Real Entries of Complex Hadamard Matrices and Mutually Unbiased Bases in Dimension Six

Abstract

We investigate the number of real entries of an $n\times n$ complex Hadamard matrix (CHM). 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 (MUBs) 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 exists then the real entries in any one of the remaining three matrices does not exceed $22$.

arXiv comment28 pages

References in corpus (13)

• On mutually unbiased bases
• Constructing Mutually Unbiased Bases in Dimension Six
• Entanglement in mutually unbiased bases
• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• Quantifying measurement incompatibility of mutually unbiased bases
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• Mutually unbiased bases in dimension six: The four most distant bases
• The Limited Role of Mutually Unbiased Product Bases in Dimension Six
• On the Impossibility to Extend Triples of Mutually Unbiased Product Bases in Dimension Six
• The Birkhoff theorem for unitary matrices of arbitrary dimensions
• Mutually unbiased bases in dimension six containing a product-vector basis
• Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras
• On properties of Karlsson Hadamards and sets of Mutually Unbiased Bases in dimension six