Practical Criteria for Entanglement and Nonlocality in Systems with Additive Observables

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 $\rho_{\left( m p\right)\left( nq\right)} \neq 0$, 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.