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: $H{H}^{†}=NI$,

where $†$ 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),

$[F_N{]}_{jk}:=\exp [2\pi i(j-1)(k-1)/N]\mathrm{f}\mathrm{o}\mathrm{r}j,k=1,2,\dots ,N$

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,

$F_4^{(1)}(a):=\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& i{e}^{ia}& -1& -i{e}^{ia}\\ 1& -1& 1& -1\\ 1& -i{e}^{ia}& -1& i{e}^{ia}\end{array}\right]\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}a\in [0,\pi ).$

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.

References

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External links

• For an explicit list of known $N=6$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices