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arXiv:1303.5326 · Physical Review A

Greenberger-Horne-Zeilinger theorem for N qudits

Junghee Ryu, Changhyoup Lee, Marek Żukowski, Jinhyoung Lee

Published Oct 2, 2013• 24 citations
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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 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 $N$.

DOI
10.1103/physreva.88.042101

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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 nnn-qudit GHZ state. By doing so we are able to construct a multisetting GHZ paradox for the nnn-qubit GHZ state, with nnn being arbitrary, which is genuine nnn-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.
Greenberger-Horne-Zeilinger (GHZ) theorem asserts that there is a set of mutually commuting nonlocal observables with a common eigenstate on which those observables assume values that refute the attempt to assign values only required to have them by the local realism of Einstein, Podolsky, and Rosen (EPR). It is known that for a three-qubit system there is only one form of the GHZ-Mermin-like argument with equivalence up to a local unitary transformation, which is exactly Mermin's version of the GHZ theorem. In this paper, however, for a four-qubit system which was originally studied by GHZ, we show that there are nine distinct forms of the GHZ-Mermin-like argument. The proof is obtained by using some geometric invariants to characterize the sets of mutually commuting nonlocal spin observables on the four-qubit system. It is proved that there are at most nine elements (except for a different sign) in a set of mutually commuting nonlocal spin observables in the four-qubit system, and each GHZ-Mermin-like argument involves a set of at least five mutually commuting four-qubit nonlocal spin observables with a GHZ state as a common eigenstate in GHZ's theorem. Therefore, we present a complete construction of the GHZ theorem for the four-qubit system.