Theory of post-selected entanglement transitions in monitored bosons

How do quantum measurements influence the entanglement dynamics of free bosonic systems with unbounded local Hilbert spaces? While measurement-induced phase transitions (MIPT) are well-characterized in qubit and fermionic systems, the non-Gaussian nature of bosonic Fock states complicates the analysis, preventing the use of standard correlation-matrix approaches.

The authors introduce a replica-free Keldysh field-theoretic framework to compute the second Renyi entanglement entropy for non-interacting bosons evolving under non-Hermitian dynamics. By utilizing the Wigner characteristic function and a derivative-based construction to handle non-Gaussian initial Fock states, they express the entanglement entropy in terms of permanents of matrices derived from single-particle Green's functions. They apply this to a one-dimensional cross-stitch lattice subject to continuous monitoring of the no-click trajectory.

The study identifies a measurement-induced transition from volume-law to logarithmic entanglement scaling. This transition is controlled by the restructuring of the non-Hermitian spectrum: at weak monitoring, an extensive manifold of long-lived modes leads to volume-law scaling, whereas strong monitoring causes bosons to dynamically condense into a microscopic number of slowest-decaying modes, resulting in logarithmic scaling. The authors also introduce a computationally efficient diagnostic—the effective rank of the overlap matrix—which allows for the identification of this transition in polynomial time, bypassing the exponential complexity of evaluating permanents.

This work establishes a distinct mechanism for MIPT in bosonic systems, where the transition is driven by the condensation of particles into a few long-lived modes rather than the critical behavior of a conformal field theory. It provides a rigorous analytical tool for studying monitored bosonic dynamics and offers a practical diagnostic for identifying entanglement phases in experimental platforms like circuit QED.