Complex Hadamard matrix explained

A complex Hadamard matrix is any complex

N x N matrix H satisfying two conditions:

• unimodularity (the modulus of each entry is unity):

|H _{jk} | = 1 for j,k = 1 , 2 , ... ,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 (1)/(\sqrt{N) }; conversely, any unitary matrix whose entries all have modulus (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.

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] { \rm for } j,k =1 , 2 , ... ,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 thatH _{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 existsa continuous, one-parameter family of inequivalent complex Hadamard matrices,F ^{( 1 )} _{4} (a): = \begin{bmatrix} 1 & 1 & 1 & 1 \ 1 & ie ^{ia} & -1 & - ie ^{ia} \\ 1 & -1 & 1 & -1 \\ 1 & - ie ^{ia} & -1 & i e ^{ia} \end{bmatrix} { \rm with } a\in [ 0 ,\pi) .

For

N =6 the following families of complex Hadamard matricesare 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