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We study dynamical phase transitions occurring in the stationary state of thedynamics of integrable many-body non-hermitian Hamiltonians, which can beeither realized as a no-click limit of a stochastic Schr\"{o}dinger equation orusing spacetime duality of quantum circuits. In two specific models, theTransverse Field Ising Chain and the Long Range Kitaev Chain, we observe thatthe entanglement phase transitions occurring in the stationary state have thesame nature as that occurring in the vacuum of the non-hermitian Hamiltonian:bounded entanglement entropy when the imaginary part of the quasi-particlespectrum is gapped and a logarithmic growth for gapless imaginary spectrum.This observation suggests the possibility to generalize the area-law theorem tonon-Hermitian Hamiltonians.

Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum