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arXiv:1809.05727 · International Journal of Quantum Information

Minimal scenario facet Bell inequalities for multi-qubit states

Arpan Das, Chandan Datta, Pankaj Agrawal

Published Jan 3, 2023• 2 citations
arXivHTMLPDF

Abstract

Facet inequalities play an important role in detecting the nonlocality of a quantum state. The number of such inequalities depends on the Bell test scenario. With the increase in the number of parties, measurement outcomes, or/and the number of measurement settings, there are more nontrivial facet inequalities. For several Bell scenarios, by involving two dichotomic measurement settings for two parties and one dichotomic measurement by other parties, we show that the local polytope has only one nontrivial facet. For three parties, we have three variants of this inequality, depending upon which party is doing one dichotomic measurement. This measurement scenario for a multipartite state may be considered as the minimal scenario involving multipartite correlations that can detect nonlocality. We show that this inequality is violated by all generalized GHZ states. Being the only facet Bell inequality, this inequality is also violated by any entangled three-qubit pure state. We also show that for noisy [Formula: see text] states, our inequality is more effective than the well-known Mermin inequality.

DOI
10.1142/s0219749923500053

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Facet inequalities play an important role in detecting the nonlocality of a quantum state. The number of such inequalities depends on the Bell test scenario. With the increase in the number of parties, measurement outcomes, or/and the number of measurement settings, there are more nontrivial facet inequalities. For several Bell scenarios, by involving two dichotomic measurement settings for two parties and one dichotomic measurement by other parties, we show that the local polytope has only one nontrivial facet. For three parties, we have three variants of this inequality, depending upon which party is doing one dichotomic measurement. This measurement scenario for a multipartite state may be considered as the minimal scenario involving multipartite correlations that can detect nonlocality. We show that this inequality is violated by all generalized GHZ states. Being the only facet Bell inequality, this inequality is also violated by any entangled three-qubit pure state. We also show that for noisy [Formula: see text] states, our inequality is more effective than the well-known Mermin inequality.
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.
Bell inequalities reveal the fundamentally nonlocal character of quantum mechanics. In this regard, one of the interesting problems is to explore all possible Bell inequalities that demonstrate a gap between local and nonlocal quantum behaviour. This is useful for the geometric characterisation of the set of nonlocal correlations achievable within quantum theory. Moreover, it provides a systematic way to construct Bell inequalities that are tailored to specific quantum information processing tasks. This characterisation is well understood in the simplest (2,2,2)(2,2,2)(2,2,2) scenario, namely two parties performing two binary outcome measurements. However, beyond this setting, relatively few Bell inequalities are known, and the situation becomes particularly scarce in scenarios involving a greater number of outcomes. Here, we consider the (2,2,3)(2,2,3)(2,2,3) scenario, or two parties performing two three-outcome measurements, and characterise all Bell inequalities that can arise from the simplest sum-of-squares decomposition and are maximally violated by the maximally entangled state of local dimension three. We then utilise them to self-test this state, along with a class of three-outcome measurements.
In multipartite Bell scenarios, we study the nonlocality robustness of the Greenberger-Horne-Zeilinger (GHZ) state. When each party performs planar measurements forming a regular polygon, we exploit the symmetry of the resulting correlation tensor to drastically accelerate the computation of (i) a Bell inequality via Frank-Wolfe algorithms and (ii) the corresponding local bound. The Bell inequalities obtained are facets of the symmetrized local polytope and they give the best-known upper bounds on the nonlocality robustness of the GHZ state for three to ten parties. Moreover, for four measurements per party, we generalize our facets and hence show, for any number of parties, an improvement on Mermin's inequality in terms of noise robustness. We also compute the detection efficiency of our inequalities and show that some give rise to the activation of nonlocality in star networks, a property that was only shown with an infinite number of measurements.
Nonlocality shapes quantum correlations, revealed through the violation of Bell inequalities. The intersection of all valid Bell inequalities is the so-called local polytope. In multipartite systems, characterizing the local polytope quickly becomes an intractable task as the system size increases. Optimizing Bell inequalities to maximize the ratio between their quantum value and classical bound is key to understanding multipartite nonlocality. We propose a gradient-based method for this optimization. Numerical results indicate that local maxima of this ratio typically correspond to facet Bell inequalities of the local polytope. This enables an iterative search for tight and robust Bell inequalities. Applied to permutation-invariant scenarios, the method provides tight Bell inequalities with large quantum violations and facilitates experimental certification of Bell correlations without full knowledge of the local polytope. Moreover, analytical results of the maximum ratio are derived in the thermodynamic limit.
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.
Abstract Characterizing the set of all Bell inequalities is a notably hard task. An insightful method of solving it in case of Bell correlation inequalities for scenarios with two dichotomic measurements per site—for arbitrary number of parties—was given in references [2001 Phys. Rev. A 64 010102(R)] and [2001 Phys. Rev. A 64 032112]. Using complex-valued correlation functions, we generalize their approach to a broader class of Bell scenarios, in which the parties may choose from more than two multi-outcome measurements. Although the resulting families of Bell inequalities are not always tight, their coefficients have an intuitively understandable structure. We probe their usefulness by numerically testing their ability to detect Bell nonclassicality in simple interferometric experiments. Moreover, we identify a similar structure in the CGLMP inequality expressed in a correlation-based form, which allows us to generalise it to three parties.