Codes and Expansions (CodEx) Seminar

Dardo Goyeneche (Universidad de Antofagasta)
Mutually unbiased frames

In this talk, mutually unbiased frames are introduced as the most general notion of unbiasedness for sets composed by linearly independent normalized vectors. It encompasses the already existing notions of unbiasedness for orthonormal bases, regular simplices, equiangular tight frames, positive operator valued measure, and also includes symmetric informationally complete quantum measurements (SIC-POVM). As an application to SIC-POVM, it is shown that real fiducial states, covariant under the standard representation of the Weyl-Heisenberg group, do not exist in any even dimension. Also, the search of \(d\)-dimensional fiducial states is reduced to determine roughly \(3d/2\) real variables only, improving the standard assumption of \(2(d-1)\) real variables, in any dimension \(d\). Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of \(d+1\) orthonormal bases are shown. These families contain all existing fiducial states in every finite dimension, and the bases include maximal sets of \(d+1\) mutually unbiased bases, when \(d\) is a prime number.