Eigenvalues of Two-Parameter Complex Hadamard Matrices of Order Six

Characterizing complex Hadamard matrices (CHMs) of order six is an open problem in both algebra and quantum information theory. Analyzing the eigenvalues of CHMs provides a novel approach to this problem. We construct a class of two-parameter CHMs of order six with two sorts of eigenvalues. All matrices in this class are H 2-reducible, and the Diţă matrix, Haagerup matrices and Hermitian matrices are both included. This class turns out to be complex equivalent to the class of all CHMs in the dephased form whose eigenvalues are of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:msqrt> <m:mn>6</m:mn> </m:msqrt> </m:mrow> </m:math> √ 6 , – <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msqrt> <m:mn>6</m:mn> </m:msqrt> </m:math> √ 6 , λ 1 , λ 1 , λ 2 , λ 2 with λ j of modulus <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msqrt> <m:mn>6</m:mn> </m:msqrt> </m:math> √ 6 excluding Tao matrix <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msubsup> <m:mi>S</m:mi> <m:mn>6</m:mn> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:msubsup> </m:math> S_6⁽⁰⁾ . We further show that some CHMs including all Hermitian CHMs in the former class have Schmidt rank at most three up to complex equivalence, and they are actually controlled unitary gates implementable in experiments. Our result demonstrates the feasibility of using eigenvalues for characterizing order-six CHMs and their application in quantum circuits.