Abstract
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.
Highlights
Quantum nonlocality was first discovered by Einstein, Podolsky and Rosen (EPR) in 1935, including quantum entanglement, quantum steering and Bell nonlocality [1]
Different from the existing discussions about the GHZ paradox, we aim to establish some theoretical methods for mathematically proving Bell nonlocality with the inexistence of the local hidden variable models (LHVMs), which is motivated by the work [46]
MAIN RESULTS The main aim of this paper is to derive some necessary conditions for a multipartite state to be Bell local in terms of "measurement expectations" of local observables without invoking Bell inequalities and obtain some sufficient conditions for a state to be Bell nonlocal
Summary
Quantum nonlocality was first discovered by Einstein, Podolsky and Rosen (EPR) in 1935, including quantum entanglement, quantum steering and Bell nonlocality [1]. They formulated an apparent paradox of quantum theory (EPR paradox) and given a “thought" experiment that argues the wave function description in quantum mechanics is incomplete. According to the EPR paradox on local realism, quantum theory allows a curious phenomenon: the so-called “spooky action at a distance". In the year 1936, Schrodinger [2] firstly introduced the terminology “entanglement" and “steering" to describe such quantum “spooky action". Quantum entanglement, originated from the EPR paradox, is the essence of quantum formalism and holistic property of compound quantum systems involves nonclassical correlations between subsystems and has many applications for many quantum processes, including quantum cryptography, quantum teleportation, dense coding and so on
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