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Necessary and Sufficient Proof of Quantum Nonlocality from Vacuum Noncommutativity

Zhen Su

Published May 16, 2026• 0 citations
Publisher

Abstract

We present a rigorous necessary and sufficient proof of quantum nonlocality within algebraic quantum field theory (AQFT). Using only the standard axioms of AQFT and fundamental properties of the vacuum state, we establish a mathematical equivalence between quantum nonlocality and the vacuum’s inherent noncommutativity. The EPR–Bell contradiction is resolved from first principles within relativistic quantum field theory.

DOI
10.5281/zenodo.20235016

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We present a rigorous necessary and sufficient proof of quantum nonlocality within algebraic quantum field theory (AQFT). Using only the standard axioms of AQFT and fundamental properties of the vacuum state, we establish a mathematical equivalence between quantum nonlocality and the vacuum’s inherent noncommutativity. The EPR–Bell contradiction is resolved from first principles within relativistic quantum field theory.
After a skeptical review of the conventional wisdom concerning locality in ordinary, non-relativistic quantum mechanics, I offer a minimally technical account of locality in Algebraic Quantum Field Theory (AQFT). I argue that whereas the observables of AQFT are as local as one likes, the theory’s states, even more than the states of ordinary quantum mechanics, exhibit a non-locality (better, a holism due to entanglement. I suggest that attempts to ramify entanglement into spooky remote steering, or invoke it to impugn the locality/independence of AQFT algebras, rely on assumptions in tension with the axioms of AQFT, including assumptions about miraculous measurement collapse.
Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyze entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter’s ability to perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard nonrelativistic quantum theory and classical relativistic field theory. “...despite its conservative way of dealing with physical principles, algebraic QFT leads to a radical change of paradigm. Instead of the Newtonian view of a space-time filled with a material content one enters the reality of Leibniz created by relation (in particular inclusions) between ‘monads ’ (∼ the hyperfinite type III1 local von Neumann factors A(O) which as single algebras are nearly void of physical meaning) ” Schroer (1998, p. 302).
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.
There has been some recent interest in applying the techniques of Algebraic Quantum Field Theory (AQFT) to entanglement problems in perturbative QFT. In particular, the Hilbert space independence of this formulation makes it particularly interesting in the context of curved spacetimes and the emphasis on the algebra of observables makes the treatment of Bell inequalities in QFT resemble such treatment in non-relativistic Quantum Mechanics. In this work, we present the mathematical structures needed for formulating AQFT in terms of the Haag-Araki-Kastler (HAK) axioms and discuss their implications. Moreover, we discuss the algebraic approach to quantum entanglement in the form of Bell inequalities. We provide an extension of this formulation to general globally hyperbolic spacetimes using the so-called Locally Covariant approach to QFT, which extends the HAK axioms to general spacetimes by means of the Category Theory language.
Bell correlations are the hallmark of quantum non-locality, and a rich context for analysing them is provided by the algebraic approach to quantum field theory (AQFT): the basic idea is to associate with each bounded region O of Minkowski spacetime an algebra A(O) of operators, of which a self-adjoint element P ∈ A(O) represents a physical quantity pertaining to that part of the field system lying in O, that is measurable by a procedure confined to O. The violation of Bell inequalities in AQFT is known to be "generic", as regards the choices of regions O, and of quantities P, and of states. Furthermore, they are typically "maximal" and "indestructible" in a sense that can be made mathematically precise. The prospects for “peaceful coexistence” between quantum non-locality and relativity theory’s requirement of no action-at-a-distance are also explored. The purpose of this Essay is to review the developments in these areas.
This paper surveys the issue of relativistic causality within the framework of algebraic quantum field theory (AQFT). In doing so, we distinguish various notions of causality formulated in the literature and study their relationships, and thereby we offer what we hope to be a useful taxonomy. We propose that the most direct expression of relativistic causality in AQFT is captured not (as has sometimes been claimed) by the spectrum condition but rather by the axiom of local primitive causality, in that it entails a form of local determinism for quantum fields which generalizes the constraint of no superluminal propagation of classical field theories to relativistic quantum field theory. We discuss the status of the axiom of micro-causality by locating its place within a large family of separability/independence/locality conditions developed for AQFT and also by relating it to so-called no-signalling theorems. And we also provide a critical survey of attempts to understand the implications for relativistic causality of the distant correlations endemic to the states in models of AQFT satisfying the standard axioms, and we provide an assessment of attempts to employ Reichenbach's common cause principle in AQFT to defuse worries that these distant correlations implicate direct causal connections between relatively spacelike events.