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

• unimodularity (the modulus of each entry is unity): $|H_{jk}|=1{\quad {\rm {for\quad }}}j,k=1,2,\dots ,N$

• orthogonality: $HH^{\dagger }=N\;{\mathbb {I} }$,

where ${\dagger }$ denotes the Hermitian transpose of H and ${\mathbb {I} }$ 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 ${\frac {1}{\sqrt {N}}}$; 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 N (compare the real case, in which existence is not known for every N). For instance the Fourier matrices

$[F_{N}]_{jk}:=\exp[(2\pi i(j-1)(k-1)/N]{\quad {\rm {for\quad }}}j,k=1,2,\dots ,N$

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,

$F_{4}^{(1)}(a):={\begin{bmatrix}1&1&1&1\\1&ie^{ia}&-1&-ie^{ia}\\1&-1&1&-1\\1&-ie^{ia}&-1&ie^{ia}\end{bmatrix}}{\quad {\rm {with\quad }}}a\in [0,\pi ).$

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.

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 $N=6$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices