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arXiv:1707.05449 · Physical Review A

Self-guided method to search maximal Bell violations for unknown quantum states

Li-Kai Yang, Geng Chen, Wen-Hao Zhang, Xing-Xiang Peng, Shang Yu, Xiang-Jun Ye, Chuan‐Feng Li, Guang‐Can Guo

Published Nov 7, 2017• 3 citations
arXivHTMLPDF

Abstract

In recent decades, a great variety of research and applications concerning Bell nonlocality have been developed with the advent of quantum information science. Providing that Bell nonlocality can be revealed by the violation of a family of Bell inequalities, finding maximal Bell violation (MBV) for unknown quantum states becomes an important and inevitable task during Bell experiments. In this paper we introduce a self-guided method to find MBVs for unknown states using a stochastic gradient ascent algorithm (SGA), by parametrizing the corresponding Bell operators. For three investigated systems (two qubit, three qubit, and two qutrit), this method can ascertain the MBV of general two-setting inequalities within 100 iterations. Furthermore, we prove SGA is also feasible when facing more complex Bell scenarios, e.g., $d$-setting $d$-outcome Bell inequality. Moreover, compared to other possible methods, SGA exhibits significant superiority in efficiency, robustness, and versatility.

DOI
10.1103/physreva.96.052310

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In recent decades, a great variety of research and applications concerning Bell nonlocality have been developed with the advent of quantum information science. Providing that Bell nonlocality can be revealed by the violation of a family of Bell inequalities, finding maximal Bell violation (MBV) for unknown quantum states becomes an important and inevitable task during Bell experiments. In this paper we introduce a self-guided method to find MBVs for unknown states using a stochastic gradient ascent algorithm (SGA), by parametrizing the corresponding Bell operators. For three investigated systems (two qubit, three qubit, and two qutrit), this method can ascertain the MBV of general two-setting inequalities within 100 iterations. Furthermore, we prove SGA is also feasible when facing more complex Bell scenarios, e.g., ddd-setting ddd-outcome Bell inequality. Moreover, compared to other possible methods, SGA exhibits significant superiority in efficiency, robustness, and versatility.
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.
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.
Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), which is defined in the simplest scenario involving two dichotomic measurements and whose all key properties are well understood. There have been many attempts to generalise the CHSH Bell inequality to higher-dimensional quantum systems, however, for most of them the maximal quantum violation---the key quantity for most device-independent applications---remains unknown. On the other hand, the constructions for which the maximal quantum violation can be computed, do not preserve the natural property of the CHSH inequality, namely, that the maximal quantum violation is achieved by the maximally entangled state and measurements corresponding to mutually unbiased bases. In this work we propose a novel family of Bell inequalities which exhibit precisely these properties, and whose maximal quantum violation can be computed analytically. In the simplest scenario it recovers the CHSH Bell inequality. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3 . We then show that in the three-outcome case our Bell inequality can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which motivates the definition of a new weak form of self-testing. The ability to certify high-dimensional MUBs makes these inequalities attractive from the device-independent cryptography point of view.
Bell's theorem shows that quantum theory is incompatible with local hidden-variable models. In recent years, research on nonlocality has moved beyond Bell's original scenario to quantum networks, where multiple independent sources distribute physical systems among distant parties, giving rise to correlations certified by nonlinear rather than standard Bell inequalities. Here, we introduce a self-guided protocol that variationally optimizes each party's measurement to maximize the violation of a network Bell inequality, with the violation evaluated efficiently at each step via local Pauli classical shadows and the search driven by the Complex Simultaneous Perturbation Stochastic Approximation (CSPSA) algorithm. Once converged, the measurement settings it returns are implemented directly and the inequality is re-evaluated without shadows. This two-stage structure separates a device-dependent search from a certificate that depends only on the observed statistics and on the causal structure of the network. We validate the protocol by numerically simulating it on the triangle network using the Wagon-Wheel inequality, recovering the violation achieved by the Fritz distribution, and extending the certification to non-maximally entangled and noisy states.
We present a general method for obtaining all Bell inequalities for a given experimental setup. Although the algorithm runs slowly, we apply it to two cases. First, the Greenberger-Horne-Zeilinger setup with three observers each performing one of two possible measurements. Second, the case of two observers each performing one of three possible experiments. In both cases we obtain hundreds of inequalities. Since this is the set of all inequalities, the one that is maximally violated in a given quantum state must be among them. We demonstrate this fact with a few examples. We also note the deep connection between the inequalities and classical logic, and their violation with quantum logic.
ABSTRACT Quantum tests directly reveal the incompatibility of quantum and classical predictions. The violation of such a test establishes a quantifiable gap between the capabilities of observers with access to classical resources and those utilizing quantum entanglement. Typically, larger violations indicate a higher degree of quantumness—for instance, enabling quantum self‐testing in Bell scenarios, where achieving maximal violation is highly desirable. The self‐guided method, initially developed for finding maximal Bell violations, is later extended to complex variables. Here, we apply and compare both variants to determine the maximal violation of the instrumental inequality. The complex‐number variant generally exhibits faster convergence, except during the initial phase. This improvement is more evident in instrumental tests, likely due to their simpler causal structure compared to Bell scenarios. Our findings establish the self‐guided method with complex variables as a generally applicable and competitive approach for robustly and efficiently identifying maximal violations across diverse quantum tests.