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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),

${\displaystyle _{jk}:=\exp{\quad {\rm {for\quad }}}j,k=1,2,\dots ,N}$

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

• 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 ${\displaystyle N=6}$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see Catalogue of Complex Hadamard Matrices

• Matrices