A complex Hadamard matrix is any complex
matrix
satisfying two conditions:
- unimodularity (the modulus of each entry is unity):
![{\displaystyle |H_{jk}|=1{\text{ for ))j,k=1,2,\dots ,N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc02229a8fd06b65dd2848bad2282b9dc1cd97b9)
- orthogonality:
,
where
denotes the Hermitian transpose of
and
is the identity matrix. The concept is a generalization of Hadamard matrices. Note that any complex Hadamard matrix
can be made into a unitary matrix by multiplying it by
; conversely, any unitary matrix whose entries all have modulus
becomes a complex Hadamard upon multiplication by
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
(compare with the real case, in which Hadamard matrices do not exist for every
and existence is not known for every permissible
). 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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06538713e6b7bb7ad70abaa3df7f0d5b22203657)
belong to this class.
Two complex Hadamard matrices are called equivalent, written
, if there exist diagonal unitary matrices
and permutation matrices
such that
![{\displaystyle H_{1}=D_{1}P_{1}H_{2}P_{2}D_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/effbee6ba09171a2889a828907740a310afccb32)
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
and
all complex Hadamard matrices are equivalent to the Fourier matrix
. For
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 ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9027092d7859e25fa1b7641cb7b7c7641a160000)
For
the following families of complex Hadamard matrices
are known:
- a single two-parameter family which includes
,
- a single one-parameter family
,
- a one-parameter orbit
, including the circulant Hadamard matrix
,
- a two-parameter orbit including the previous two examples
,
- a one-parameter orbit
of symmetric matrices,
- a two-parameter orbit including the previous example
,
- a three-parameter orbit including all the previous examples
,
- a further construction with four degrees of freedom,
, yielding other examples than
,
- a single point - one of the Butson-type Hadamard matrices,
.
It is not known, however, if this list is complete, but it is conjectured that
is an exhaustive (but not necessarily irredundant) list of all complex Hadamard matrices of order 6.
- 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)