Locality, correlations, information, and non-Hermitian quantum systems

Local non-Hermitian (NH) quantum systems generically exhibit breakdown of Lieb-Robinson (LR) bounds, motivating study of whether new locality measures might shed light not seen by existing measures. In this paper we discuss extensions of the connected correlation function (CC) as measures of locality and information spreading in both Hermitian and NH systems. We find that in Hermitian systems, $\delta \rho =\rho -{\rho }_A\otimes {\rho }_B$ can be written as a linear combination of CCs, allowing placement of an LR bound on ${\parallel \delta \rho \parallel }_2$, which we show generically extends to an LR bound on mutual information. Additionally, we extend the CC to NH systems in a form that recovers locality, and use the metric formalism to derive a modified CC which recovers not just locality but even LR bounds in local $PT$-Symmetric systems. We find that even with these CCs, the bound on ${\parallel \delta \rho \parallel }_2$ breaks down in certain NH cases, which can be used to place a necessary condition on which local NH Hamiltonians are capable of nonlocal entanglement generation. Numerical simulations are provided by means of exact diagonalization for the NH Transverse-Field Ising Model, demonstrating both breakdown and recovery of LR bounds.