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Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
Combescure M.
Journal of Mathematical Physics 50 (2009) 032104 - http://hal.in2p3.fr/in2p3-00184037
Mathématiques/Physique mathématique
Physique/Physique mathématique
Physique/Physique Quantique
Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
M. Combescure1
1 :  IPNL - Institut de Physique Nucléaire de Lyon
http://www.ipnl.in2p3.fr/
CNRS : UMR5822 – IN2P3 – Université Claude Bernard - Lyon I
France
In our previous paper \cite{co1} we have shown that the theory of circulant matrices allows to recover the result that there exists $p+1$ Mutually Unbiased Bases in dimension $p$, $p$ being an arbitrary prime number. Two orthonormal bases $\mathcal B,\ \mathcal B'$ of $\mathbb C^d$ are said mutually unbiased if $\forall b\in \mathcal B, \ \forall b' \in \mathcal B'$ one has that $$\vert b\cdot b'\vert = \frac{1}{\sqrt d}$$ ($b\cdot b'$ hermitian scalar product in $\mathbb C^d$). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if $d=p^n$ ($p$ a prime number, $n$ any integer) there exists $d+1$ mutually Unbiased Bases in $\mathbb C^d$. Our result relies heavily on an idea of Klimov, Munoz, Romero \cite{klimuro}. As a subproduct we recover properties of quadratic Weil sums for $p\ge 3$, which generalizes the fact that in the prime case the quadratic Gauss sums properties follow from our results.

Articles dans des revues avec comité de lecture
2009
Journal of Mathematical Physics (J. Math. Phys.)
Publisher American Institute of Physics (AIP)
ISSN 0022-2488 
50
032104

LYCEN 2007-21
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