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arXiv:1206.2718 · Physical Review Letters

Greenberger-Horne-Zeilinger Paradoxes from Qudit Graph States

Weidong Tang, Sixia Yu, C. H. Oh

Published Mar 6, 2013• 40 citations
arXivHTMLPDF

Abstract

One fascinating way of revealing quantum nonlocality is the all-versus-nothing test due to Greenberger, Horne, and Zeilinger (GHZ) known as the GHZ paradox. So far genuine multipartite and multilevel GHZ paradoxes are known to exist only in systems containing an odd number of particles. Here we shall construct GHZ paradoxes for an arbitrary number (greater than 3) of particles with the help of qudit graph states on a special kind of graphs, called GHZ graphs. Furthermore, based on the GHZ paradox arising from a GHZ graph, we derive a Bell inequality with two d-outcome observables for each observer, whose maximal violation attained by the corresponding graph state, and a Kochen-Specker inequality testing the quantum contextuality in a state-independent fashion.

DOI
10.1103/physrevlett.110.100403

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One fascinating way of revealing quantum nonlocality is the all-versus-nothing test due to Greenberger, Horne, and Zeilinger (GHZ) known as the GHZ paradox. So far genuine multipartite and multilevel GHZ paradoxes are known to exist only in systems containing an odd number of particles. Here we shall construct GHZ paradoxes for an arbitrary number (greater than 3) of particles with the help of qudit graph states on a special kind of graphs, called GHZ graphs. Furthermore, based on the GHZ paradox arising from a GHZ graph, we derive a Bell inequality with two d-outcome observables for each observer, whose maximal violation attained by the corresponding graph state, and a Kochen-Specker inequality testing the quantum contextuality in a state-independent fashion.
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.
The paradox of Greenberger-Horne-Zeilinger (GHZ) disproves directly the concept of EPR elements of reality, based on the EPR correlations, in an all-versus-nothing way. A three-qubit experimental demonstration of the GHZ paradox was achieved nearly 20 years ago, followed by demonstrations for more qubits. Still, the GHZ contradictions underlying the tests can be reduced to a three-qubit one. We show an irreducible four-qubit GHZ paradox, and report its experimental demonstration. The bound of a three-setting-per-party Bell-GHZ inequality is violated by $7\ensuremath{\sigma}$. The fidelity of the GHZ state was around 8181%81, and an entanglement witness reveals a violation of the separability threshold by $19\ensuremath{\sigma}$.
Nonlocality is the defining feature of quantum entanglement. Entangled states with multiple particles are of crucial importance in fundamental tests of quantum physics as well as in many quantum information tasks. One of the archetypal multipartite quantum states, Greenberger-Horne-Zeilinger (GHZ) state, allows one to observe the striking conflict of quantum physics to local realism in the so-called all-versus-nothing way. This is profoundly different from Bell's theorem for two particles, which relies on statistical predictions. Here, we demonstrate an integrated photonic chip capable of generating and manipulating the four-photon GHZ state. We perform a complete characterization of the four-photon GHZ state using quantum state tomography and obtain a state fidelity of 0.729±0.006. We further use the all-versus-nothing test and the Mermin inequalities to witness the quantum nonlocality of GHZ entanglement. Our work paves the way to perform fundamental tests of quantum physics with complex integrated quantum devices.
Abstract The Greenberger–Horne–Zeilinger (GHZ) paradox is an exquisite no-go theorem that shows the sharp contradiction between classical theory and quantum mechanics by ruling out any local realistic description of quantum theory. The investigation of GHZ-type paradoxes has been carried out in a variety of systems and led to fruitful discoveries. However, its range of applicability still remains unknown and a unified construction is yet to be discovered. In this work, we present a unified construction of GHZ-type paradoxes for graph states, and show that the existence of GHZ-type paradox is not limited to graph states. The results have important applications in quantum state verification for graph states, entanglement detection, and construction of GHZ-type steering paradox for mixed states. We perform a photonic experiment to test the GHZ-type paradoxes via measuring the success probability of their corresponding perfect Hardy-type paradoxes, and demonstrate the proposed applications. Our work deepens the comprehension of quantum paradoxes in quantum foundations, and may have applications in a broad spectrum of quantum information tasks.
Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.
The Greenberger-Horne-Zeilinger (GHZ) paradox, involving quantum systems with three or more subsystems, offers an 'all-vs-nothing' test of quantum nonlocality. Unlike Bell tests for bipartite systems, which reveal statistical contradictions, the GHZ paradox demonstrates a definitive (i.e. 100%) conflict between local hidden variable theories and quantum mechanics. Given this, how can the claim made in the title be justified? The key lies in recognising that GHZ games are typically played under a predefined promise condition for input distribution. By altering this promise, different GHZ games can be constructed. Here, we introduce a randomized variant of GHZ game, where the promise condition is randomly selected from multiple possibilities and revealed to only one of the parties chosen randomly. We demonstrate that this randomized GHZ paradox can also be perfectly resolved using a GHZ state, revealing a potentially stronger form of nonlocality than the original paradox. The claim of enhanced nonlocality is supported by its operational implications: correlations yielding perfect success in the randomized game offer a greater communication advantage than traditional GHZ correlations in a distributed multi-party communication complexity task.