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arXiv:1012.1737 · Physical Review A

Generating nonclassical correlations without fully aligning measurements

Joel J. Wallman, Yeong-Cherng Liang, Stephen D. Bartlett

Published Feb 28, 2011• 38 citations
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Abstract

We investigate the scenario where spatially separated parties perform measurements in randomly chosen bases on an $N$-partite Greenberger-Horne-Zeilinger state. We show that without any alignment of the measurements, the observers will obtain correlations that violate a Bell inequality with a probability that rapidly approaches 1 as $N$ increases and that this probability is robust against noise. We also prove that restricting these randomly chosen measurements to a plane perpendicular to a common direction will always generate correlations that violate some Bell inequality. Specifically, if each observer chooses their two measurements to be locally orthogonal, then the $N$ observers will violate one of two Bell inequalities by an amount that increases exponentially with $N$. These results are also robust against noise and perturbations of each observer's reference direction from the common direction.

DOI
10.1103/physreva.83.022110

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We investigate the scenario where spatially separated parties perform measurements in randomly chosen bases on an NNN-partite Greenberger-Horne-Zeilinger state. We show that without any alignment of the measurements, the observers will obtain correlations that violate a Bell inequality with a probability that rapidly approaches 1 as NNN increases and that this probability is robust against noise. We also prove that restricting these randomly chosen measurements to a plane perpendicular to a common direction will always generate correlations that violate some Bell inequality. Specifically, if each observer chooses their two measurements to be locally orthogonal, then the NNN observers will violate one of two Bell inequalities by an amount that increases exponentially with NNN. These results are also robust against noise and perturbations of each observer's reference direction from the common direction.
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.
Quantum theory allows for correlations between the outcomes of distant measurements that are inconsistent with any locally causal model, as demonstrated by the violation of a Bell inequality. Typical demonstrations of these correlations require careful alignment between the measurements, which requires distant parties to share a reference frame. Here, we prove, following a numerical observation by Shadbolt et al., that if two parties share a Bell state and each party preforms measurements along three perpendicular directions on the Bloch sphere, then the parties will always violate a Bell inequality. Furthermore, we prove that this probability is highly robust against local depolarizing noise, in that small levels of noise only decrease the probability of violating a Bell inequality by a small amount. We also show that generalizing to NNN parties can increase the robustness against noise. These results improve on previous ones that only allowed a high probability of violating a Bell inequality for large numbers of parties.
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.
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.
A common experimental strategy for demonstrating nonclassical correlations is to show violation of a Bell inequality by measuring a continuously emitted stream of entangled photon pairs. The measurements involve the detection of photons by two spatially separated parties. The detection times are recorded and compared to quantify the violation. The violation critically depends on determining which detections are coincident. Because the recorded detection times have ``jitter,'' coincidences cannot be inferred perfectly. In the presence of settings-dependent timing errors, this can allow a local-realistic system to show apparent violation, the so-called ``coincidence loophole.'' Here, we introduce a family of Bell inequalities based on signed, directed distances between the parties' sequences of recorded time tags. Given that the time tags are recorded for synchronized, fixed observation periods and that the settings choices are random and independent of the source, violation of these inequalities unambiguously shows nonclassical correlations violating local realism. Distance-based Bell inequalities are generally useful for two-party configurations where the effective size of the measurement outcome space is large or infinite. We show how to systematically modify the underlying Bell functions to improve the signal-to-noise ratio and to quantify the significance of the violation.
Abstract Greenberger–Horne–Zeilinger (GHZ) states are intuitively known to be the most nonclassical ones. They lead to the most radically nonclassical behavior of three or more entangled quantum subsystems. In the case of two-dimensional systems, it has been shown that GHZ states lead to an exponentially higher robustness of Bell nonclassicality against the white noise for geometrical inequalities than in the case of Weinfurter–Werner–Wolf–Żukowski–Brukner ones. We introduce geometrical Bell inequalities for collections of arbitrarily many systems of any dimensionality. We show that the violation factor of these inequalities grows exponentially with the number of parties and study their behavior in terms of dimensionality of subsystems and number of local measurements. We also investigate various strategies of assigning mathematical objects to events in the experiment, each leading to different violation ratios.