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ABSTRACT:
Recently the mathematical structure of the quantum information theory was better understood
by establishing a one-to-one correspondence between quantum teleportation schemes,
dense coding schemes, orthogonal bases of maximally entangled vectors, bases of unitary operators and unitary depolarizers by showing that given any object of any one
of the above types one can construct any object of each of these types by using a precise procedure
(R.F. Werner, All teleportation and dense coding schemes,
quantum-ph/0003070).
The construction procedure will be efficient only to the extent that the unitary
bases can be generated and the construction of these bases makes explicit use of the
complex Hadamard matrices and the Latin squares.
The aim of this paper is to provide a procedure for the parametrisation of the complex Hadamard
matrices for an arbitrary integer n starting from our previous results on parametrisation of unitary matrices ( P. Dita, Parametrisation of unitary matrices by moduli of their elements,
Commun.Math.Phys. 159 (1994) 581-591, Factorization of unitary matrices,
math-ph/0103005).
More precisely we obtain an elementary two-dimensional array
| A
B |
| A -B |
which is unitary and Hadamard when the matrices A, B are unitary and Hadamard,
array which can be used to obtain complex Hadamard matrices depending on arbitary phases.
We also obtain a set of (n-2)^2 equations whose solutions will give all the complex Hadamard
matrices of size n.
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