Complex Hadamard matrix


A complex Hadamard matrix is any complex
matrix satisfying two conditions:
where denotes the Hermitian transpose of and is the identity matrix. The concept is a generalization of Hadamard matrices. 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 number . For instance the Fourier matrices,
belong to this class.

Equivalency

Two complex Hadamard matrices are called equivalent, written, if there exist diagonal unitary matrices and permutation 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.
For and all complex Hadamard matrices are equivalent to the Fourier matrix. For there exists
a continuous, one-parameter family of inequivalent complex Hadamard matrices,
For the following families of complex Hadamard matrices
are known:
  • a single two-parameter family which includes,
  • a single one-parameter family,
  • a one-parameter orbit, including the circulant Hadamard matrix,
  • a two-parameter orbit including the previous two examples,
  • a one-parameter orbit of symmetric matrices,
  • a two-parameter orbit including the previous example,
  • a three-parameter orbit including all the previous examples,
  • a further construction with four degrees of freedom,, yielding other examples than,
  • a single point - one of the Butson-type Hadamard matrices,.
It is not known, however, if this list is complete, but it is conjectured that is an exhaustive list of all complex Hadamard matrices of order 6.