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The breakdown of Lieb-Robinson bounds in local, non-Hermitian quantum systemsopens up the possibility for a rich landscape of quantum many-bodyphenomenology. We elucidate this by studying information scrambling and quantumchaos in non-Hermitian variants of paradigmatic local quantum spin-chainmodels. We utilize a mixture of exact diagonalization and tensor networktechniques for our numerical results and focus on three dynamical quantities:(i) out-of-time-ordered correlators (OTOCs), (ii) operator entanglement of thedynamics, and (iii) entanglement growth following a quench from product initialstates. We show that while OTOCs fail to capture information scrambling in asimple, local, non-Hermitian transverse-field Ising model, the closely relatedoperator entanglement is a robust measure of dynamical properties of interest.Moreover, we show that the short-time growth of operator entanglement cangenerically detect ``entanglement phase transitions'' in these systems whileits long-time average is shown to be a reliable indicator of quantum chaos andentanglement phases. This allows us to extend operator entanglement baseddiagnostics from previous works on closed and open quantum systems, to the newarena of monitored quantum dynamics. Finally, we remark on the efficacy ofthese dynamical quantities in detecting integrability/chaos in the presence ofcontinuous monitoring.

Scrambling and operator entanglement in local non-Hermitian quantum systems