On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states

We address the problem of constructing positive operator-valued measures(POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one whichhave an inner product close to uniform. This is motivated by the relatedquestion of constructing symmetric informationally complete POVMs (SIC-POVMs)for which the inner products are perfectly uniform. However, SIC-POVMs arenotoriously hard to construct and despite some success of constructing themnumerically, there is no analytic construction known. We present twoconstructions of approximate versions of SIC-POVMs, where a small deviationfrom uniformity of the inner products is allowed. The first construction isbased on selecting vectors from a maximal collection of mutually unbiased basesand works whenever the dimension of the system is a prime power. The secondconstruction is based on perturbing the matrix elements of a subset of mutuallyunbiased bases. Moreover, we construct vector systems in $\C^n$ which are almost orthogonaland which might turn out to be useful for quantum computation. Ourconstructions are based on results of analytic number theory.Comment: 29 pages, LaTe