SIC-POVMs and MUBs: Geometrical Relationships in Prime Dimension

TL;DR

This paper explores the geometric relationships between SIC-POVMs and MUBs in prime dimensions, revealing simple structures and interpretations in generalized Bloch space that deepen understanding of quantum measurement frameworks.

Contribution

It demonstrates specific geometric relationships between SIC-POVMs and MUBs in prime dimensions and provides new interpretations of minimum uncertainty states and fiduciality conditions.

Findings

SIC-POVM forms a regular simplex in Bloch space

MUBs form mutually orthogonal simplices

Geometrical insights into uncertainty and fiduciality conditions

Abstract

The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d^2-1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d+1 mutually orthogonal d-1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d+1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs.