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arXiv:1703.01841 · Physical Review Letters

Ubiquitous Nonlocal Entanglement with Majorana Zero Modes

Alessandro Romito, Yuval Gefen

Published Oct 13, 2017• 16 citations
arXivHTMLPDF

Abstract

Entanglement in quantum mechanics contradicts local realism and is a manifestation of quantum nonlocality. Its presence can be detected through the violation of Bell, or Clauser-Horne-Shimony-Holt (CHSH) inequalities. Paradigmatic quantum systems provide examples of both, nonentangled and entangled states. Here, we consider a minimal complexity setup consisting of six Majorana zero modes. We find that any allowed state in the degenerate Majorana space is nonlocally entangled. We show how to measure (with available techniques) the CHSH-violating correlations using either intermediate strength or weak measurement protocols.

DOI
10.1103/physrevlett.119.157702

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Entanglement in quantum mechanics contradicts local realism and is a manifestation of quantum nonlocality. Its presence can be detected through the violation of Bell, or Clauser-Horne-Shimony-Holt (CHSH) inequalities. Paradigmatic quantum systems provide examples of both, nonentangled and entangled states. Here, we consider a minimal complexity setup consisting of six Majorana zero modes. We find that any allowed state in the degenerate Majorana space is nonlocally entangled. We show how to measure (with available techniques) the CHSH-violating correlations using either intermediate strength or weak measurement protocols.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.
We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between two measurement settings, with each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the four total measurements are carried out by homodyne detection. The largest violation of the CHSH inequality is obtained when all four measurements are squeezed-and-displaced on-off detections. We test the robustness of violations against imperfections in on-off detectors and single-photon sources, finding that the squeezed-and-displaced measurement schemes perform better than the displacement-only measurement schemes.
Quantum nonlocality is presented often as the most remarkable and inexplicable phenomenon known to modern science. It has been known already for a long time that the probabilistic models used to prove Bell and Clauser-Horn-Shimony-Holt inequalities (BI-CHSH) for spin polarization correlation experiments (SPCE) are incompatible with the experimental protocols of SPCE. In particular these models use the same common probability space, joint probability distributions and/or conditional independence to describe coincidence experiments in incompatible experimental settings. Strangely enough these results are not known or simply neglected. This is why we will once again reanalyze Bell locality assumptions and show that they have nothing to do with the notion of Einsteinian locality therefore their violation should not be called quantum nonlocality but rather quantum non-Kolmogorovness or quantum contextuality. Moreover if local variables describing the measuring instruments are correctly taken into account then BI-CHSH can no longer be proven and one can easily construct non-signaling probabilistic models able to reproduce the predictions of QT. The violation of BI-CHSH is considered usually as a proof that a quantum state is entangled. Since BI-CHSH are violated also in some experiments from outside the domain of quantum physics therefore the entanglement is not exclusively a quantum phenomenon. In order to further demystify these notions we show that one can prepare two macroscopic systems in such a way that simple realizable local experiments on these systems violate BI. In view of these arguments the further testing of BI-CHSH inequalities in search for the loopholes does not seem to be necessary.
Nonlocality is a fascinating and counterintuitive aspect of nature, revealed by the violation of a Bell inequality. The standard and easiest configuration in which Bell inequalities can be measured has been proposed by Clauser-Horne-Shimony-Holt (CHSH). However, alternative nonlocality tests can also be carried out. In particular, Bell inequalities requiring multiple measurement settings can provide deeper fundamental insights about quantum nonlocality, as well as offering advantages in the presence of noise and detection inefficiency. In this paper we show how these nonlocality tests can be performed using a commercially available source of entangled photon pairs. We report the violation of a series of these nonlocality tests (I3322{I}_{3322}I3322​, I4422{I}_{4422}I4422​, and chained inequalities). With the violation of the chained inequality with 4 settings per side we put an upper limit at 0.49 on the local content of the states prepared by the source (instead of 0.63 attainable with CHSH). We also quantify the amount of true randomness that has been created during our experiment (assuming fair sampling of the detected events).
The key feature in correlations established by multi-party quantum entangled states is nonlocality. A quantity to measure the average nonlocality, distinguishing it from shared randomness and in a direct relation with no-signaling stochastic processes (which provide an operational interpretation of quantum correlations, without involving information transmission between the parties as to sustain causality), is proposed and resolved exhaustively for the quantum correlations established by a Clauser-Horne-Shimony-Holt setup (or CHSH box). The amount of nonlocality that is available in a CHSH box is measured by its proximity to the nearest Popescu-Rohrlich set of causal stochastic processes (aka a PR box) in the no-signaling polytope, related by polyhedral duality to Bell's correlation function. The proposed amount of average nonlocality is an entanglement monotone with a simple relation to concurrence. We provide the optimal setup vectors of a maximally nonlocal CHSH box for any entangled pair. The strongest nonlocality is the fraction [Formula: see text] of a PR box, attained by maximally entangled qubit pairs. The most economical causal stochastic process reproducing any maximally nonlocal CHSH box is developed. Data produced by a computer implementation of the simulator agrees with the quantum mechanical formulas.
The assumption of local realism imposes constraints, such as Bell inequalities, on quantities obtained from measurements. In recent years, various tests of local realism have gained popularity in undergraduate laboratories, giving students the exciting opportunity to experimentally contradict this philosophical assumption. The standard test of the CHSH (Clauser–Horne–Shimony–Holt) Bell inequality requires 16 measurements, whereas a test of Freedman's inequality requires only three measurements. The calculations required to test Freedman's inequality are correspondingly simpler and the theory is less abstract. We suggest that students may benefit from testing Freedman's inequality before proceeding to the CHSH inequality and other more complicated experiments. Our measured data violated Freedman's inequality by more than six standard deviations.