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arXiv:1812.10017 · Physical Review A

Bell inequalities from no-signaling distributions

Thomas Cope, Roger Colbeck

Published Aug 12, 2019• 25 citations
arXivHTMLPDF

Abstract

A Bell inequality is a constraint on a set of correlations whose violation can be used to certify nonlocality. They are instrumental for device-independent tasks such as key distribution or randomness expansion. In this work, we consider bipartite Bell inequalities where two parties have ${m}_{A}$ and ${m}_{B}$ possible inputs and give ${n}_{A}$ and ${n}_{B}$ possible outputs, referring to this as the $({m}_{A},{m}_{B},{n}_{A},{n}_{B})$ scenario. By exploiting knowledge of the set of extremal no-signaling distributions, we find all 175 Bell inequality classes in the (4,4,2,2) scenario as well as provide a partial list of 18 277 classes in the (4,5,2,2) scenario. We also use a probabilistic algorithm to obtain 5 classes of inequality in the (2,3,3,2) scenario, which we confirmed to be complete, 25 classes in the (3,3,2,3) scenario, and a partial list of 21 170 classes in the (3,3,3,3) scenario. Our inequalities are given as Supplemental Material. Finally, we discuss the application of these inequalities to the detection loophole problem and provide lower bounds on the detection efficiency threshold for small numbers of inputs and outputs.

DOI
10.1103/physreva.100.022114

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A Bell inequality is a constraint on a set of correlations whose violation can be used to certify nonlocality. They are instrumental for device-independent tasks such as key distribution or randomness expansion. In this work, we consider bipartite Bell inequalities where two parties have mA{m}_{A}mA​ and mB{m}_{B}mB​ possible inputs and give nA{n}_{A}nA​ and nB{n}_{B}nB​ possible outputs, referring to this as the (mA,mB,nA,nB)({m}_{A},{m}_{B},{n}_{A},{n}_{B})(mA​,mB​,nA​,nB​) scenario. By exploiting knowledge of the set of extremal no-signaling distributions, we find all 175 Bell inequality classes in the (4,4,2,2) scenario as well as provide a partial list of 18 277 classes in the (4,5,2,2) scenario. We also use a probabilistic algorithm to obtain 5 classes of inequality in the (2,3,3,2) scenario, which we confirmed to be complete, 25 classes in the (3,3,2,3) scenario, and a partial list of 21 170 classes in the (3,3,3,3) scenario. Our inequalities are given as Supplemental Material. Finally, we discuss the application of these inequalities to the detection loophole problem and provide lower bounds on the detection efficiency threshold for small numbers of inputs and outputs.
Nonlocal tests on multi-partite quantum correlations form the basis of protocols that certify randomness in a device-independent (DI) way. Such correlations admit a rich structure, making the task of choosing an appropriate test difficult. For example, extremal Bell inequalities are tight witnesses of nonlocality, but achieving their maximum violation places constraints on the underlying quantum system, which can reduce the rate of randomness generation. As a result there is often a trade-off between maximum randomness and the amount of violation of a given Bell inequality. Here, we explore this trade-off for more than two parties. More precisely, we study the maximum amount of randomness that can be certified by correlations with a particular violation of the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequality. For any even number of parties, we find that maximum randomness cannot occur beyond a threshold quantum violation, which increases with the number of parties, and we give a conjectured form of the maximum randomness in terms of the MABK value. We also show that maximum randomness can be obtained for any MABK violation for odd numbers of parties. To obtain our results, we derive new families of Bell inequalities certifying maximum randomness from a technique for randomness certification, which we call "expanding Bell inequalities''. Our technique allows a bipartite Bell expression to be used as a seed, and transformed into a multi-partite Bell inequality tailored for randomness certification, showing how intuition learned in the bipartite case can find use in more complex scenarios.
Physical principles constrain the way nonlocal correlations can be distributed among distant parties in a Bell-type experiment. These constraints are usually expressed by monogamy relations that bound the amount of Bell inequality violation observed by a set of parties by the violation observed by a different set of parties. Here we show that the no-signaling principle yields simple and tight monogamy relations for an important class of bipartite and multipartite Bell inequalities. We also link these trade-offs to the guessing probability---a key quantity in device-independent information processing.
Multipartite Bell nonlocality provides a device-independent probe of many-body quantum correlations, but its characterization is limited by the rapid growth of the underlying classical and quantum optimization problems. We develop a scalable method for constructing and certifying permutationally invariant Bell inequalities using only one- and two-body correlators. The construction gives families of inequalities with robust quantum violations for general mmm measurements as the number of parties NNN becomes large. To improve robustness against noise, we optimize the ratio of the quantum value to the classical bound for these families in the large-NNN limit. We then certify the resulting quantum violation using semidefinite programming. For the broad class of Bell inequalities studied here, the infinite-NNN ratios take simple rational values for finite mmm and converge to coth⁡(1)\coth(1)coth(1) as m→∞m\to\inftym→∞. The optimized inequalities efficiently detect many-body Bell nonlocality with collective measurements, with more measurement settings leading to stronger violations.
We derive tight bipartite Bell inequalities for various scenarios. A bipartite Bell scenario (X,Y,A,B)(X,Y,A,B)(X,Y,A,B) is defined by the numbers of settings and outcomes per party, X,AX,AX,A, and Y,BY,BY,B for Alice and Bob, respectively. We derive the complete set of facets of the local polytopes of (6,3,2,2), (3,3,3,2), (3,2,3,3), and (2,2,3,5). We provide extensive lists of facets for (2,2,4,4), (3,3,4,2), and (4,3,3,2). For each inequality we compute the maximum quantum violation, the resistance to noise, and the minimal symmetric detection efficiency required to close the detection loophole, for qubits, qutrits, and ququarts. Based on these results, we identify scenarios which perform better in terms of visibility, resistance to noise, or both, when compared to the Clauser-Horne-Shimony-Holt inequality. Such scenarios could find important applications in quantum communication.
Abstract The detection loophole problem arises when quantum devices fail to provide an output for some runs. If treating these devices in a device-independent manner, failure to include the unsuccessful runs in the output statistics can lead to an adversary falsifying security i.e. Bell inequality violation. If the devices fail with too high frequency, known as the detection threshold, then no security is possible, as the full statistics cannot violate a Bell inequality. In this work we provide an intuitive local hidden-variable strategy that the devices may use to falsify any two-party, binary-outcome no-signalling distribution up to a threshold of 2(m A + m B − 8)/(m A m B − 16), where m A, m B refer to the number of available inputs choices to the two parties. This value is the largest analytically predicted lower bound for no-signalling distributions. We strongly conjecture it gives the true detection threshold for m A = m B, and for computationally tractable scenarios we provide the Bell inequality which verifies this. We also prove that a non-trivial detection threshold remains, even when allowing one party an arbitrary number of input choices.
In this work, we study a particular class of Bell inequalities involving only direct equality-comparisons of outcomes. In the bipartite case, the scenario can be interpreted as a natural generalization of full-correlator in- equalities (XOR games) beyond binary outputs. We define the sub-polytope of the local polytope corresponding to this scenario and solve it for several bipartite and multipartite scenarios by leveraging some structural prop- erties. In doing so, we obtain thousands of new tight inequalities, many of which are also facets of the standard local polytope. We also define unanimous Bell inequalities, a particular case of the previous class applied to the multipartite setting in which only full-equality events (all outcomes equal) are considered. We show that such inequalities can always be written as deterministic nonlocal games, and we give a simple multipartite unanimous family and prove its local bound. We show that most of these inequalities admit quantum violations, and we also display aspects of their importance for nonlocality. For instance, we identify examples where such inequalities can act as dimension witnesses, outcome witnesses, witnesses of genuine multipartite nonlocality, as well as being relevant to CHSH. These results show that these simple and elegant inequalities by themselves provide a powerful tool for discovering new Bell inequalities and device-independent witnesses.