Symbolic integration with respect to the Haar measure on the unitary groups

• Autorzy: Puchała Z. , Miszczak J. A.

Identyfikatory

DOI

10.1515/bpasts-2017-0003

• Języki publikacji: EN

Abstrakty

EN

We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.

Słowa kluczowe

EN

unitary group; Haar measure; circular unitary ensemble; symbolic integration

PL

macierze unitarne; Miara Haara; integracja symboliczna

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2017
• Tom: Vol. 65, nr 1
• Strony: 21--27
• Opis fizyczny: Bibliogr. 15 poz.

Twórcy

autor

Puchała Z.

• Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, 5 Bałtycka Str., 44-100 Gliwice, Poland

autor

Miszczak J. A.

• Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, 5 Bałtycka Str., 44-100 Gliwice, Poland

Bibliografia

• [1] B. Collins and P. Śniady, “Integration with respect to the Haar measure on unitary, orthogonal and symplectic group”, Commun. Math. Phys. 264, 773-795 (2006).
• [2] N. Ullah and C. Porter. “Expectation value fluctuations in the unitary ensemble”, Physical Review 132 (2), 948 (1963).
• [3] D. Weingarten, “Asymptotic behavior of group integrals in the limit of infinite rank”, Journal of Mathematical Physics 19, 999 (1978).
• [4] Z. Puchała and J. A. Miszczak, “IntU package for Mathematica” (2011). Software available at http://zksi.iitis.pl/wiki/projects:intu.
• [5] W. Fulton and J. Harris, Representation Theory: A First Course - Graduate Texts in Mathematics vol. 129, Springer Verlag (1991).
• [6] G. James and A. Kerber, “The representation theory of the symmetric group”, Encyclopaedia of Mathematics, vol .16 (1981).
• [7] B. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer Verlag (2001).
• [8] D. Bernstein, “The computational complexity of rules for the character table of Sn”, Journal of Symbolic Computation 37 (6), 727-748 (2004).
• [9] F. Hiai and D. Petz, “The semicircle law, free random variables and entropy”, Amer. Mathematical Society 77 (2006).
• [10] C. Donati-Martin and A. Rouault, “Truncations of Haar unitary matrices, traces and bivariate Brownian bridge”. Arxiv preprint, arXiv:1007.1366 (2010).
• [11] Z. Puchała, J. A. Miszczak, P. Gawron, C. F. Dunkl, J. A. Holbrook, and K. Życzkowski, “Restricted numerical shadow and the geometry of quantum entanglement”, Journal of Physics A: Mathematical and Theoretical 45 (41), 415309 (2012).
• [12] J. Miszczak, “Generating and using truly random quantum states in Mathematica”, Comput. Phys. Commun. 183 (1), 118-124 (2012).
• [13] C. F. Dunkl, P. Gawron, J. A. Holbrook, Z. Puchała, and K. Życzkowski, “Numerical shadows: measures and densities on the numerical range”, Linear Algebra Appl. 434, 2042-2080 (2011).
• [14] C. F. Dunkl, P. Gawron, J. A. Holbrook, J. A. Miszczak, Z. Puchała, and K. Życzkowski, “Numerical shadow and geometry of quantum states”, J. Phys. A: Math. Theor. 44 (33), 335301 (2011).
• [15] M. Enríquez, Z. Puchała, and K. Życzkowski, “Minimal Rényi- -Ingarden-Urbanik entropy of multipartite quantum states”, Entropy 17 (7), 5063 (2015).

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).