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arXiv:2201.01886 · New Journal of Physics

Deterministic all-versus-nothing proofs of Bell nonlocality based on non-stabilizer states

Weidong Tang

Published Jun 1, 2023• 1 citation
arXivHTMLPDF

Abstract

Abstract The all-versus-nothing proof of Bell nonlocality is a prominent demonstration of Bell’s theorem without inequalities. There are two kinds of such proofs: the deterministic all-versus-nothing proof and the probabilistic all-versus-nothing proof, which have received extensive research attention. Traditionally, all previous deterministic all-versus-nothing proofs are constructed based on stabilizer states. However, this work presents new deterministic proofs derived from non-stabilizer states, thereby breaking away from this conventional approach. These novel results not only significantly broaden the range of demonstrations of Bell nonlocality without inequalities but also offer valuable resources for certain quantum information processing applications.

DOI
10.1088/1367-2630/acdd8b

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Abstract The all-versus-nothing proof of Bell nonlocality is a prominent demonstration of Bell’s theorem without inequalities. There are two kinds of such proofs: the deterministic all-versus-nothing proof and the probabilistic all-versus-nothing proof, which have received extensive research attention. Traditionally, all previous deterministic all-versus-nothing proofs are constructed based on stabilizer states. However, this work presents new deterministic proofs derived from non-stabilizer states, thereby breaking away from this conventional approach. These novel results not only significantly broaden the range of demonstrations of Bell nonlocality without inequalities but also offer valuable resources for certain quantum information processing applications.
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.
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's theorem is 50 years old. Still there is a controversy about its implications. Much of it has its roots in confusion regarding the premises from which the theorem can be derived. Some claim that a derivation of Bell's inequalities requires just locality assumption, and nothing more. Violations of the inequalities are then interpreted as ``nonlocality'' or ``quantum nonlocality''. We show that such claims are unfounded and that every derivation of Bell's inequalities requires a premise -- in addition to locality and freedom of choice -- which is either assumed tacitly, or unconsciously, or is embedded in a single compound condition (like Bell's ``local causality''). The premise is equivalent to the assumption of existence of additional variables which do not appear in the quantum formalism (in form of determinism, or joint probability for outcomes of all conceivable measurements, or ``additional causes`'', or ``hidden variables'', ``complete description of the state'' or counterfactual definiteness, etc.). A certain irony is that perhaps the main message of violation of Bell's inequalities is that our notion of locality should be based on an operationally well-defined no-signalling condition, rather than on local causality.
If n qubits were distributed between 2 parties, which quantum pure states and distributions of qubits would allow all-versus-nothing (or Greenberger-Horne-Zeilinger-like) proofs of Bell's theorem using only single-qubit measurements? We show a necessary and sufficient condition for the existence of these proofs for any number of qubits, and provide all distinct proofs up to n=7 qubits. Remarkably, there is only one distribution of a state of n=4 qubits, and six distributions, each for a different state of n=6 qubits, which allow these proofs.
We introduce a two-observer all-versus-nothing proof of Bell's theorem which reduces the number of required quantum predictions from 9 [A. Cabello, Phys. Rev. Lett. 87, 010403 (2001); Z.-B. Chen et al., Phys. Rev. Lett. 90, 160408 (2003)] to 4, provides a greater amount of evidence against local realism, reduces the detection efficiency requirements for a conclusive experimental test of Bell's theorem, and leads to a Bell inequality which resembles Mermin's inequality for three observers [N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990)] but requires only two observers.