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arXiv:2302.06320 · International Journal of Quantum Information

Revisited aspects of the local set in CHSH Bell scenario

N. Gigena, Giovanni Scala, Antonio Mandarino

Published Dec 20, 2022• 5 citations
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Abstract

The Bell inequalities stand at the cornerstone of the developments of quantum theory on both the foundational and applied side. The discussion started as a way to test whether the quantum description of reality is complete or not, but it developed in such a way that a new research area stemmed from it, namely quantum information. Far from being and exhausted topic, in this paper, we present a constructive and geometrically intuitive description of the local polytope and its facets in a bipartite Bell scenario with two dichotomic measurements per party.

DOI
10.1142/s0219749923400051

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The Bell inequalities stand at the cornerstone of the developments of quantum theory on both the foundational and applied side. The discussion started as a way to test whether the quantum description of reality is complete or not, but it developed in such a way that a new research area stemmed from it, namely quantum information. Far from being and exhausted topic, in this paper, we present a constructive and geometrically intuitive description of the local polytope and its facets in a bipartite Bell scenario with two dichotomic measurements per party.
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.
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.
Non-trivial facet inequalities play important role in detecting and quantifying the nonolocality of a state -- specially a pure state. Such inequalities are expected to be tight. Number of such inequalities depends on the Bell test scenario. With the increase in the number of parties, dimensionality of the Hilbert space, or/and the number of measurements, there are more nontrivial facet inequalities. By considering a specific measurement scenario, we find that for any multipartite qubit state, local polytope can have only one nontrivial facet. Therefore there exist a possibility that only one Bell inequality, and its permutations, would be able to detect the nonlocality of a pure state. The scenario involves two dichotomic measurement settings for two parties and one dichotomic measurement by other parties. This measurement scenario for a multipartite state may be considered as minimal scenario involving multipartite correlations that can detect nonlocality. We present detailed results for three-qubit states.
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.
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.
We study the set of almost-quantum correlations and their refinements in the simplest tripartite Bell scenario where each party is allowed to perform two dichotomic measurements. In contrast to its bipartite counterpart, we find that there already exist facet Bell inequalities that witness almost-quantum correlations beyond quantum theory in this simplest tripartite Bell scenario. Furthermore, we study the relation between the almost-quantum set and the hierarchy of supersets to the quantum set due to Navascu\'es-Pironio-Ac\'{\i}n (NPA) [Phys. Rev. Lett. 98, 010401 (2007)]. While the former lies between the first and the third level of the NPA hierarchy, we find that its second level does not contain and is not contained within the almost-quantum set. Finally, we investigate the hierarchy of refinements to the almost-quantum set due to Moroder et al. [Phys. Rev. Lett. 111, 030501 (2013)], which converges to the set of quantum correlations producible by quantum states having positive partial transposition. This allows us to consider (approximations of) the ``biseparable'' subsets of the almost-quantum set as well as of the quantum set of correlations, and thereby gain further insights into the subtle similarities and differences between the two sets. In addition, they allow us to identify candidate Bell-like inequalities that can serve as device-independent witnesses for genuine tripartite entanglement.