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Strong local disorder in interacting quantum spin chains can turn delocalizedeigenmodes into localized eigenstates, giving rise to many-body localized (MBL)phases. This is accompanied by distinct spectral statistics: chaotic for thedelocalized phase and integrable for the localized phase. In isolated systems,localization and chaos are defined through a web of relations amongeigenvalues, eigenvectors, and real-time dynamics. These may change as thesystem is made open. We ask whether random dissipation (without randomdisorder) can induce chaotic or localized behavior in an otherwise integrablesystem. The dissipation is described using non-Hermitian Hamiltonians, whichcan effectively be obtained from Markovian dynamics conditioned on nullmeasurement. Through the use of the singular value decomposition and theintroduction of new diagnostic tools complementing the singular-valuestatistics, namely, the singular form factor, the inverse participation ratio,and entanglement entropy for singular vectors, we provide a positive answer.Our method is illustrated in an XXZ Hamiltonian with random local dissipation.

Diagnosing non-Hermitian Many-Body Localization and Quantum Chaos via Singular Value Decomposition