The study of Mutually Unbiased Bases continues to be developed vigorously,
and presents several challenges in the Quantum Information Theory. Two
orthonormal bases in Cd,BandB′ are said mutually unbiased if
∀b∈B,b′∈B′ the scalar product b⋅b′ has modulus
d−1/2. In particular this property has been introduced in order to allow
an optimization of the measurement-driven quantum evolution process of any
state ψ∈Cd when measured in the mutually unbiased bases
B_jofCd. At present it is an open problem to find the maximal
umber of mutually Unbiased Bases when d is not a power of a prime number.
\noindent In this article, we revisit the problem of finding Mutually Unbiased
Bases (MUB's) in any dimension d. The method is very elementary, using the
simple unitary matrices introduced by Schwinger in 1960, together with their
diagonalizations. The Vandermonde matrix based on the d-th roots of unity
plays a major role. This allows us to show the existence of a set of 3 MUB's in
any dimension, to give conditions for existence of more than 3 MUB's for d
even or odd number, and to recover the known result of existence of d+1 MUB's
for d a prime number. Furthermore the construction of these MUB's is very
explicit. As a by-product, we recover results about Gauss Sums, known in number
theory, but which have apparently not been previously derived from MUB
properties.Comment: International Conference on Transport and Spectral Problems in
Quantum Mechanics held in Honor of Jean-Michel Combes, Cergy Pontoise :
France (2006