  The breakdown of Lieb-Robinson bounds in local, non-Hermitian quantum systems
opens up the possibility for a rich landscape of quantum many-body
phenomenology. We elucidate this by studying information scrambling and quantum
chaos in non-Hermitian variants of paradigmatic local quantum spin-chain
models. We utilize a mixture of exact diagonalization and tensor network
techniques for our numerical results and focus on three dynamical quantities:
(i) out-of-time-ordered correlators (OTOCs), (ii) operator entanglement of the
dynamics, and (iii) entanglement growth following a quench from product initial
states. We show that while OTOCs fail to capture information scrambling in a
simple, local, non-Hermitian transverse-field Ising model, the closely related
operator entanglement is a robust measure of dynamical properties of interest.
Moreover, we show that the short-time growth of operator entanglement can
generically detect ``entanglement phase transitions'' in these systems while
its long-time average is shown to be a reliable indicator of quantum chaos and
entanglement phases. This allows us to extend operator entanglement based
diagnostics from previous works on closed and open quantum systems, to the new
arena of monitored quantum dynamics. Finally, we remark on the efficacy of
these dynamical quantities in detecting integrability/chaos in the presence of
continuous monitoring.
