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,

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

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

• Haagerup, U. (1997). "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. ISBN 1-57146-047-0. OCLC 1409082233.
• Dita, P. (2004). "Some results on the parametrization of complex Hadamard matrices". J. Phys. A: Math. Gen. 37 (20): 5355–74. doi:10.1088/0305-4470/37/20/008.
• Szöllősi, F. (2010). "A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids". Proceedings of the American Mathematical Society. 138 (3): 921–8. arXiv:0811.3930v2. JSTOR 40590684.
• Tadej, W.; Życzkowski, K. (2006). "A concise guide to complex Hadamard matrices". Open Systems & Infor. Dyn. 13 (2): 133–177. arXiv:quant-ph/0512154. doi:10.1007/s11080-006-8220-2.

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