# Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case

Abstract : 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.
docType_s : Journal articles
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http://hal.in2p3.fr/in2p3-00184037
Contributor : Florès Sylvie <>
Submitted on : Tuesday, October 30, 2007 - 1:00:18 PM
Last modification on : Tuesday, October 30, 2007 - 1:00:18 PM

### Files

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### Citation

M. Combescure. Block circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case. Journal of Mathematical Physics, American Institute of Physics (AIP), 2009, 50, pp.032104. <10.1063/1.3078420>. <in2p3-00184037>