Rotational covariance and Greenberger-Horne-Zeilinger theorems for three or more particles of any dimension

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 the predicted measurement outcomes for covariant transformations, and we extract specific GHZ contradictions from discrete subgroups, with no restrictions on particle number $N$ or dimension $d$ except for the general requirement that $N\ensuremath{\ge}3$ for GHZ states. However, the specific contradictions fall into three regimes distinguished by increasing demands on the number of measurement operators required for the proofs. The first regime consists of proofs found recently by Ryu et al. [Phys. Rev. A 88, 042101 (2013)], the first operator-based theorems for all odd dimensions $d$, covering many (but not all) particle numbers $N$ for each $d$. We introduce alternative methods of proof that define second and third regimes and produce 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.