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

TL;DR

This paper demonstrates that block circulant matrices with circulant blocks can be used to prove the existence of $d+1$ mutually unbiased bases in complex spaces of prime power dimensions, extending previous results and connecting to Weil sums.

Contribution

The paper introduces a simplified proof technique using block-circulant matrices to establish the existence of mutually unbiased bases in prime power dimensions.

Findings

Existence of $d+1$ mutually unbiased bases in $ ext{C}^d$ for $d=p^n$

Connection between mutually unbiased bases and properties of Weil sums

Generalization of quadratic Gauss sums properties for prime $p extgreater 3$

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 $$| b\cdot b'| = \frac{1}{\sqrt d}$$ ($b\cdot b'$ 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.