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

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dc.creatorCombescure, M.
dc.date2007-10-30
dc.date.accessioned2026-07-07T13:07:26Z
dc.date.available2026-07-07T13:07:26Z
dc.descriptionIn 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 $$| b\cdot b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ 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.
dc.identifierhttps://arxiv.org/abs/0710.5643
dc.identifierhttp://arxiv.org/abs/0710.5643
dc.identifierJournal of Mathematical Physics 50 (2009) 032104
dc.identifierdoi:10.1063/1.3078420
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/228092
dc.subjectMathematical Physics
dc.subjectQuantum Physics
dc.titleBlock circulant matrices with circulant blocks, weil sums and mutually unbiased bases, II. The prime power case
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