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.
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Journal of Mathematical Physics, American Institute of Physics (AIP), 2009, 50, pp.032104. 〈10.1063/1.3078420〉
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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〉

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