  Strong local disorder in interacting quantum spin chains can turn delocalized
eigenmodes into localized eigenstates, giving rise to many-body localized (MBL)
phases. This is accompanied by distinct spectral statistics: chaotic for the
delocalized phase and integrable for the localized phase. In isolated systems,
localization and chaos are defined through a web of relations among
eigenvalues, eigenvectors, and real-time dynamics. These may change as the
system is made open. We ask whether random dissipation (without random
disorder) can induce chaotic or localized behavior in an otherwise integrable
system. The dissipation is described using non-Hermitian Hamiltonians, which
can effectively be obtained from Markovian dynamics conditioned on null
measurement. Through the use of the singular value decomposition and the
introduction of new diagnostic tools complementing the singular-value
statistics, 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.
