Mutually unbiased frames

TL;DR

This paper introduces the concept of mutually unbiased frames, a broad generalization of unbiasedness in quantum measurements, and applies it to derive new results about fiducial states and mutually unbiased bases in quantum information theory.

Contribution

It defines mutually unbiased frames as a unifying framework and uses it to prove the non-existence of real fiducial states in even dimensions and to parameterize unknown fiducial states with minimal variables.

Findings

Real fiducial states do not exist in even dimensions.

Unknown $d$-dimensional fiducial states are parameterized with roughly $3d/2$ real variables.

Multi-parametric families of states with minimum uncertainty are constructed in every dimension.

Abstract

In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and 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. After introducing the tool, its power is shown by finding the following results about the last mentioned class of constellations: (i) real fiducial states do not exist in any even dimension, and (ii) unknown $d$-dimensional fiducial states are parameterized, a priori, with roughly $3d/2$ real variables only, without loss of generality. Furthermore, multi-parametric families of pure quantum states having minimum uncertainty with regard to several choices of $d+1$ orthonormal bases are shown, in every dimension $d$. These last 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.