Greenberger-Horne-Zeilinger theorem for N qudits

TL;DR

This paper extends the GHZ theorem to N qudits of arbitrary dimension D using incompatible observables, demonstrating genuine N-partite and D-dimensional nonlocality under specific divisibility conditions.

Contribution

It introduces a novel approach employing incompatible and concurrent observables to generalize the GHZ theorem for arbitrary N and D, beyond conventional methods.

Findings

GHZ theorem applies to N qudits with D dimensions when N is not divisible by all nonunit divisors of D

Uses incompatible and concurrent observables to establish nonlocality

Generalizes previous bipartite and qubit results to multi-qudit systems

Abstract

We generalize Greenberger-Horne-Zeilinger (GHZ) theorem to an arbitrary number of D-dimensional systems. Contrary to conventional approaches using compatible composite observables, we employ incompatible and concurrent observables, whose common eigenstate is still a generalized GHZ state. It is these concurrent observables which enable to prove a genuinely N-partite and D-dimensional GHZ theorem. Our principal idea is illustrated for a four-partite system with D which is an arbitrary multiple of 3. By extending to N qudits, we show that GHZ theorem holds as long as N is not divisible by all nonunit divisors of D, smaller than N.