Complex Hadamard matrix

A complex Hadamard matrix is any complex $$N \times N$$ matrix $$H$$ satisfying two conditions:


 * unimodularity (the modulus of each entry is unity): $$|H_{jk}| = 1 \text{ for } j,k = 1,2,\dots,N $$
 * orthogonality: $$HH^{\dagger} = NI$$,

where $$\dagger$$ denotes the Hermitian transpose of $$H$$ and $$I$$ is the identity matrix. The concept is a generalization of Hadamard matrices. Note that any complex Hadamard matrix $$H$$ can be made into a unitary matrix by multiplying it by $$\frac{1}{\sqrt{N}}$$; conversely, any unitary matrix whose entries all have modulus $$\frac{1}{\sqrt{N}}$$ becomes a complex Hadamard upon multiplication by $$\sqrt{N}.$$

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 $$N$$ (compare with the real case, in which Hadamard matrices do not exist for every $$N$$ and existence is not known for every permissible $N$). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor),


 * $$[F_N]_{jk}:= \exp[2\pi i (j-1)(k-1)/N]

{\quad \rm for \quad} j,k=1,2,\dots,N $$

belong to this class.

Equivalency
Two complex Hadamard matrices are called equivalent, written $$H_1 \simeq H_2$$, if there exist diagonal unitary matrices $$D_1, D_2$$ and permutation matrices $$P_1, P_2$$ such that


 * $$H_1 = D_1 P_1 H_2 P_2 D_2.$$

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 $$N=2,3$$ and $$5$$ all complex Hadamard matrices are equivalent to the Fourier matrix $$F_{N}$$. For $$N=4$$ there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,


 * $$ F_{4}^{(1)}(a):=

\begin{bmatrix} 1 & 1      & 1  & 1 \\ 1 & ie^{ia} & -1 & -ie^{ia} \\ 1 & -1     & 1  &-1 \\                1 & -ie^{ia}& -1 & i e^{ia} \end{bmatrix} {\quad \rm with \quad } a\in [0,\pi) . $$

For $$N=6$$ the following families of complex Hadamard matrices are known:


 * a single two-parameter family which includes $$F_6$$,
 * a single one-parameter family $$D_6(t)$$,
 * a one-parameter orbit $$B_6(\theta)$$, including the circulant Hadamard matrix $$C_6$$,
 * a two-parameter orbit including the previous two examples $$X_6(\alpha)$$,
 * a one-parameter orbit $$M_6(x)$$ of symmetric matrices,
 * a two-parameter orbit including the previous example $$K_6(x,y)$$,
 * a three-parameter orbit including all the previous examples $$K_6(x,y,z)$$,
 * a further construction with four degrees of freedom, $$G_6$$, yielding other examples than $$K_6(x,y,z)$$,
 * a single point - one of the Butson-type Hadamard matrices, $$S_6 \in H(3,6)$$.

It is not known, however, if this list is complete, but it is conjectured that $$K_6(x,y,z),G_6,S_6$$ is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.