A small frame and a certificate of its injectivity

TL;DR

This paper constructs a specific set of eleven vectors in four-dimensional space to demonstrate injective measurements, disproving a recent conjecture and using algebraic methods for proof.

Contribution

It introduces a concrete example of an injective measurement frame in 4-space, challenging previous conjectures with algebraic proof techniques.

Findings

Constructed a complex frame of eleven vectors in 4-space.

Proved that this frame defines injective measurements.

Disproved a recent conjecture on injective measurement frames.

Abstract

We present a complex frame of eleven vectors in 4-space and prove that it defines injective measurements. That is, any rank-one $4\times 4$ Hermitian matrix is uniquely determined by its values as a Hermitian form on this collection of eleven vectors. This disproves a recent conjecture of Bandeira, Cahill, Mixon, and Nelson. We use algebraic computations and certificates in order to prove injectivity.