From stabilizer states to SIC-POVM fiducial states

Abstract

In the stabilizer formalism of quantum computation, the Gottesman-Knill theorem shows that universal fault-tolerant quantum computation requires the resource called non-stabilizerness. Characterization, detection, and quantification of non-stabilizer states are basic issues in this context. In the paradigm of quantum measurement, symmetric informationally complete positive operator valued measuresplay a prominent role due to their structural symmetry and remarkable features. However, their existence in all dimensions, although strongly supported by a plethora of theoretical and numerical evidences, remains an elusive open problem. A standard method for constructing SICs is via the orbit of Heisenberg-Weyl group on a fiducial state. A natural question arises as the relation between stabilizer states and fiducial states. In this talk, we connect them by showing that they are on two extremes with respect to the p-norm of characteristic functions of quantum states. This not only reveals a simple path from stabilizer states to SIC fiducial states which shows in a quantitative fashion that they are as far away as possible from each other, but also provides a simple reformulation of Zauner's conjecture. A convenient criterion for non-stabilizerness and some open problems are also presented.