Complex Hadamard Matrices

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

*unimodularity (the modulus of each entry is unity): $, H_, =1 j,k=1,2,\dots,N$
*orthogonality: $HH^ = NI$,

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

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 (the complex conjugate of the DFT matrices without the normalizing factor),

: _N:= \exp \pi i(j - 1)(k - 1) / N
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 $F_$. For $N=4$ there exists
a continuous, one-parameter family of inequivalent complex Hadamard matrices,

:$F_^(a):= \begin 1 & 1 & 1 & 1 \\ 1 & ie^ & -1 & -ie^ \\ 1 & -1 & 1 &-1 \\ 1 & -ie^& -1 & i e^ \end a\in [0,\pi) .$

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

*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 Karol Życzkowski, 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 $N=6$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 se
Catalogue of Complex Hadamard Matrices

{{Matrix classes

Matrices