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Maximal lexicographic spectra and ranks for states with fixed uniform margins

Author: Xin Li
Journal: Proc. Amer. Math. Soc. 147 (2019), 3303-3315
MSC (2010): Primary 20C30; Secondary 15A18
Published electronically: April 3, 2019
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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.

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Xin Li
Affiliation: Department of Mathematics, Zhejiang University of Technology, Hangzhou 310023, People’s Republic of China

Keywords: Quantum marginal problem, maximal lexicographic spectrum, rank, rectangular Kronecker coefficients, generalized discrete Weyl operators
Received by editor(s): January 8, 2018
Received by editor(s) in revised form: November 7, 2018
Published electronically: April 3, 2019
Additional Notes: The research was supported by National Natural Science Foundation of China (Grant No.11626211).
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2019 American Mathematical Society