Complex Hadamard matrix

Note that any complex Hadamard matrix

can be made into a unitary matrix by multiplying it by

; conversely, any unitary matrix whose entries all have modulus

Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation.

Complex Hadamard matrices exist for any natural number

(compare with the real case, in which Hadamard matrices do not exist for every

For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor), belong to this class.

Two complex Hadamard matrices are called equivalent, written

, if there exist diagonal unitary matrices

such that 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.

all complex Hadamard matrices are equivalent to the Fourier matrix

there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices, For

the following families of complex Hadamard matrices are known: It is not known, however, if this list is complete, but it is conjectured that

is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.