A complex Hadamard matrix is any complex matrix satisfying two conditions:
- unimodularity (the modulus of each entry is unity):
- orthogonality: ,
where denotes the Hermitian transpose of H and is the identity matrix. The concept is a generalization of the 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 becomes a complex Hadamard upon multiplication by .
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
belong to this class.
Read more about Complex Hadamard Matrix: Equivalency
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