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.

Figure 1

Phase diagram of purification rate—i.e., the gap between the slowest and second-slowest decaying eigenvalues of the non-Hermitian Hamiltonian (7)—vs measurement strength $\gamma$ and interaction strength $J$ for $h_0=1.25$ and $\epsilon =0$. The dashed line is the mean-field prediction, which is in good agreement with the numerical data (color map) on one-dimensional spin chains. The model is solvable at $\gamma =1$ (dotted line). Inset: size dependence of gap for $J=0.95$.

Figure 2

(a) Dynamics of the purity $\mathrm{\Pi }(t)$ of an initially fully mixed state and (b) the bipartite entanglement entropy $S_1$ of a random initial pure state (averaged over 100 random product states) under the Hamiltonian (4), showing a clear qualitative distinction between the area-law–pure and volume-law–mixed phases. (c) In the mixed phase, the time-averaged density matrix (see main text) has a finite entropy density that continuously decreases as $\gamma$ is tuned toward the transition. The quantity $\alpha$ (which is an estimate of the accessible Hilbert space) tracks the von Neumann entropy density ($s_1$) for small $\gamma$. (d) On the area-law side, any initial state converges to the slowest decaying eigenstate, which is area-law entangled with (inset) a coefficient that grows as the transition is approached. (e) Eigenvalue statistics in the mixed phase. Inset: the deviation from random matrix theory—i.e., the Gaussian orthogonal ensemble (GOE)—persists with system size. (f) Eigenvector entanglement for a single realization at $\gamma =0.9$. Inset: the mean half-chain eigenvector entanglement grows as a volume law. (a),(b),(d) $\epsilon =0$; (c),(e),(f) $\epsilon =0.05$, to lift degeneracies in the spectrum and make the problem well defined. Results are insensitive to the precise choice of $\epsilon$. Error bars in (c) and (e) denote one standard deviation due to variations over disorder realizations.