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arXiv:2503.17297 · Physical review. D/Physical review. D.

Practical criteria for entanglement and nonlocality in systems with additive observables

Alexander Bernal, J.A. Casas, Juan Falceto

Published Jul 2, 2025• 3 citations

Abstract

For general bipartite mixed states, a sufficient and necessary mathematical condition for certifying entanglement and/or (Bell) non-locality remains unknown. In this paper, we examine this question for a broad and physically relevant class of bipartite systems, specifically those possessing an additive observable with a definite value. Such systems include, for example, final states of particle decays or bipartitions of spin chains with well-defined magnetization. We derive very simple, handy criteria for detecting entanglement or non-locality in many cases. For instance, if $ρ_{\left( m p\right)\left( nq\right)} \neq 0$, where the eigenstates $|np\rangle$ or $|mq\rangle$ do not correspond to the given definite value of the additive observable, then the state is necessarily entangled-this condition is very easy to check in practice. If, in addition, the partitioned Hilbert space has dimension 2xd, the condition becomes necessary. Furthermore, if the sectors associated with the eigenstates $|mp\rangle$ or $|nq\rangle$ are non-degenerate, there exists a CHSH inequality that is violated. We illustrate these results by analyzing the potential detection of entanglement and nonlocality in Higgs to ZZ decays at the LHC.

DOI: 10.1103/r436-6jnb

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For general bipartite mixed states, a sufficient and necessary mathematical condition for certifying entanglement and/or (Bell) non-locality remains unknown. In this paper, we examine this question for a broad and physically relevant class of bipartite systems, specifically those possessing an additive observable with a definite value. Such systems include, for example, final states of particle decays or bipartitions of spin chains with well-defined magnetization. We derive very simple, handy criteria for detecting entanglement or non-locality in many cases. For instance, if

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, where the eigenstates

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or

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do not correspond to the given definite value of the additive observable, then the state is necessarily entangled-this condition is very easy to check in practice. If, in addition, the partitioned Hilbert space has dimension 2xd, the condition becomes necessary. Furthermore, if the sectors associated with the eigenstates

|mp\rangle

or

|nq\rangle

are non-degenerate, there exists a CHSH inequality that is violated. We illustrate these results by analyzing the potential detection of entanglement and nonlocality in Higgs to ZZ decays at the LHC.

We develop criteria to detect three classes of nonlocality that have been shown by Wiseman et al. [Phys. Rev. Lett. 98, 140402 (2007)] to be nonequivalent: entanglement, EPR steering, and the failure of local hidden-variable theories. We use the approach of Cavalcanti et al. [Phys. Rev. Lett. 99, 210405 (2007)] for continuous variables to develop the nonlocality criteria for arbitrary spin observables defined on a discrete Hilbert space. The criteria thus apply to multisite qudits, i.e., systems of fixed dimension

d

, and take the form of inequalities. We find that the spin moment inequalities that test local hidden variables (Bell inequalities) can be violated for arbitrary

d

by optimized highly correlated nonmaximally entangled states provided the number of sites

N

is high enough. On the other hand, the spin inequalities for entanglement are violated and thus detect entanglement for such states, for arbitrary

d

and

N

, and with a violation that increases with

N

. We show that one of the moment entanglement inequalities can detect the entanglement of an arbitrary generalized multipartite Greenberger-Horne-Zeilinger state. Because they involve the natural observables for atomic systems, the relevant spin-operator correlations should be readily observable in trapped ultracold atomic gases and ion traps.

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