GHZ (Greenberger—Horne—Zeilinger) Theorem and GHZ States

A. Fahmi

Adán Cabello

The proof of Bell's theorem without inequalities by Greenberger, Horne, and Zeilinger (GHZ) is extended to multiparticle multilevel systems. The proposed procedure generalizes previous partial results and provides an operational characterization of the so-called GHZ states for multiparticle multilevel systems.

A. Fahmi, Mojtaba Golshani

Junghee Ryu, Changhyoup Lee, Marek Żukowski +1 more

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 one 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 $N$.

Junghee Ryu, Changhyoup Lee, Zhi Yin +4 more

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.

Xiao-Hua Wu, Hong-Shi Zong, Pang Hou-Rong