# The Mutually Unbiased Bases Revisited

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1 IPNL - Institut de Physique Nucléaire de Lyon

Abstract : 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.\\ oindent 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.

Autor: M. Combescure -

Fuente: https://hal.archives-ouvertes.fr/

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