Skip to Main content Skip to Navigation
Journal articles

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.
Complete list of metadatas

Cited literature [15 references]  Display  Hide  Download

http://hal.in2p3.fr/in2p3-00184037
Contributor : Sylvie Flores <>
Submitted on : Tuesday, October 30, 2007 - 1:18:22 PM
Last modification on : Tuesday, November 19, 2019 - 2:37:55 AM
Long-term archiving on: : Monday, April 12, 2010 - 12:59:52 AM

Files

powerofprime.pdf
Files produced by the author(s)

Identifiers

Collections

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⟩

Share

Metrics

Record views

809

Files downloads

1031