Entanglement and Purification Transitions in Non-Hermitian Quantum Mechanics

A quantum system subject to continuous measurement and postselection evolves according to a non-Hermitian Hamiltonian. We show that, as one increases the strength of postselection, this non-Hermitian Hamiltonian can undergo a spectral phase transition. On one side of this phase transition (for weak postselection), 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.

Automated summary

With increasing postselection strength, a quantum system can undergo a spectral phase transition of the non-Hermitian Hamiltonian, where one phase retains a mixed state and develops volume-law entanglement, while the other phase approaches a unique pure state with low entanglement from an arbitrary initial state. The transition is identified with an exceptional point in the spectrum of the non-Hermitian Hamiltonian, where PT symmetry is spontaneously broken, and is characterized using exact diagonalization and an approximate mean-field theory.