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On Mutually Unbiased Bases and Hadamard Matrices : An exploration from quantum mechanics to number theory

Daniel C. Qin

Published May 3, 2021• 0 citations
OpenAlex

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•Jan 1, 2014•0 citations
We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22.
Mutually unbiased unit Hadamard (MUUH) matrices have been studied for almost 40 years. Recent interest in such matrices has been motivated by their applications to quantum information theory. In this paper, we introduce the class of mutually unbiased complex Hadamard (MUCH) matrices having as sntries fourth roots of unity. The number of MUCH matrices of order 2n, n odd, is at most 2, and the bound is attained for n = 1,5,9. Certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. We then turn our attention to the class of mutually unbiased real Hadamard matrices of order 16n2which seem to be the most interesting ones. Mutually unbiased weighing (MUW) matrices are introduced for the first time. Further study of these objects look quite promising.