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.

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Cited by in corpus (10)

• On mutually unbiased bases
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• Upper bounds on the error probabilities and asymptotic error exponents in quantum multiple state discrimination
• Recovery of Future Data via Convolution Nuclear Norm Minimization
• Quantum state discrimination bounds for finite sample size
• Time Series Forecasting via Learning Convolutionally Low-Rank Models
• $BCCB$ complex Hadamard matrices of order 9, and MUBs
• Efficient quantum circuits for dense and non-unitary operators
• The problem of mutually unbiased bases in dimension 6
• Towards a classification of incomplete Gabor POVMs in ${\mathbb{C}}^d$