3] 4. [4] 5. [5] 6. [6] 7. [7] 8. [8] 9. [9] 10. [10] 11. [11] 12. [12] 13. [13] 14. [14] 15. [15] 16. [16] 17. [17] 18. [18] 19. [19] 20. [20] 21. [21] 22. [22]

1. O. Toeplitz, 1918. Das algebraische Analogon zu einem Satze von Fejer. Mathematische Zeitschrift, 2, Springer, pp.187–197.
2. F. Hausdorff, 1919. Der Wertevorrat einer Bilinearform. Mathematische Zeitschrift, 3, Springer, pp.314–316.
3. F. D Murnaghan, 1932. On the field of values of a square matrix. Proceedings of the National Academy of Sciences of the United States of America, 18, National Academy of Sciences, pp.246.
4. R. Kippenhahn, 1951. Uber den Wertevorrat einer matrix. Mathematische Nachrichten, 6, Wiley Online Library, pp.193–228.
5. R. Horn, C. Johnson, 1994. Topics in matrix analysis. Cambridge university press.
6. K. E. Gustafson, D. K. M. Rao, 1997. Numerical range: The Field of Values of Linear Operators and Matrices. Springer.
7. E. Gutkin, 2004. The Toeplitz-Hausdorff theorem revisited: relating linear algebra and geometry. The Mathematical Intelligencer, 26, Springer, pp.8–14.
8. C. K. Li, 1996. A simple proof of the elliptical range theorem. Proceedings of the American Mathematical Society, 124, pp.1985–1986.
9. D. S. Keeler, L. Rodman, I. M. Spitkovsky, 1997. The numerical range of 3x3 matrices. Linear Algebra and its Applications, 252, pp.115 - 139.
10. M. Crouzeix, 2006. Open problems on Numerical range and functional calculus.
11. C. R. Johnson, 1981. Numerical ranges of principal sumbatrices. Linear Algebra and its Applications, 1, pp.23-34.
12. C. R. Johnson, 1976. Normality and the numerical range. Linear Algebra and its Applications, 37, Elsevier, pp.89–94.
13. P. Nylen, T. Y. Tam, 1991. Numerical range of a doubly stochastic matrix. Linear Algebra and Its Applications, 153, Elsevier, pp.161–176.
14. Panayiotis J Psarrakos, Michael J Tsatsomeros, 2002. Numerical ​range:(in) a matrix nutshell. Department of Mathematics,​ Washington State University.
15. Ch. Chorianopoulos, S. Karanasios, P. Psarrakos, 2009. A definition of numerical range of rectangular matrices. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.459–475.
16. J. H. Shapiro, 2004. Notes on the numerical range. Lecture Notes, Michigan State University.
17. P. Skoufranis, 2012. Numerical Ranges of Operators.
18. Russell Carden, 2009. A simple algorithm for the inverse field of values problem. Inverse Problems, 25, IOP Publishing, pp.115019.
19. Moshe Goldberg, Ernst Straus, 1977. On characterizations and integrals of generalized numerical ranges. Pacific Journal of Mathematics, 69, pp.45–54.
20. Moshe Goldberg, 1979. On certain finite dimensional numerical ranges and numerical radii†. Linear and Multilinear Algebra, 7, Taylor & Francis, pp.329–342.
21. Joseph Stampfli, 1970. The norm of a derivation. Pacific journal of mathematics, 33, Mathematical Sciences Publishers, pp.737–747.

Geometry of numerical range

1. [23] 2. [24] 3. [25] 4. [26] 5. [27] 6. [28] 7. [29] 8. [30] 9. [31] 10. [32]

22. M. Fiedler, 1981. Geometry of the numerical range of matrices. Linear Algebra and its Applications, 37, Elsevier, pp.81–96.
23. E. A. Jonckheere, F. Ahmad, E. Gutkin, 1998. Differential topology of numerical range. Linear algebra and its applications, 279, Elsevier, pp.227–254.
24. D. Henrion, 2010. Semidefinite geometry of the numerical range. Electronic Journal of Linear Algebra, 20, pp.322-332.
25. J. W. Helton, I. M. Spitkovsky, 2011. The possible shapes of numerical ranges. arXiv preprint arXiv:1104.4587.
26. M. T. Chien, Y. H. Lin, 2000. On the area of numerical range. Ssoochow Journal of Mathematics, 26, Soochow University, pp.255–270.
27. Jeffrey Eldred, Leiba Rodman, Ilya Spitkovsky, 2012. Numerical ranges of companion matrices: flat portions on the boundary. Linear and Multilinear Algebra, 60, pp.1295–1311.
28. J Maroulas, P Psarrakos, 1996. Geometrical properties of numerical range of matrix polynomials. Computers & Mathematics with Applications, 31, pp.41–47.
29. Moshe Goldberg, EG Straus, 1977. Elementary inclusion relations for generalized numerical ranges. Linear Algebra and Its Applications, 18, Elsevier, pp.1–24.
30. Ilya M Spitkovsky, Stephan Weis, 2015. Pre-images of extreme points of the numerical range, and applications. arXiv preprint arXiv:1509.05676.
31. Erin Militzer, Linda J Patton, Ilya M Spitkovsky, Ming-Cheng Tsai, 2017. Numerical Ranges of 4-by-4 Nilpotent Matrices: Flat Portions on the Boundary. Large Truncated Toeplitz Matrices, Toeplitz Operators, and Related Topics, Springer, pp.561–591.

Joint numerical range

1. [33] 2. [34] 3. [35] 4. [36]

32. E. Gutkin, E.A. Jonckheere, M. Karow, 2004. Convexity of the joint numerical range: topological and differential geometric viewpoints. Linear algebra and its applications, 376, Elsevier, pp.143–171.
33. N. Krupnik, I. M. Spitkovsky, 2006. Sets of matrices with given joint numerical range. Linear algebra and its applications, 419, Elsevier, pp.569–585.
34. Konrad Szymański, 2017. Uncertainty relations and joint numerical ranges. arXiv preprint arXiv:1707.03464.
35. Konrad Szymański, Stephan Weis, Karol Życzkowski, 2017. Classification of joint numerical ranges of three hermitian matrices of size three. Linear Algebra and its Applications, Elsevier.

Restricted numerical ranges

1. [37] 2. [38] 3. [39] 4. [40] 5. [41] 6. [42]

36. A. Abdollahi, 2006. The polynomial numerical hull of a matrix and algorithms for computing the numerical range. Applied Mathematics and Computation, 180, pp.635-640.
37. H. Gau, 2006. Elliptic numerical ranges of 4x4 matrices. Taiwanese Journal of Mathematics, 10, pp.117-128.
38. Z. Puchała, P. Gawron, J.A. Miszczak, Ł. Skowronek, M.D. Choi, K. Życzkowski, 2011. Product numerical range in a space with tensor product structure. Linear Algebra and its Applications, 434, Elsevier, pp.327–342.
39. J. W. Helton, I. M. Spitkovsky, 2011. The possible shapes of numerical ranges. arXiv:1104.4587, 1, pp.1-4.
40. W. Cheung, C. Li, 2012. Elementary proofs for some results on the circular symmetry of the numerical range. Linear & Multilinear Algebra, ahead-of, pp.1-7.
41. Jacek Jurkowski, Adam Rutkowski, D Chruściński, 2010. Local numerical range for a class of 2⊗ d Hermitian operators. Open Systems & Information Dynamics, 17, pp.347–359.

Higher order numerical ranges

1. [43] 2. [44] 3. [45] 4. [46] 5. [47] 6. [48] 7. [49] 8. [50] 9. [51] 10. [52] 11. [53]

42. M. D. Choi, J. A. Holbrook, D. W, Kribs, K. Życzkowski, 2007. Higher-rank numerical ranges of unitary and normal matrices. Operators and Matrices, 1, pp.409–426.
43. M. D. Choi, M. Giesinger, J. A. Holbrook, D. W. Kribs, 2008. Geometry of higher-rank numerical ranges. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.53–64.
44. H. J. Woerdeman, 2008. The higher rank numerical range is convex. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.65–67.
45. C. K, Li, N. S. Sze, 2008. Canonical forms, higher rank numerical ranges, totally isotropic subspaces, and matrix equations. Proceedings of the American Mathematical Society, 136, pp.3013–3023.
46. C. K. Li, Y. T. Poon, N. S. Sze, 2009. Condition for the higher rank numerical range to be non-empty. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.365–368.
47. H. L. Gau, C. K. Li, P. Y. Wu, 2010. Higher-rank numerical ranges and dilations. Journal of Operator Theory, 63, pp.181.
48. M. D. Choi, D. W. Kribs, K. Życzkowski, 2006. Higher-rank numerical ranges and compression problems. Linear algebra and its applications, 418, pp.828–839.
49. Mao-Ting Chien, Hiroshi Nakazato, 2011. The boundary of higher rank numerical ranges. Linear Algebra and its Applications, 435, pp.2971–2985.
50. Hwa-Long Gau, Pei Yuan Wu, 2013. Higher-rank numerical ranges and Kippenhahn polynomials. Linear Algebra and its Applications, 438, pp.3054–3061.
51. Hwa-Long Gau, Chi-Kwong Li, Yiu-Tung Poon, Nung-Sing Sze, 2011. Higher rank numerical ranges of normal matrices. SIAM Journal on Matrix Analysis and Applications, 32, SIAM, pp.23–43.
52. John Holbrook, Nishan Mudalige, Mike Newman, Rajesh Pereira, 2015. Bounds on polygons of higher rank numerical ranges. Linear Algebra and its Applications, 474, Elsevier, pp.230–242.

Quaternion numerical ranges

1. [54]

53. G Najarbashi, S Ahadpour, MA Fasihi, Y Tavakoli, 2007. Geometry of a two-qubit state and intertwining quaternionic conformal mapping under local unitary transformations. Journal of Physics A: Mathematical and Theoretical, 40, pp.6481.

Applications in quantum physics

1. [55] 2. [56] 3. [57]] 4. [58]

54. D. W. Kribs, A. Pasieka, M. Laforest, C. Ryan, M. P. da Silva, 2009. Research problems on numerical ranges in quantum computing. Linear and Multilinear Algebra, 57, Taylor & Francis, pp.491–502.
55. T. Schulte-Herbruggen, G. Dirr, U. Helmke, S. J. Glaser, 2008. The significance of the C-numerical range and the local C-numerical range in quantum control and quantum information. Linear and Multilinear Algebra, 56, Taylor & Francis, pp.3–26.
56. P. Gawron, Z. Puchała, J.A. Miszczak, Ł. Skowronek, K. Życzkowski, 2010. Restricted numerical range: A versatile tool in the theory of quantum information. Journal of Mathematical Physics, 51, pp.102204.
57. K. Życzkowski, M.-D. Choi, C. Dunkl, J. Holbrook, P. Gawron, J. A. Miszczak, Z. Puchala, Ł. Skowronek, 2009. Generalized numerical range as a versatile tool to study quantum entanglement. Oberwolfach Report, 59, pp.34-37.

Numerical shadow

General

1. [59] 2. [60] 3. [61] 4. [62] 5. [63]

58. C.F. Dunkl, P. Gawron, J.A. Holbrook, J.A. Miszczak, Z. Puchała, K. Życzkowski, 2011. Numerical shadow and geometry of quantum states. Journal of Physics A: Mathematical and Theoretical, 44, IOP Publishing, pp.335301.
59. C.F. Dunkl, P. Gawron, J.A. Holbrook, Z. Puchała, K. Zyczkowski, 2011. Numerical shadows: measures and densities on the numerical range. Linear Algebra and its Applications, 434, North-Holland, pp.2042–2080.
60. I. Bengtsson, S. Weis, K. Życzkowski, 2011. Geometry of the set of mixed quantum states: An apophatic approach. arxiv, 1, pp.1-13.
61. E. Gutkin, K. Zyczkowski, 2013. Joint numerical ranges, quantum maps, and joint numerical shadows. Linear Algebra Appl., 438, pp.2394-2404.
62. T. Gallay, D. Serre, 2010. The numerical measure of a complex matrix. arXiv:1009.1522, 1, pp.1-41.

Restricted numerical shadow

1. [64] 2. [65]

63. Z. Puchała, J.A. Miszczak, P. Gawron, C.F. Dunkl, J.A. Holbrook, K. Życzkowski, 2012. Restricted numerical shadow and geometry of quantum entanglement. Journal of Physics A: Mathematical and Theoretical, 45, pp.415309.
64. Charles F Dunkl, Piotr Gawron, Łukasz Pawela, Zbigniew Puchała, Karol Życzkowski, 2015. Real numerical shadow and generalized B-splines. Linear Algebra and its Applications, 479, pp.12–51.

Quantum canonical ensemble

1. [66] 2. [67] 3. [68] 4. [69] 5. [70]

65. D. C. Brody, L. P. Hughston, 1998. The quantum canonical ensemble. Journal of Mathematical Physics, 39, pp.6502-6508.
66. D. C. Brody, D. W. Hook, L. P. Hughston, 2005. Microcanonical distributions for quantum systems. arxiv, 1, pp.1-8.
67. D. C. Brody, D. W. Hook, L. P. Hughston, 2007. On quantum microcanonical equilibrium. Journal of Physics: Conference Series, 67, pp.012025.
68. D. C. Brody, D. W. Hook, L. P. Hughston, 2007. Quantum phase transitions without thermodynamic limits. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 463, pp.2021-2030.
69. L. C. V. P. Zanardi, 2012. Probability density of quantum expectation values. arXiv preprint, 1, pp.1-8.
literature.txt · Last modified: 2018/06/06 12:58 by lpawela

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