Complex Hadamard matrix

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 {\quad \rm for \quad} j,k=1,2,\dots,N

where  {\dagger} denotes the Hermitian transpose of H and  {\mathbb I} is the identity matrix. The concept is a generalization of the Hadamard matrix.

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 N (compare the real case, in which existence is not known for every N). For instance the Fourier matrices

 [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 D1,D2 and permutation matrices P1,P2 such that

H1 = D1P1H2P2D2.

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 FN. 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 F6,
  • a single one-parameter family D6(t),
  • a one-parameter orbit B6(θ), including the circulant Hadamard matrix C6,
  • a two-parameter orbit including the previous two examples X6(α),
  • a one-parameter orbit M6(x) of symmetric matrices,
  • a two-parameter orbit including the previous example K6(x,y),
  • a three-parameter orbit including all the previous examples K6(x,y,z),
  • a further construction with four degrees of freedom, G6, yielding other examples than K6(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 K6(x,y,z),G6,S6 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. Zyczkowski, A concise guide to complex Hadamard matrices Open Systems & Infor. Dyn. 13 133-177 (2006)

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