A complex Hadamard matrix is any complex N × N matrix H satisfying two conditions:
unimodularity (the modulus of each entry is unity): | H j k | = 1 f o r j , k = 1 , 2 , … , N orthogonality: H H † = N I ,where † denotes the Hermitian transpose of H and I is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix H can be made into a unitary matrix by multiplying it by 1 N ; conversely, any unitary matrix whose entries all have modulus 1 N becomes a complex Hadamard upon multiplication by N .
Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation. Real Hadamard matrices and Butson-type Hadamard matrices form particular cases of complex Hadamard matrices.
Complex Hadamard matrices exist for any natural N (compare the real case, in which existence is not known for every N). For instance the Fourier matrices
[ F N ] j k := exp [ ( 2 π i ( j − 1 ) ( k − 1 ) / N ] f o r j , k = 1 , 2 , … , N belong to this class.
Two complex Hadamard matrices are called equivalent, written H 1 ≃ H 2 , if there exist diagonal unitary matrices D 1 , D 2 and permutation matrices P 1 , P 2 such that
H 1 = D 1 P 1 H 2 P 2 D 2 . Any complex Hadamard matrix is equivalent to a dephased Hadamard matrix, in which all elements in the first row and first column are equal to unity.
For N = 2 , 3 and 5 all complex Hadamard matrices are equivalent to the Fourier matrix F N . For N = 4 there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,
F 4 ( 1 ) ( a ) := [ 1 1 1 1 1 i e i a − 1 − i e i a 1 − 1 1 − 1 1 − i e i a − 1 i e i a ] w i t h a ∈ [ 0 , π ) . For N = 6 the following families of complex Hadamard matrices are known:
a single two-parameter family which includes F 6 ,a single one-parameter family D 6 ( t ) ,a one-parameter orbit B 6 ( θ ) , including the circulant Hadamard matrix C 6 ,a two-parameter orbit including the previous two examples X 6 ( α ) ,a one-parameter orbit M 6 ( x ) of symmetric matrices,a two-parameter orbit including the previous example K 6 ( x , y ) ,a three-parameter orbit including all the previous examples K 6 ( x , y , z ) ,a further construction with four degrees of freedom, G 6 , yielding other examples than K 6 ( x , y , z ) ,a single point - one of the Butson-type Hadamard matrices, S 6 ∈ H ( 3 , 6 ) .It is not known, however, if this list is complete, but it is conjectured that K 6 ( x , y , z ) , G 6 , S 6 is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.
