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 extend the standard connected correlation function (CC) to NH systems in a form that recovers locality. Additionally, we use the metric formalism to derive a modified CC which recovers not just locality but even LR bounds in local $PT$-Symmetric systems, and discuss extensions of both CCs to the $n$-partite case. We show that in Hermitian systems $\delta\rho = \rho-\rho_A\otimes\rho_B$ can be written as a linear combination of CCs, allowing us to place an LR bound on $\Vert\delta\rho\Vert_2$. We show this generically extends to an LR bound on mutual information as well. We then extend this to NH systems, where we show its violations can be used to place a necessary condition on which 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.