Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum-Uncertainty States

TL;DR

This paper explores the properties and existence of symmetric informationally complete (SIC) quantum states, showing their close relation to orthonormal bases and minimum-uncertainty states, especially in prime dimensions, and proposes a conjecture on their mathematical structure.

Contribution

It demonstrates that if SIC-sets exist, they closely resemble orthonormal bases and are related to mutually unbiased bases and minimum-uncertainty states, providing new insights and a conjecture on their mathematical equations.

Findings

SIC-sets are nearly orthonormal in the space of density operators.

In prime dimensions, SIC-sets relate to mutually unbiased bases.

A conjecture on quadratic redundancy in SIC equations is proposed.

Abstract

Since Renes et al. [J. Math. Phys. 45, 2171 (2004)], there has been much effort in the quantum information community to prove (or disprove) the existence of symmetric informationally complete (SIC) sets of quantum states in arbitrary finite dimension. This paper strengthens the urgency of this question by showing that if SIC-sets exist: 1) by a natural measure of orthonormality, they are as close to being an orthonormal basis for the space of density operators as possible, and 2) in prime dimensions, the standard construction for complete sets of mutually unbiased bases and Weyl-Heisenberg covariant SIC-sets are intimately related: The latter represent minimum uncertainty states for the former in the sense of Wootters and Sussman. Finally, we contribute to the question of existence by conjecturing a quadratic redundancy in the equations for Weyl-Heisenberg SIC-sets.