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,

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

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

{{cite conference |first=U. |last=Haagerup |title=Orthogonal maximal abelian *-subalgebras of the n×n matrices and cyclic n-roots |book-title=Operator Algebras and Quantum Field Theory (Rome), 1996 |publisher=International Press |location=Cambridge MA |date=1997 |isbn=1-57146-047-0 |oclc=1409082233 |pages=296–322 }}
{{cite journal |first=P. |last=Dita |title=Some results on the parametrization of complex Hadamard matrices |journal=J. Phys. A: Math. Gen. |volume=37 |issue=20 |pages=5355–74 |date=2004 |doi=10.1088/0305-4470/37/20/008 |url=https://iopscience.iop.org/article/10.1088/0305-4470/37/20/008|url-access=subscription }}
{{cite journal |first=F. |last=Szöllősi |title=A two-parameter family of complex Hadamard matrices of order 6 induced by hypocycloids |journal=Proceedings of the American Mathematical Society |volume=138 |issue=3 |pages=921–8 |date=2010 |doi= |url=https://www.ams.org/proc/2010-138-03/S0002-9939-09-10102-8/ |arxiv=0811.3930v2 |jstor=40590684}}
{{cite journal |first=W. |last=Tadej |author2-link=Karol Życzkowski |first2=K. |last2=Życzkowski |title=A concise guide to complex Hadamard matrices |journal=Open Systems & Infor. Dyn. |volume=13 |issue=2 |pages=133–177 |date=2006 |doi=10.1007/s11080-006-8220-2 |url=https://link.springer.com/article/10.1007/s11080-006-8220-2 |arxiv=quant-ph/0512154}}

External links

For an explicit list of known $N=6$ complex Hadamard matrices and several examples of Hadamard matrices of size 7-16 see [http://chaos.if.uj.edu.pl/~karol/hadamard/ Catalogue of Complex Hadamard Matrices]

Category:Matrices (mathematics)