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International Journal of Theoretical Physics

Detecting Bell Nonlocality Based on the Hardy Paradox

Zhenzhu Dong, Ying Zhen Yang, Huaixin Cao

Published Mar 13, 2020• 10 citations
Publisher
DOI
10.1007/s10773-020-04432-1

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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.
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.
In this paper, we introduce post-selection games, a generalization of nonlocal games where each round can be not only won or lost by the players, but also discarded by the referee. Such games naturally formalize possibilistic proofs of nonlocality, such as Hardy's paradox. We develop algorithms for computing the local and Tsirelson bounds of post-selection games. Furthermore, we show that they have an unbounded advantage in statistical power over traditional nonlocal games, making them ideally suited for analysing Bell tests with low detection efficiency.