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 $\mathcal{B},{\mathcal{B}}^{\prime }$ of ${\mathbb{C}}^d$ are said mutually unbiased if $\forall b\in \mathcal{B},\forall b^{\prime }\in {\mathcal{B}}^{\prime }$ one has that $$|b\cdot b^{\prime }|=\frac{1}{\sqrt{d}}$$ ($b\cdot b^{\prime }$ 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.

References in corpus (4)

• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• Discrete phase-space structure of $n$-qubit mutually unbiased bases
• SU(2) nonstandard bases: the case of mutually unbiased bases
• Encryption via Entangled states belonging to Mutually Unbiased Bases