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

摘要

In our previous paper [Combescure, M., "Circulant matrices, Gauss sums and the mutually unbiased bases. I. The prime number case," Cubo A Mathematical Journal (unpublished)] we have shown that the theory of circulant matrices allows to recover the result that there exists p+1 mutually unbiased bases in dimension p, p being an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 _, _' of C~d are said mutually unbiased if A　b ∈B, Vb' ∈_' one has that |b_ b'| =1 /√d(b_ b'Hermitian scalar product in 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 and n any integer) there exists d+1 mutually unbiased bases in C~d. Our result relies heavily on an idea of Klimov et al. ["Geometrical approach to the discrete Wigner function," J. Phys. A 39, 14471 (2006)]. As a subproduct we recover properties of quadratic Weil sums for p ≥3, which generalizes the fact that in the prime case the quadratic Gauss sum properties follow from our results.