Abstract
A wave function exposed to measurements undergoes pure state dynamics, with deterministic unitary and probabilistic measurement induced state updates, defining a quantum trajectory. For many-particle systems, the competition of these different elements of dynamics can give rise to a scenario similar to quantum phase transitions. To access it despite the randomness of single quantum trajectories, we construct an $n$-replica Keldysh field theory for the ensemble average of the $n$-th moment of the trajectory projector. A key finding is that this field theory decouples into one set of degrees of freedom that heats up indefinitely, while $n-1$ others can be cast into the form of pure state evolutions generated by an effective non-Hermitian Hamiltonian. This decoupling is exact for free theories, and useful for interacting ones. In particular, we study locally measured Dirac fermions in $(1+1)$ dimensions, which can be bosonized to a monitored interacting Luttinger liquid at long wavelengths. For this model, the non-Hermitian Hamiltonian corresponds to a quantum Sine-Gordon model with complex coefficients. A renormalization group analysis reveals a gapless critical phase with logarithmic entanglement entropy growth, and a gapped area law phase, separated by a Berezinskii-Kosterlitz-Thouless transition. The physical picture emerging here is a pinning of the trajectory wave function into eigenstates of the measurement operators upon increasing the monitoring rate.
Highlights
In quantum mechanics, there are two fundamentally distinct dynamical evolutions
The physical picture emerging here is a measurement-induced pinning of the trajectory wave function into eigenstates of the measurement operators, which succeeds upon increasing the monitoring rate across a critical threshold
The masking effect of such trajectory ensemble average can be mitigated by considering correlation functions which are nonlinear in the state—such as, for example, the equal-time product hψtjOijψ ti × hψtjOjjψti for the same stochastic wave function jψti in both quantum mechanical expectation values
Summary
There are two fundamentally distinct dynamical evolutions. First, a pure state can evolve deterministically according to the Schrödinger equation, with dynamics generated by a Hamiltonian operator H. [2,11] that the respective entanglement growth averaged over the ensemble of trajectories is a good witness for the phase transition between volume and area law growth at a finite competition ratio between unitary and measurement dynamics This discovery has sparked significant research on the nature of this transition, its proper description, and its generality in terms of models hosting such behavior [17,18,19,20,21]. For free-fermion models, a general correspondence between nonunitary circuit dynamics and unitary but random Hamiltonian dynamics in (d þ 1) dimensions has been shown to enable a classification of the measurement dynamics in terms of symmetries [30] These results suggest a finer structure in the phenomenology of measurement-induced phase transitions. Several technical details of our analysis are presented in the Appendixes
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