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International Conference on Transport and Spectral Problems in Quantum Mechanics held in Honor of Jean-Michel Combes, Cergy Pontoise : France (2006)
The Mutually Unbiased Bases Revisited
M. Combescure1

The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d,\ B\ \mbox{and}\ B'$ are said mutually unbiased if $\forall b\in B,\ b'\in B'$ the scalar product $b\cdot b'$ 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 $\psi \in \mathbb C^d$ when measured in the mutually unbiased bases $B_{j}\ \mbox{of}\ \mathbb C^d$.\\ 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.
1 :  IPNL - Institut de Physique Nucléaire de Lyon
Physique/Physique Quantique

Physique/Physique mathématique
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MUB3.ps(154.3 KB)
MUB3.pdf(209.6 KB)