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Physica Scripta

Hardy-Bell inequalities and fault-tolerant Hardy paradoxes

Kanyuan Han, Wenqian Zhu, Zhihua Guo, Huaixin Cao

Published Nov 24, 2023• 0 citations
Publisher

Abstract

Abstract Usually, the verification of Bell nonlocality involves two main approaches: violation of specific inequalities and utilization of no-inequality methods. In this paper, we continue to develop the inequality methods by deducing the so-called ‘Hardy-Bell inequalities (HBIs)’ and ‘fault-tolerant Hardy paradoxes (FTHPs)’ for correlation tensors (CTs) with two inputs and general outcomes. We prove that the HBIs are necessary conditions for a CT to be Bell local and one of the FTHPs is sufficient condition for a CT to be Bell nonlocal. We demonstrate the effectiveness of HBIs in determining the nonlocality of CTs or quantum states when the classical Hardy paradox does not appear or a Bell inequality is not violated. Consequently, our methods can be utilized to explore more correlations having Bell nonlocality. Based on the obtained results, we find a neighborhood of a Hardy nonlocal state, in which all states are all Bell nonlocal.

DOI
10.1088/1402-4896/ad0fc8

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Abstract Usually, the verification of Bell nonlocality involves two main approaches: violation of specific inequalities and utilization of no-inequality methods. In this paper, we continue to develop the inequality methods by deducing the so-called ‘Hardy-Bell inequalities (HBIs)’ and ‘fault-tolerant Hardy paradoxes (FTHPs)’ for correlation tensors (CTs) with two inputs and general outcomes. We prove that the HBIs are necessary conditions for a CT to be Bell local and one of the FTHPs is sufficient condition for a CT to be Bell nonlocal. We demonstrate the effectiveness of HBIs in determining the nonlocality of CTs or quantum states when the classical Hardy paradox does not appear or a Bell inequality is not violated. Consequently, our methods can be utilized to explore more correlations having Bell nonlocality. Based on the obtained results, we find a neighborhood of a Hardy nonlocal state, in which all states are all Bell nonlocal.
Bell nonlocality is a special quantum nonlocality and has emerged as an important resource in quantum information processing tasks. Typically, there are two types of strategies for testing the Bell nonlocality: the Bell inequality method and the “all-versus-nothing proof", the later includes methods such as the GHZ argument and Hardy paradox. The objective of this work is to provide new methods for detecting Bell nonlocality by understanding Hardy-like paradoxes (HLPs) and Hardy-Bell inequalities (HBIs) as well as Hardy inequality. First, the weak HLP (WHLP) is established, and an HBI is obtained for tripartite correlation tensors (CTs) with two inputs and two outcomes. Based on an existing Hardy inequality for an n-qubit pure state, a Hardy inequality is proven for n-partite Bell local CTs. Second, the WHLP and HBI are proven for tripartite states in terms of conditional probabilities. Our WHLP is theoretically and practically easier to construct and more efficient in checking Bell nonlocality compared to the usual HLP.
Bell nonlocality of quantum states is an important resource in quantum information and then has various applications. It is usually detected by the violation of some Bell’s inequalities and the all-versus-nothing test. In the present paper, we aim to establish some mathematical methods for proving Bell nonlocality without inequalities, inspired by the work [Phys. Rev. Lett., 89, 080402 (2002)] regarding the GHZ paradox. For self-containedness, we recall the mathematical definition of Bell nonlocality proposed in [Sci. China-Phys. Mech. Astron. 62, 030311 (2019)] and then give some basic properties on it. Then we derive some necessary conditions for a multipartite state to be Bell local and obtain some sufficient conditions for a state to be Bell nonlocal in terms of “expectations” of local observables without invoking Bell inequalities. Unlike the standard approach to nonlocality detection based on violation of Bell inequalities, the obtained criteria are formulated in terms of certain relations for expectation values of local observables that are constructed from the well-known GHZ paradoxes.
Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as ``the simplest form of Bell's theorem.'' In this work, we advance the study of Hardy's paradox based on Bell's inequality. Our formalism is essentially equivalent to the ``logical Bell inequality'' formalism developed by Abramsky and Hardy, but ours is more easily applied to experimental tests. Remarkably, we find that not only for a two-qubit two-setting case but also for a two-qubit four-setting case, we can construct stronger Hardy-type paradoxes based on Bell's inequality, whose successful probabilities can attain four times and seven times larger than the original one, respectively, thus providing more friendly tests for experiment. Meanwhile, we experimentally test the stronger Hardy-type paradoxes in a two-qubit system. Within the experimental errors, the experimental results coincide with the theoretical predictions.
Homogenization proposed in [Y.-C Wu and M. Żukowski, Phys. Rev. A 85 (2012) 022119] is a procedure to transform a tight Bell inequality with partial correlations into a full-correlation form that is also tight. In this paper, we check the homogenizations of two families of n-partite Bell inequalities: the Hardy inequality and the tight Bell inequality without quantum violation. For Hardy's inequalities, their homogenizations bear stronger quantum violation for the maximally entangled state; the tight Bell inequalities without quantum violation give the boundary of quantum and supra-quantum, but their homogenizations do not have the similar properties. We find their homogenization are violated by the maximally entangled state. Numerically computation shows the the domains of quantum violation of homogenized Hardy's inequalities for the generalized GHZ states are smaller than those of Hardy's inequalities.
Logical nonlocality is completely characterized by Hardy's ``paradox'' in (2,2,l)(2,2,l)(2,2,l) and (2,k,2)(2,k,2)(2,k,2) scenarios. We consider a variety of consequences and applications of this fact. (i) Polynomial algorithms may be given for deciding logical nonlocality in these scenarios. (ii) Bell states are the only entangled two-qubit states which are not logically nonlocal under projective measurements. (iii) It is possible to witness Hardy nonlocality with certainty in a simple tripartite quantum system. (iv) Noncommutativity of observables is necessary and sufficient for enabling logical nonlocality.
Measurement incompatibility and quantum nonlocality are two key features of quantum theory.Violations of Bell inequalities require quantum entanglement and incompatibility of the measurements used by the two parties involved in the protocol.We analyze the converse question: for which Bell inequalities is the incompatibility of measurements enough to ensure a quantum violation?We relate the two questions by comparing two tensor norms on the space of dichotomic quantum measurements, one characterizing measurement compatibility and the other characterizing violations of a given Bell inequality.We provide sufficient conditions for the equivalence of the two notions in terms of the matrix describing the correlation Bell inequality.We show that the Clauser-Horne-Shimony-Holt inequality and its variants are the only ones that satisfy it.