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arXiv:2004.07673 · Physical Review Research

Violation of all two-party facet Bell inequalities by almost-quantum correlations

Ravishankar Ramanathan

Published Jul 29, 2021• 4 citations
arXivHTMLPDF

Abstract

The characterization of the set of quantum correlations is a problem of fundamental importance in quantum information. The question whether every proper (tight) Bell inequality is violated in quantum theory is an intriguing one in this regard. Here we make significant progress in answering this question, by showing that every tight Bell inequality is violated by ``almost-quantum'' correlations, a semidefinite programming relaxation of the set of quantum correlations. As a consequence, we show that many (classes of) Bell inequalities including two-party correlation Bell inequalities and multioutcome nonlocal computation games, which do not admit quantum violations, are not facets of the classical Bell polytope. To do this, we make use of the intriguing connections between Bell correlations and the graph-theoretic Lov\'asz-theta set, discovered by Cabello-Severini-Winter (CSW). We also exploit connections between the cut polytope of graph theory and the classical correlation Bell polytope, to show that correlation Bell inequalities that define facets of the lower dimensional correlation polytope are violated in quantum theory.

DOI
10.1103/physrevresearch.3.033100

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The characterization of the set of quantum correlations is a problem of fundamental importance in quantum information. The question whether every proper (tight) Bell inequality is violated in quantum theory is an intriguing one in this regard. Here we make significant progress in answering this question, by showing that every tight Bell inequality is violated by ``almost-quantum'' correlations, a semidefinite programming relaxation of the set of quantum correlations. As a consequence, we show that many (classes of) Bell inequalities including two-party correlation Bell inequalities and multioutcome nonlocal computation games, which do not admit quantum violations, are not facets of the classical Bell polytope. To do this, we make use of the intriguing connections between Bell correlations and the graph-theoretic Lov'asz-theta set, discovered by Cabello-Severini-Winter (CSW). We also exploit connections between the cut polytope of graph theory and the classical correlation Bell polytope, to show that correlation Bell inequalities that define facets of the lower dimensional correlation polytope are violated in quantum theory.
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.
FINDING: Buscemi's framework unifies Bell inequalities with entanglement witnesses via "semidefinite programming" and "facet structure" of correlation polytopes, showing quantum violations correspond to specific geometric facets. | MATH: Bell inequalities are linear inequalities over the correlation polytope P \mathcal{P} P; quantum correlations form a convex set Q \mathcal{Q} Q strictly larger than P \mathcal{P} P (local hidden variable polytope). Buscemi's unification maps entanglement witnesses (Hermitian operators W≥0 W \geq 0 W≥0 with Tr(Wρ)≥0 \mathrm{Tr}(W\rho) \geq 0 Tr(Wρ)≥0 for separable ρ \rho ρ) to Bell-type inequalities via a "quantum channel" transformation. Key constants: the Tsirelson bound 22 2\sqrt{2} 22​ for CHSH, and the ratio 22/2=2≈1.414 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 22​/2=2​≈1.414 (not a golden ratio, but a quadratic irrational). Facet structure: the polytope P \mathcal{P} P has facets defined by inequalities like \( \langle A_1B_1 \rangle + \langle A_1B_2 \rangle + \langle A_2B_1 \r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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.
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.
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.