  A quantum system subject to continuous measurement and post-selection evolves
according to a non-Hermitian Hamiltonian. We show that, as one increases the
rate of post-selection, this non-Hermitian Hamiltonian undergoes a spectral
phase transition. On one side of this phase transition (for weak
post-selection) an initially mixed density matrix remains mixed at all times,
and an initially unentangled state develops volume-law entanglement; on the
other side, an arbitrary initial state approaches a unique pure state with low
entanglement. We identify this transition with an exceptional point in the
spectrum of the non-Hermitian Hamiltonian, at which PT symmetry is
spontaneously broken. We characterize the transition as well as the nontrivial
steady state that emerges at late times in the mixed phase using exact
diagonalization and an approximate, analytically tractable mean-field theory;
these methods yield consistent conclusions.
