Complex Hadamard matrix

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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. 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 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 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&ie^{{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

  • 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

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