The Lie Algebraic Significance of Symmetric Informationally Complete Measurements

TL;DR

This paper explores the connection between symmetric informationally complete measurements in quantum mechanics and Lie algebra structures, proposing a new algebraic approach to address the existence problem of SIC-POVMs.

Contribution

It introduces a novel Lie algebraic framework for analyzing SIC-POVMs, linking their existence to specific structures in the adjoint representation of gl(d,C).

Findings

Structure constants characterize SIC-POVMs up to unitary equivalence.

Existence of SIC-POVMs is equivalent to certain Lie algebraic structures.

Transforming the problem may aid in solving the existence question.

Abstract

Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than or equal to 67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.