Complex Hadamard matrix

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A complex Hadamard matrix is any complex N \times N matrix H satisfying two conditions:

  • unimodularity: |H_{jk}|=1 {\quad \rm for \quad} j,k=1,2,\dots,N
  • orthogonality: HH^{\dagger} = N \; {\mathbb I},

where {\dagger} denotes the Hermitian transpose of H and {\mathbb I} is the identity 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 exists for any natural 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.

[edit] 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 non-equivalent complex Hadamard matrices are known:

  • a single two-parameter family which includes F6,
  • a single one-parameter family
  • a one-parameter orbit including the circulant Hadamard matrix C6,
  • 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.

[edit] 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).

[edit] External links

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