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arXiv:1011.5206 · Physical Review A

More nonlocality with less entanglement

Thomas Vidick, Stephanie Wehner

Published May 13, 2011• 65 citations
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Abstract

Recent numerical investigations [K. P\'al and T. V\'ertesi, Phys. Rev. A 82, 022116 (2010)] suggest that the I3322 inequality, arguably the simplest extremal Bell inequality after the CHSH inequality, has a very rich structure in terms of the entangled states and measurements that maximally violate it. Here we show that for this inequality the maximally entangled state of any dimension achieves the same violation than just a single EPR pair. In contrast, stronger violations can be achieved using higher dimensional states which are less entangled. This shows that the maximally entangled state is not the most nonlocal resource, even when one restricts attention to the most simple extremal Bell inequalities.

DOI
10.1103/physreva.83.052310

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Recent numerical investigations [K. P'al and T. V'ertesi, Phys. Rev. A 82, 022116 (2010)] suggest that the I3322 inequality, arguably the simplest extremal Bell inequality after the CHSH inequality, has a very rich structure in terms of the entangled states and measurements that maximally violate it. Here we show that for this inequality the maximally entangled state of any dimension achieves the same violation than just a single EPR pair. In contrast, stronger violations can be achieved using higher dimensional states which are less entangled. This shows that the maximally entangled state is not the most nonlocal resource, even when one restricts attention to the most simple extremal Bell inequalities.
Motivated by the recent proposal for the quantification of the Bell nonlocality [Phys. Rev. A 92, 030101(R) (2015)], we provide an extensive analysis of this concept, namely, the volume of violation. This new measure of nonlocality has been proposed to deliver an evidence that the anomaly between maximally entangled states and states that maximally violate a Bell inequality is caused by the method, which has been applied to quantify nonlocality and the anomaly disappears when the volume of violation in used. We prove that such conclusion is not true for all bipartite quantum systems with dimension $d\ensuremath{\bigotimes}d$. In fact, if one assumes the same condition as in Phys. Rev. A 92, 030101(R) (2015), it is limited to $d\ensuremath{\le}7$. Furthermore, we discuss several type of local measurements and their influence on the quantification of nonlocality by mean of volume of violation. In particular, we propose a set of local observers that significantly enhance the volume of violation, which may have important meaning in a real-world application. Finally, we present a comparison between the volume of violation and the maximal violation of a Bell inequality.
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.
Recently, Wildfeuer et al. [Phys. Rev. A 76, 052101 (2007)] studied possible experiments demonstrating nonlocal correlation effects through the violation of various Bell-type inequalities by maximally path-entangled number states of the form $(|N⟩|0⟩+{e}^{i\ensuremath{\varphi}}|0⟩|N⟩)/\sqrt{2}$, the so-called N00NN00NN00N states, and some strong violations were found. In this paper, we re-examine the same Bell-type inequalities with respect to maximally path-entangled coherent states of the form $\mathcal{N}({|\ensuremath{\alpha}⟩}_{a}{|0⟩}_{b}+{e}^{i\ensuremath{\Phi}}{|0⟩}_{a}{|\ensuremath{\alpha}\text{ }{e}^{i\ensuremath{\theta}}⟩}_{b})$. We find in many cases even stronger violations of the Bell-type inequalities than appear to be possible with the N00NN00NN00N states.
While all bipartite pure entangled states are known to generate correlations violating a Bell inequality, and are therefore nonlocal, the quantitative relation between pure state entanglement and nonlocality is poorly understood. In fact, some Bell inequalities are maximally violated by non-maximally entangled states and this phenomenon is also observed for other operational measures of nonlocality. In this work, we study a recently proposed measure of nonlocality defined as the probability that a pure state displays nonlocal correlations when subjected to random measurements. We first prove that this measure satisfies some natural properties for an operational measure of nonlocality. Then, we show that for pure states of two qubits the measure is monotonic with entanglement for all correlation two-outcome Bell inequalities: for all these inequalities, the more the state is entangled, the larger the probability to violate them when random measurements are performed. Finally, we extend our results to the multipartite setting.
There is no doubt about the fact that entanglement and nonlocality are distinct resources. It is acknowledged that a clear illustration of this point is the difference between maximally entangled states and states that maximally violate a Bell inequality. We give strong evidence that this anomaly may be an artifact of the measures that have been used to quantify nonlocality. By reasoning that the numeric value of a Bell function is akin to a witness rather than a quantifier, we define a measure of nonlocality and show that, for pairs of qutrits and of four-level systems, maximal entanglement does correspond to maximal nonlocality in the same scenario that gave rise to the discrepancy.
Recently a new Bell inequality has been introduced by Collins et al. [Phys. Rev. Lett. 88, 040404 (2002)], which is strongly resistant to noise for maximally entangled states of two d-dimensional quantum systems. We prove that a larger violation, or equivalently a stronger resistance to noise, is found for a nonmaximally entangled state. It is shown that the resistance to noise is not a good measure of nonlocality and we introduce some other possible measures. The nonmaximally entangled state turns out to be more robust also for these alternative measures. From these results it follows that two von Neumann measurements per party may be not optimal for detecting nonlocality. For d=3,4,d=3,4,d=3,4, we point out some connections between this inequality and distillability. Indeed, we demonstrate that any state violating it, with the optimal von Neumann settings, is distillable.