Rotational covariance and GHZ contradictions for three or more particles of any dimension

TL;DR

This paper explores how GHZ states' transformation properties under symmetry groups lead to contradictions with local hidden variable theories, introducing new proofs that cover all particle numbers and dimensions.

Contribution

The paper introduces new methods of proof for GHZ contradictions that apply to any number of particles and dimensions, filling gaps in previous theorems.

Findings

GHZ contradictions arise from different transformation representations of states and operators.

New proofs cover all particle numbers N ≥ 3 for any dimension d.

The methods unify and extend previous results, including those by Ryu et al.

Abstract

Greenberger-Horne-Zeilinger (GHZ) states are characterized by their transformation properties under a continuous symmetry group, and $N$-body operators that transform covariantly exhibit a wealth of GHZ contradictions. We show that local or noncontextual hidden variables cannot duplicate this covariance for any state-changing transformations, and we extract specific GHZ contradictions from discrete subgroups, with no restrictions on particle number $N$ or dimension $d$ except for the fundamental requirement that $N \geq 3$ for nonprobabilistic contradictions. However, the specific contradictions fall into three regimes distinguished by increasing demands on the number of measurement operators required for the proofs. We introduce new methods of proof that define these regimes. The first recovers theorems equivalent to those found recently by Ryu et. al. \cite{RLZL}, the first operator-based theorems for all odd dimensions, $d$, covering many (but not all) particle numbers $N$ for each $d$. The second and third produce new theorems that fill all remaining gaps down to $N=3$, for every $d$. The common origin of all such GHZ contradictions is that the GHZ states and measurement operators transform according to different representations of the symmetry group, which has an intuitive physical interpretation.