Construction of all general symmetric informationally complete measurements

TL;DR

This paper constructs the complete set of all general symmetric informationally complete measurements (SIC-POVMs), establishing a correspondence with orthonormal bases in real vector spaces and exploring properties of weak SIC-POVMs.

Contribution

It provides a comprehensive construction of all general SIC-POVMs and links them to orthonormal bases, expanding understanding of measurement structures in quantum information.

Findings

Any orthonormal basis of a real vector space of dimension d^2-1 corresponds to a general SIC-POVM.

The set includes weak SIC-POVMs where elements can approach multiples of the identity.

Open question remains whether the set contains rank 1 SIC-POVMs in all finite dimensions.

Abstract

We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC-POVMs contains weak SIC-POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC-POVM.