Locally free sheaf

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A complex Hadamard matrix is any complex matrix satisfying two conditions:

where denotes the Hermitian transpose of H and is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix can be made into a unitary matrix by multiplying it by ; conversely, any unitary matrix whose entries all have modulus becomes a complex Hadamard upon multiplication by .

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

belong to this class.

Equivalency

Two complex Hadamard matrices are called equivalent, written , if there exist diagonal unitary matrices and permutation matrices such that

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 and all complex Hadamard matrices are equivalent to the Fourier matrix . For there exists a continuous, one-parameter family of inequivalent complex Hadamard matrices,

For the following families of complex Hadamard matrices are known:

It is not known, however, if this list is complete, but it is conjectured that 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)

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