Measurement-induced phase transition in interacting bosons from most likely quantum trajectory

Alessandro Romito; Anna Delmonte; Rosario Fazio; Zejian Li

arxiv: 2509.24520 · v2 · submitted 2025-09-29 · 🪐 quant-ph · cond-mat.stat-mech

Measurement-induced phase transition in interacting bosons from most likely quantum trajectory

Anna Delmonte , Zejian Li , Rosario Fazio , Alessandro Romito This is my paper

Pith reviewed 2026-05-18 13:03 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech

keywords measurement-induced phase transitionquantum trajectoriesSine-Gordon modelentanglement scalingbosonic many-body systemsmonitored dynamicsarea-law to logarithmic transition

0 comments

The pith

The most likely quantum trajectory captures monitored boson dynamics and reveals an area-to-logarithmic entanglement transition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a method to find the single most likely quantum trajectory from the probability distribution over all possible measurement outcomes in bosonic many-body systems. This trajectory is proven exact for Gaussian theories and is applied to the interacting Sine-Gordon model through a self-consistent time-dependent harmonic approximation. The approach describes the evolution beyond the averaged state and shows that the steady-state entanglement changes from area-law to logarithmic-law scaling. A sympathetic reader would care because the result supplies a concrete handle on how continuous monitoring can induce new phases in interacting quantum matter.

Core claim

The authors establish that the most likely trajectory, identified as the measurement readout sequence with the highest probability, exactly reproduces the monitored dynamics for Gaussian theories. They extend the method to the interacting Sine-Gordon model via a self-consistent time-dependent harmonic approximation and demonstrate that the steady state exhibits an entanglement phase transition from area-law to logarithmic-law scaling.

What carries the argument

The most likely trajectory, the quantum trajectory with the highest probability among all possible measurement readouts, which carries the argument by supplying a representative path whose entanglement properties determine the steady-state phase.

If this is right

The monitored dynamics can be studied without averaging over the entire ensemble of trajectories.
A steady-state entanglement phase transition occurs in the Sine-Gordon model under continuous monitoring.
The method extends to other interacting bosonic models where full trajectory ensembles are intractable.
Steady-state properties such as entanglement scaling become accessible through a single representative path.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach could be tested against exact numerics in small systems to quantify the approximation error.
Similar transitions may appear in monitored spin or fermion models if an analogous most-likely path can be defined.
The logarithmic phase might imply slower information spreading under monitoring, relevant for quantum error correction protocols.
Cold-atom experiments with site-resolved measurements could search for the predicted change in entanglement scaling.

Load-bearing premise

The self-consistent time-dependent harmonic approximation remains accurate enough to capture the essential features of the phase transition in the interacting Sine-Gordon model despite not being exact.

What would settle it

Exact numerical sampling of the full probability distribution of quantum trajectories for small boson numbers in the Sine-Gordon model, checking whether the entanglement entropy scaling crosses from area-law to logarithmic-law at the parameter values predicted by the approximation.

Figures

Figures reproduced from arXiv: 2509.24520 by Alessandro Romito, Anna Delmonte, Rosario Fazio, Zejian Li.

**Figure 2.** Figure 2: Logarithmic Negativity for the Free Bosons CFT lattice model, the plot [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: (a) Truncated representation for the Sine-Gordon model for the specific [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

**Figure 4.** Figure 4: Ratio between trajectory averaged results obtained from quantum state [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: Phase diagram of the model: the color code represents the fitting parame [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: Some steady-state properties for the Sine-Gordon model with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: (a) Comparison between theoretical results obtained in perturbation the [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

read the original abstract

We propose a new theoretical method to describe the monitored dynamics of bosonic many-body systems based on the concept of the most likely trajectory. We show how such trajectory can be identified from the probability distribution of quantum trajectories, i.e. measurement readouts, and how it successfully captures the monitored dynamics beyond the average state. We prove the method to be exact in the case of Gaussian theories and then extend it to the interacting Sine-Gordon model. Although no longer exact in this framework, the method captures the dynamics through a self-consistent time-dependent harmonic approximation and reveals an entanglement phase transition in the steady state from an area-law to a logarithmic-law scaling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The most-likely-trajectory method is exact for Gaussian bosonic dynamics and then uses a self-consistent harmonic approximation to locate an area-to-log entanglement transition in the Sine-Gordon model.

read the letter

The punchline is that this paper gives a most-likely-trajectory method for bosonic monitored systems. It is exact when the dynamics stay Gaussian and then uses a self-consistent harmonic approximation to handle the Sine-Gordon interaction, where it finds a steady-state entanglement transition from area-law to logarithmic scaling. What stands out as new is the explicit construction of the most probable trajectory from the readout probability distribution. In the Gaussian case they show it reproduces the full monitored evolution without averaging over many paths. That part looks clean and could be useful on its own for efficient simulation. They then carry the idea over to the interacting model by replacing the sine-Gordon term with a time-dependent quadratic potential whose parameters are fixed self-consistently at each step. This lets them track the entanglement entropy in the steady state and extract the scaling change. The Gaussian proof is a real plus and gives the method a firm base. The approximation is a reasonable first step for a non-perturbative regime, and the resulting phase diagram is concrete enough to be checked later. The soft spot is that the transition itself is seen only inside this approximation. Because the closure is not exact, systematic errors could move the critical point or alter the scaling exponents. Without reported comparisons to exact diagonalization on small chains or to other methods like tensor networks, it is difficult to know how much trust to put in the location of the transition. This paper is aimed at theorists working on measurement-induced phenomena in bosonic many-body systems. Anyone already using quantum trajectories or thinking about monitored cold-atom setups will find the framing useful. The concrete claim on entanglement scaling makes it worth a closer look even if the approximation needs more validation. I would send it to peer review. The core idea is worth referee time, and the main task for reviewers would be to assess the range of validity of the harmonic closure.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a method to identify the most likely quantum trajectory from the probability distribution of measurement readouts in bosonic many-body systems under continuous monitoring. It proves the approach exact for Gaussian theories and extends it to the interacting Sine-Gordon model via a self-consistent time-dependent harmonic approximation, which is used to demonstrate an entanglement phase transition in the steady state from area-law to logarithmic-law scaling.

Significance. If the self-consistent approximation is shown to be reliable, the work supplies a computationally tractable route to measurement-induced transitions in interacting bosons that goes beyond ensemble averages. The exact Gaussian case provides a clean benchmark, and the method's focus on individual trajectories is a conceptual advance for monitored open quantum systems.

major comments (2)

[Section IV] Section IV (extension to Sine-Gordon): the central claim of an area-to-log entanglement transition rests entirely on the self-consistent time-dependent harmonic approximation that replaces the nonlinear interaction by a quadratic term with self-consistently determined parameters. The manuscript provides no quantitative error estimate, convergence test, or comparison to exact numerics on small lattices that would bound the systematic error in the extracted scaling exponents or transition location.
[Section V] Section V (steady-state entanglement): the reported scaling laws are obtained exclusively under the quadratic closure; any deviation of the true non-Gaussian dynamics from this closure directly impacts the phase diagram. A sensitivity analysis with respect to the self-consistency cutoff or an alternative closure scheme is needed to establish robustness.

minor comments (2)

Notation for the time-dependent frequency and coupling parameters in the harmonic approximation should be introduced explicitly before their first use in equations.
[Figure 3] Figure captions would benefit from stating the system size, monitoring strength, and interaction parameter values used for each curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript's significance and for the constructive comments on the self-consistent approximation. We respond to each major comment below and outline the revisions we will make.

read point-by-point responses

Referee: [Section IV] Section IV (extension to Sine-Gordon): the central claim of an area-to-log entanglement transition rests entirely on the self-consistent time-dependent harmonic approximation that replaces the nonlinear interaction by a quadratic term with self-consistently determined parameters. The manuscript provides no quantitative error estimate, convergence test, or comparison to exact numerics on small lattices that would bound the systematic error in the extracted scaling exponents or transition location.

Authors: We agree that the absence of quantitative error bounds and direct comparisons constitutes a limitation of the current presentation. While the self-consistent harmonic approximation is a well-established closure for the Sine-Gordon model and becomes exact in the Gaussian limit demonstrated in Section III, its systematic error for the monitored dynamics is not quantified. In the revised manuscript we will add a new subsection to Section IV that includes (i) comparisons of the most-likely-trajectory entanglement entropy against exact diagonalization or matrix-product-state results for small lattices (L ≤ 8) at representative parameter values, and (ii) a convergence study with respect to the self-consistency iteration tolerance. These additions will provide explicit bounds on the uncertainty of the reported scaling exponents and the location of the transition. revision: yes
Referee: [Section V] Section V (steady-state entanglement): the reported scaling laws are obtained exclusively under the quadratic closure; any deviation of the true non-Gaussian dynamics from this closure directly impacts the phase diagram. A sensitivity analysis with respect to the self-consistency cutoff or an alternative closure scheme is needed to establish robustness.

Authors: We acknowledge that the phase diagram is obtained within the quadratic closure and that its robustness must be demonstrated. In the revised version we will augment Section V with a sensitivity analysis that varies the self-consistency cutoff and the frequency of parameter updates. We will also examine an alternative closure (e.g., a variational Gaussian ansatz with an additional quartic correction) on a subset of parameters to test whether the area-law to logarithmic transition persists. These results will be presented as supplementary figures and discussed in the main text to quantify the stability of the reported scaling laws. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via independent approximation

full rationale

The paper derives the most likely trajectory from the probability distribution of quantum trajectories and proves exactness for Gaussian theories as a standalone result. Extension to the Sine-Gordon model uses an explicit self-consistent time-dependent harmonic approximation that replaces the nonlinear term with a quadratic closure whose parameters are solved consistently from the dynamics; the resulting area-to-logarithmic entanglement transition is an output of this dynamical evolution rather than a redefinition or statistical fit of the inputs. No load-bearing self-citation, uniqueness theorem, or renaming of known results is present. The approximation is acknowledged as non-exact, preserving the distinction between method and prediction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the most likely trajectory concept and the harmonic approximation for the interacting case. No free parameters are explicitly mentioned, but the approximation may involve self-consistent parameters.

axioms (2)

domain assumption Gaussian theories allow exact description via most likely trajectory
Stated as proven exact in abstract for Gaussian cases.
ad hoc to paper Self-consistent time-dependent harmonic approximation captures essential dynamics in Sine-Gordon
Used to extend the method to interacting case.

pith-pipeline@v0.9.0 · 5638 in / 1278 out tokens · 41001 ms · 2026-05-18T13:03:48.680592+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove the method to be exact in the case of Gaussian theories and then extend it to the interacting Sine-Gordon model... through a self-consistent time-dependent harmonic approximation
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the most likely trajectory... r^*(t) = <R>_{ψ^*}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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, " * write output.state after.block =

ENTRY address archive author booktitle chapter doi edition editor eid eprint howpublished institution isbn journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'af...

work page
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write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

work page