Entanglement in Open and Monitored Field Dynamics

Open and monitored dynamics require two distinct states. The unconditional density operator averages over environmental or measurement records. A quantum trajectory is conditioned on a particular record. Because entropy and entanglement are nonlinear, the entanglement of the average is not the average trajectory entanglement.

Required background. Causal quantum channels supply localized completely positive dynamics and record handling.

Helpful background. Entanglement growth supplies the closed-unitary baseline.

Unconditional and conditioned evolution

An unconditional Markovian model has

$$\frac{d\rho}{dt}=\mathcal L(\rho) =-i[H,\rho] +\sum_j\left(L_j\rho L_j^\dagger -\frac12\{L_j^\dagger L_j,\rho\}\right).$$

A chosen unraveling produces stochastic conditioned states $\rho_c(t)$ and records $r$. They satisfy $\rho(t)=\mathbb E_r[\rho_c(t)]$, but generally

$$E\!\left(\rho(t)\right) \ne \mathbb E_r\!\left[E(\rho_c(t))\right]$$

for an entanglement measure $E$. Detector efficiency, which record is retained, and whether trajectories are postselected are therefore part of the observable.

Different unravelings can realize the same Lindblad generator while producing different trajectory ensembles. A trajectory phase is not a property of $\mathcal L$ alone.

Monitored Gaussian-field comparison

Take a regulated bosonic chain with local Gaussian measurements. Run:

1. stochastic conditional covariance evolution with the full record;
2. the unconditional Gaussian channel obtained by averaging outcomes;
3. a second unraveling that yields the same master equation;
4. a closed-system control with measurement strength zero.

Compare interval entropy, logarithmic negativity, and mutual information at matched time. Report trajectory medians and distributions, not only an average dominated by rare records. If postselection is used, report its probability and scaling with size.

Measurement-induced entanglement transitions were established in monitored circuit models by Li, Chen, and Fisher 2018, §§ II–IV and Skinner, Ruhman, and Nahum 2019, §§ II–IV. Their critical behavior is not automatically inherited by a continuum monitored field.

Monitored evolution is a separate dynamical class. The direct diagnostic must state whether it is evaluated on a conditioned trajectory, an ensemble of trajectories, or the unconditional state. The map is schematic.

Record and postselection checks

Keep the physical monitoring rate fixed while refining the timestep. Vary efficiency and missed outcomes. Compare two unravelings of the same $\mathcal L$; agreement of unconditional observables and disagreement of trajectory entanglement is expected, not a contradiction. A claimed transition needs size scaling, a stable crossing or collapse, and a stated trajectory sampling cost.

Discarding the record changes conditioned pure-state dynamics into an unconditional mixed-state channel. Equating the two can reverse the qualitative entanglement conclusion. The map is schematic.

Evidence boundary

This account reflects results available through 10 August 2026. Measurement-induced transitions are firmly established in important circuit and solvable-model classes. Hardware studies now include direct small-system observations and Kamakari et al. 2025, Eqs. (1)–(4), a cross-entropy benchmark on up to 22 superconducting qubits, but continuum-QFT universality, detector locality, and finite-efficiency scaling remain model-specific questions.

References

• Kamakari, Hirsh, Jiace Sun, Yaodong Li, Jonathan J. Thio, Tanvi P. Gujarati, Matthew P. A. Fisher, Mario Motta, and Austin J. Minnich. “Experimental Demonstration of Scalable Cross-Entropy Benchmarking to Detect Measurement-Induced Phase Transitions on a Superconducting Quantum Processor.” Physical Review Letters 134 (2025): 120401. DOI.
• Li, Yaodong, Xiao Chen, and Matthew P. A. Fisher. “Quantum Zeno Effect and the Many-Body Entanglement Transition.” Physical Review B 98 (2018): 205136. DOI.
• Skinner, Brian, Jonathan Ruhman, and Adam Nahum. “Measurement-Induced Phase Transitions in the Dynamics of Entanglement.” Physical Review X 9 (2019): 031009. DOI.