We show that the continuous-variable analogues to the multipartite entangled Greenberger-Horne-Zeilinger states of qubits violate Bell-type inequalities imposed by local realistic theories. Our results suggest that the degree of nonlocality of these nonmaximally entangled continuous-variable states, represented by the maximum violation, grows with increasing number of parties. This growth does not appear to be exponentially large as for the maximally entangled qubit states, but rather decreases for larger numbers of parties.
The Greenberger-Horne-Zeilinger (GHZ) paradox provides an all-versus-nothing test for the quantum nonlocality. In most of the GHZ paradoxes known so far each observer is allowed to measure only two alternative observables. Here we present a general construction for GHZ paradoxes in which each observer measures more than two observables given that the system is prepared in the
n-qudit GHZ state. By doing so we are able to construct a multisetting GHZ paradox for the
n-qubit GHZ state, with
n being arbitrary, which is genuine
n-partite; i.e., no GHZ paradox exists when restricted to a subset of a number of observers for a given set of Mermin observables. Our result fills up the gap of the absence of a genuine GHZ paradox for the GHZ state of an even number of qubits, especially the four-qubit GHZ state as used in GHZ's original proposal.
We present a generalized Greenberger-Horne-Zeilinger (GHZ) theorem, which involves more than two local measurement settings for some parties, and cannot be reduced to one with less settings. Our results hold for an odd number of parties. We use a set of observables, which are incompatible but share a common eigenstate, here a GHZ state. Such observables are called concurrent. The idea is illustrated with an example of a three-qutrit system and then generalized to systems of higher dimensions, and more parties. The GHZ paradoxes can lead to, e.g., secret sharing protocols.
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.
In a gedanken experiment N particles in a generalized Greenberger-Horne-Zeilinger-(GHZ)-type beam-entangled state (each particle can be in one of M beams) are fed into N symmetric
2M-port beam splitters (spatially separated). Correlation functions for such a process (using the Bell numbers value assignment approach) reveal a remarkable symmetry. For
N=M+1> 4 a series of GHZ paradoxes are shown.