Complex Hadamard matrix

A complex Hadamard matrix is any complex
formula_1 matrix formula_2 satisfying two conditions:
where formula_5 denotes the Hermitian transpose of formula_2 and formula_7 is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix formula_2 can be made into a unitary matrix by multiplying it by formula_9; conversely, any unitary matrix whose entries all have modulus formula_9 becomes a complex Hadamard upon multiplication by formula_11.
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 formula_12 (compare the real case, in which existence is not known for every formula_12). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor),
belong to this class.
Equivalency.
Two complex Hadamard matrices are called equivalent, written formula_15, if there exist diagonal unitary matrices formula_16 and permutation matrices formula_17 
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.
For formula_19 and formula_20 all complex Hadamard matrices are equivalent to the Fourier matrix formula_21. For formula_22 there exists
a continuous, one-parameter family of inequivalent complex Hadamard matrices,
For formula_24 the following families of complex Hadamard matrices
are known:
It is not known, however, if this list is complete, but it is conjectured that formula_36 is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.