Maximal lexicographic spectra and ranks for states with fixed uniform margins

Li, Xin

doi:10.1090/proc/14494

Maximal lexicographic spectra and ranks for states with fixed uniform margins
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Proc. Amer. Math. Soc. 147 (2019), 3303-3315 Request permission

Abstract:

We find the spectrum in maximal lexicographic order for quantum states $\rho _{AB}\in \mathcal {H}_A\otimes \mathcal {H}_B$ with margins $\rho _A=\frac {1}{n}I_n$ and $\rho _B=\frac {1}{m}I_m$ and discuss the construction of $\rho _{AB}$. By nonzero rectangular Kronecker coefficients, we give counterexamples for Klyachko’s conjecture which says that a quantum state with maximal lexicographical spectrum has minimal rank among all states with given margins. Moreover, we show that quantum states with the maximal lexicographical spectrum are extreme points.

References

Christine Bessenrodt and Stephanie van Willigenburg, On (almost) extreme components in Kronecker products of characters of the symmetric groups, J. Algebra 410 (2014), 460–500. MR 3201060, DOI 10.1016/j.jalgebra.2014.01.035
J. Bryana, S. Leutheusser, Z. Reichstein, and M. V. Raamsdonk, Locally Maximally Entangled States of Multipart Quantum Systems, arXiv:1801.03508v2, (2018).
Peter Bürgisser, Matthias Christandl, and Christian Ikenmeyer, Nonvanishing of Kronecker coefficients for rectangular shapes, Adv. Math. 227 (2011), no. 5, 2082–2091. MR 2803795, DOI 10.1016/j.aim.2011.04.012
Matthias Christandl and Graeme Mitchison, The spectra of quantum states and the Kronecker coefficients of the symmetric group, Comm. Math. Phys. 261 (2006), no. 3, 789–797. MR 2197548, DOI 10.1007/s00220-005-1435-1
M.Christandl, The Structure of Bipartite Quantum States-Insights from Group Theory and Cryptography, arXiv:quant-ph/0604183, (2006).
Matthias Christandl, Aram W. Harrow, and Graeme Mitchison, Nonzero Kronecker coefficients and what they tell us about spectra, Comm. Math. Phys. 270 (2007), no. 3, 575–585. MR 2276458, DOI 10.1007/s00220-006-0157-3
Michael Clausen and Helga Meier, Extreme irreduzible Konstituenten in Tensordarstellungen symmetrischer Gruppen, Bayreuth. Math. Schr. 45 (1993), 1–17 (German, with German summary). MR 1244185
Sumit Daftuar and Patrick Hayden, Quantum state transformations and the Schubert calculus, Ann. Physics 315 (2005), no. 1, 80–122. MR 2115656, DOI 10.1016/j.aop.2004.09.012
Roger A. Horn and Charles R. Johnson, Matrix analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. MR 2978290
Christian Ikenmeyer and Greta Panova, Rectangular Kronecker coefficients and plethysms in geometric complexity theory, Adv. Math. 319 (2017), 40–66. MR 3695867, DOI 10.1016/j.aim.2017.08.024
Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
A. Klyachko, Quantum marginal problem and representations of the symmetric group, arXiv:quant-ph/0409113v1, 2004.
A. Klyachko, Quantum marginal problem and N-representability, Journal of Physics: Conference Series 36(2006) 72-86.
Chi-Kwong Li, Yiu-Tung Poon, and Xuefeng Wang, Ranks of quantum states with prescribed reduced states, Electron. J. Linear Algebra 34 (2018), 331–342. MR 3856464, DOI 10.13001/1081-3810.3654
Albert W. Marshall, Ingram Olkin, and Barry C. Arnold, Inequalities: theory of majorization and its applications, 2nd ed., Springer Series in Statistics, Springer, New York, 2011. MR 2759813, DOI 10.1007/978-0-387-68276-1
Bruce E. Sagan, The symmetric group, 2nd ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001. Representations, combinatorial algorithms, and symmetric functions. MR 1824028, DOI 10.1007/978-1-4757-6804-6
Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
Vasu V. Tewari, Kronecker coefficients for some near-rectangular partitions, J. Algebra 429 (2015), 287–317. MR 3320625, DOI 10.1016/j.jalgebra.2015.01.018
Ernesto Vallejo, Multiple hooks in Kronecker products of $S_n$ characters and a conjecture of Berele and Imbo, J. Algebra 259 (2003), no. 1, 300–309. MR 1953721, DOI 10.1016/S0021-8693(02)00554-9