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arXiv:1408.1837 · Physical Review A

Demonstrating genuine multipartite entanglement and nonseparability without shared reference frames

Celal Furkan Şenel, Thomas Lawson, Marc A. Kaplan, Damian Markham, Eleni Diamanti

Published May 26, 2015• 13 citations
arXivHTMLPDF

Abstract

Multipartite nonlocality is of great fundamental interest and constitutes a useful resource for many quantum information protocols. However, demonstrating it in practice, by violating a Bell inequality, can be difficult. In particular, standard experimental setups require the alignment of distant parties' reference frames, which can be challenging or impossible in practice. In this work we study the violation of certain Bell inequalities, namely the Mermin, Mermin-Klyshko, and Svetlichny inequalities, without shared reference frames, when parties share a Greenberger-Horne-Zeilinger state. Furthermore, we analyze how these violations demonstrate genuine multipartite features of entanglement and nonlocality. For three, four, and five parties, we show that it is possible to violate these inequalities with high probability, when the parties choose their measurements from the three Pauli operators, defined only with respect to their local frames. Moreover, the probability of violation, and the amount of violation, are increased when each party chooses their measurements from the four operators describing the vertices of a tetrahedron. We also consider how many randomly chosen measurement directions are needed to violate the Bell inequalities with high probability. We see that the obtained levels of violation are sufficient to also demonstrate genuine multipartite entanglement and nonseparability. Finally, we show analytically that choosing from two measurement settings per party is sufficient to demonstrate the maximum degree of genuine multipartite entanglement and nonseparability with certainty when the parties' reference frames are aligned in one direction so that they differ only in rotations around one axis.

DOI
10.1103/physreva.91.052118

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Multipartite nonlocality is of great fundamental interest and constitutes a useful resource for many quantum information protocols. However, demonstrating it in practice, by violating a Bell inequality, can be difficult. In particular, standard experimental setups require the alignment of distant parties' reference frames, which can be challenging or impossible in practice. In this work we study the violation of certain Bell inequalities, namely the Mermin, Mermin-Klyshko, and Svetlichny inequalities, without shared reference frames, when parties share a Greenberger-Horne-Zeilinger state. Furthermore, we analyze how these violations demonstrate genuine multipartite features of entanglement and nonlocality. For three, four, and five parties, we show that it is possible to violate these inequalities with high probability, when the parties choose their measurements from the three Pauli operators, defined only with respect to their local frames. Moreover, the probability of violation, and the amount of violation, are increased when each party chooses their measurements from the four operators describing the vertices of a tetrahedron. We also consider how many randomly chosen measurement directions are needed to violate the Bell inequalities with high probability. We see that the obtained levels of violation are sufficient to also demonstrate genuine multipartite entanglement and nonseparability. Finally, we show analytically that choosing from two measurement settings per party is sufficient to demonstrate the maximum degree of genuine multipartite entanglement and nonseparability with certainty when the parties' reference frames are aligned in one direction so that they differ only in rotations around one axis.
Multipartite quantum nonlocality is an important diagnostic tool and resource for both researches in fundamental quantum mechanics and applications in quantum information protocols. Shared reference frames among all parties are usually required for experimentally observing quantum nonlocality, which is not possible in many circumstances. Previous results have shown violations of bipartite Bell inequalities with approaching unit probability, without shared reference frames. Here we experimentally demonstrate genuine multipartite quantum nonlocality without shared reference frames, using the Svetlichny inequality. A significant violation probability of 0.58 is observed with a high-fidelity three-photon Greenberger-Horne-Zeilinger state. Furthermore, when there is one shared axis among all the parties, which is the usual case in fiber-optic or earth-satellite links, the experimental results demonstrate the genuine three-partite nonlocality with certainty. Our experiment will be helpful for applications in multipartite quantum communication protocols.
We experimentally demonstrate, using qubits encoded in photon polarization, that two parties who share a single reference direction and use locally orthogonal measurements will always violate a Bell inequality, up to experimental deficiencies. This contrasts with the standard view of Bell inequalities, in which the parties need to completely align their measurements. Furthermore, we experimentally demonstrate that as the reference direction degrades the probability of the observers randomly choosing measurements that violate a Bell inequality decreases gradually and smoothly to 39.739.7%\ifmmode\pm\else\textpm\fi{}0.1%39.7 in the limiting case that the observers do not share a reference direction. This result promises simplified distribution of entanglement between separated parties, with applications in fundamental investigations of quantum physics and tasks such as quantum communication.
Local measurements acting on entangled quantum states give rise to a rich correlation structure in the multipartite scenario. We introduce a versatile technique to build families of Bell inequalities witnessing different notions of multipartite nonlocality for any number of parties. The idea behind our method is simple: a known Bell inequality satisfying certain constraints, for example the Clauser–Horne–Shimony–Holt inequality, serves as the seed to build new families of inequalities for more parties. The constructed inequalities have a clear operational meaning, capturing an essential feature of multipartite correlations: their violation implies that numerous subgroups of parties violate the inequality chosen as seed. The more multipartite nonlocal the correlations, the more subgroups can violate the seed. We illustrate our construction using different seeds and designing Bell inequalities to detect k -way nonlocal multipartite correlations, in particular, genuine multipartite nonlocal correlations—the strongest notion of multipartite nonlocality. For one of our inequalities we prove analytically that a large class of pure states that are genuine multipartite entangled (GME) exhibit genuine multipartite nonlocality for any number of parties, even for some states that are almost product. We also provide numerical evidence that this family is violated by all GME pure states of three and four qubits. Our results make us conjecture that this family of Bell inequalities can be used to prove the equivalence between genuine multipartite pure-state entanglement and nonlocality for any number of parties.
We consider the problem of demonstrating non-Bell-local correlations by performing local measurements in randomly chosen triads, i.e., three mutually unbiased bases, on a multipartite Greenberger-Horne-Zeilinger state. Our main interest lies in investigating the feasibility of using these correlations to certify multipartite entanglement in a device-independent setting. In contrast with previous works, our numerical results up to the eight-partite scenario suggest that if each triad is randomly but uniformly chosen according to the Haar measure, one always (except possibly for a set of measure zero) finds Bell-inequality-violating correlations. In fact, a substantial fraction of these is even sufficient to reveal, in a device-independent manner, various higher-order entanglement. In particular, for the specific cases of three parties and four parties, our results---obtained from semidefinite programming---suggest that these randomly generated correlations always reveal, even in the presence of a non-negligible amount of white noise, the genuine multipartite entanglement possessed by these states. In other words, provided local calibration can be carried out to good precision, a device-independent certification of the genuine multipartite entanglement contained in these states can, in principle, also be carried out in an experimental situation without sharing a global reference frame.
The nonlocal correlations of multipartite entangled states can be reproduced by a classical model if sufficiently many parties join together or if sufficiently many parties broadcast their measurement inputs. The maximal number m of groups and the minimal number k of broadcasting parties that allow for the reproduction of a given set of correlations quantify their multipartite nonlocal content. We show how upper bounds on m and lower bounds on k can be computed from the violation of the Mermin-Svetlichny inequalities. While n-partite Greenberger-Horne-Zeilinger states violate these inequalities maximally, we find that W states violate them only by a very small amount.
We show that correlations inconsistent with any locally causal description can be a generic feature of measurements on entangled quantum states. Specifically, spatially separated parties who perform local measurements on a maximally entangled state using randomly chosen measurement bases can, with significant probability, generate nonclassical correlations that violate a Bell inequality. For n parties using a Greenberger-Horne-Zeilinger state, this probability of violation rapidly tends to unity as the number of parties increases. We also show that, even with both a randomly chosen two-qubit pure state and randomly chosen measurement bases, a violation can be found about 10% of the time. Among other applications, our work provides a feasible alternative for the demonstration of Bell inequality violation without a shared reference frame.