Complex Hadamard matrix

A complex Hadamard matrix is any complex matrix $\text{[math]}$ satisfying two conditions:

unimodularity (the modulus of each entry is unity): $\text{[math]}$
orthogonality: $\text{[math]}$ ,

where $\text{[math]}$ denotes the Hermitian transpose of H and $\text{[math]}$ is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix $\text{[math]}$ can be made into a unitary matrix by multiplying it by $\text{[math]}$ ; conversely, any unitary matrix whose entries all have modulus $\text{[math]}$ becomes a complex Hadamard upon multiplication by $\text{[math]}$ .

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

\text{[math]}

belong to this class.

Equivalency

Two complex Hadamard matrices are called equivalent, written $\text{[math]}$ , if there exist diagonal unitary matrices $\text{[math]}$ and permutation matrices $\text{[math]}$ such that

\text{[math]}

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 $\text{[math]}$ and $\text{[math]}$ all complex Hadamard matrices are equivalent to the Fourier matrix $\text{[math]}$ . For $\text{[math]}$ there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,

\text{[math]}

For $\text{[math]}$ the following families of complex Hadamard matrices are known:

a single two-parameter family which includes $\text{[math]}$ ,
a single one-parameter family $\text{[math]}$ ,
a one-parameter orbit $\text{[math]}$ , including the circulant Hadamard matrix $\text{[math]}$ ,
a two-parameter orbit including the previous two examples $\text{[math]}$ ,
a one-parameter orbit $\text{[math]}$ of symmetric matrices,
a two-parameter orbit including the previous example $\text{[math]}$ ,
a three-parameter orbit including all the previous examples $\text{[math]}$ ,
a further construction with four degrees of freedom, $\text{[math]}$ , yielding other examples than $\text{[math]}$ ,
a single point - one of the Butson-type Hadamard matrices, $\text{[math]}$ .

It is not known, however, if this list is complete, but it is conjectured that $\text{[math]}$ is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.

References

U. Haagerup, Orthogonal maximal abelian *-subalgebras of the n×n matrices and cyclic n-roots, Operator Algebras and Quantum Field Theory (Rome), 1996 (Cambridge, MA: International Press) pp 296–322.
P. Dita, Some results on the parametrization of complex Hadamard matrices, J. Phys. A: Math. Gen. 37, 5355-5374 (2004).
F. Szollosi, A two-parametric family of complex Hadamard matrices of order 6 induced by hypocycloids, preprint, arXiv:0811.3930v2 [math.OA]
W. Tadej and K. Życzkowski, A concise guide to complex Hadamard matrices Open Systems & Infor. Dyn. 13 133-177 (2006)

External links

For an explicit list of known $\text{[math]}$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices

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