Complex Hadamard matrix

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

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

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

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

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

belong to this class.

Equivalency

Two complex Hadamard matrices are called equivalent, written ${\displaystyle H_{1}\simeq H_{2}}$, if there exist diagonal unitary matrices ${\displaystyle D_{1},D_{2}}$ and permutation matrices ${\displaystyle P_{1},P_{2}}$ such that

${\displaystyle H_{1}=D_{1}P_{1}H_{2}P_{2}D_{2}.}$

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

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

For ${\displaystyle N=6}$ the following families of complex Hadamard matrices are known:

• a single two-parameter family which includes ${\displaystyle F_{6}}$,
• a single one-parameter family ${\displaystyle D_{6}(t)}$,
• a one-parameter orbit ${\displaystyle B_{6}(\theta )}$, including the circulant Hadamard matrix ${\displaystyle C_{6}}$,
• a two-parameter orbit including the previous two examples ${\displaystyle X_{6}(\alpha )}$,
• a one-parameter orbit ${\displaystyle M_{6}(x)}$ of symmetric matrices,
• a two-parameter orbit including the previous example ${\displaystyle K_{6}(x,y)}$,
• a three-parameter orbit including all the previous examples ${\displaystyle K_{6}(x,y,z)}$,
• a further construction with four degrees of freedom, ${\displaystyle G_{6}}$, yielding other examples than ${\displaystyle K_{6}(x,y,z)}$,
• a single point - one of the Butson-type Hadamard matrices, ${\displaystyle S_{6}\in H(3,6)}$.

It is not known, however, if this list is complete, but it is conjectured that ${\displaystyle K_{6}(x,y,z),G_{6},S_{6}}$ 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 ${\displaystyle N=6}$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices