Symmetric informationally complete quantum measurements

TLDR

SIC‑POVMs are rank‑one POVMs with \(d^2\) equiangular elements in dimension \(d\), relevant for quantum tomography, cryptography, and foundational studies. The authors aim to construct SIC‑POVMs in low dimensions and conjecture a group‑covariant family exists in all dimensions. They successfully construct SIC‑POVMs in dimensions 2, 3, and 4, and provide numerical evidence supporting the conjecture up to dimension 45.

Abstract

We consider the existence in arbitrary finite dimensions d of a POVM comprised of d^2 rank-one operators all of whose operator inner products are equal. Such a set is called a ``symmetric, informationally complete'' POVM (SIC-POVM) and is equivalent to a set of d^2 equiangular lines in C^d. SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.