arxiv: 2605.12783 · v1 · submitted 2026-05-12 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: unknown

Purification of a monitored qubit: exact path-integral solution

Matheus M. R. Poltronieri Martins , Henrique Santos Lima

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:56 UTC · model grok-4.3

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

keywords monitored qubitpurification dynamicsmultiplicative Langevin equationOnsager-Machlup path integralstochastic master equationcollisional modelprobability distributionmeasurement regimes

0 comments

The pith

Purification of a monitored qubit reduces to an exactly solvable multiplicative Langevin equation for its purity.

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

This paper establishes that when a qubit begins in a mixed state and undergoes continuous monitoring through sequential collisions with ancilla qubits, its entire conditioned evolution is captured by a single scalar: the state's purity. The purity obeys a multiplicative Langevin equation whose solution is obtained exactly via the Onsager-Machlup path-integral method, yielding the full probability distribution over possible trajectories. A sympathetic reader would care because the result supplies an analytic benchmark for how information is extracted from a quantum system under monitoring, replacing numerical simulation of the full density matrix with a closed-form description of the purification process. The analysis further shows that the trajectories exhibit a crossover between diffusion-dominated and measurement-dominated regimes, marked by the emergence of a bimodal distribution.

Core claim

For initial mixed states, the dynamics reduce to a multiplicative Langevin equation for a single scalar parameter representing the state's purity. This stochastic process is solved exactly using the Onsager-Machlup path integral formalism, allowing us to derive the full probability distribution for the qubit's trajectories. Our analytical results reveal that purification is characterized by a dynamical crossover from a diffusion dominated regime to a measurement dominated regime, visible in the emergence of a bimodal state distribution. The analytical solutions are in strong agreement with numerical simulations.

What carries the argument

The multiplicative Langevin equation for the purity scalar, solved exactly by the Onsager-Machlup path-integral formalism.

Load-bearing premise

The collisional model of sequential interactions with ancillary qubits faithfully represents continuous monitoring and the conditioned density-matrix evolution reduces to a single purity scalar without loss of essential information.

What would settle it

Direct numerical integration of the stochastic master equation for the full qubit density matrix would produce statistics of purity trajectories that deviate from the analytically derived probability distribution.

Figures

Figures reproduced from arXiv: 2605.12783 by Henrique Santos Lima, Matheus M. R. Poltronieri Martins.

**Figure 2.** Figure 2: FIG. 2. Individual trajectories [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Left: solutions of the extremal condition Ω [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Left: probability density [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Full trajectory distributions for the monitored qubit. Top: probability density [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

read the original abstract

We investigate the purification dynamics of a single qubit under continuous in time monitoring. By employing a collisional model framework where the system interacts sequentially with ancillary qubits, we describe the conditioned evolution of the density matrix through a stochastic master equation. We show that for initial mixed states, the dynamics reduce to a multiplicative Langevin equation for a single scalar parameter representing the state's purity. This stochastic process is solved exactly using the Onsager-Machlup path integral formalism, allowing us to derive the full probability distribution for the qubit's trajectories. Our analytical results reveal that purification is characterized by a dynamical crossover from a diffusion dominated regime to a measurement dominated regime, visible in the emergence of a bimodal state distribution. The analytical solutions are in strong agreement with numerical simulations, providing a robust theoretical benchmark for the study of information extraction in monitored quantum systems.

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 exact path-integral solution for qubit purification trajectories is a useful benchmark if the single-variable reduction holds, but that step needs careful verification against standard Bloch dynamics.

read the letter

The paper's main contribution is an exact analytical solution for the probability distribution of purity trajectories in a continuously monitored qubit, obtained by reducing the stochastic master equation to a multiplicative Langevin equation and then applying the Onsager-Machlup path integral. If the reduction step is valid, this gives a clean benchmark for how information is extracted over time, including the crossover from diffusion-dominated to measurement-dominated regimes that shows up as a bimodal distribution. The work does a few things well. It starts from a collisional model with ancillary qubits, which is a standard way to model continuous monitoring, and it shows strong agreement between the analytical distribution and direct numerical simulations of the master equation. That match lends credibility to the path-integral approach here. The explicit form of the distribution and the identification of the dynamical crossover are concrete results that people in the field can use to test approximations or other models. The main soft spot is the claim that the dynamics reduce exactly to a single scalar Langevin equation for the purity, even for general initial mixed states. In the usual Bloch-vector picture for monitoring along z, the equation for p = Tr(ρ²) involves the z-component separately through terms like r_x² + r_y², which do not automatically express themselves in terms of p alone. Unless the initial state has no transverse components or the model forces them to zero quickly, an extra variable is needed and the single-variable path integral does not apply directly. The abstract presents the reduction as exact, so the full paper needs to show where that closure comes from—perhaps through a specific choice of initial conditions or a property of the collisional setup. If that assumption is hidden, it limits how general the result is. This paper is for researchers in open quantum systems who study continuous monitoring and want analytical tools beyond numerics. A reader interested in exact solutions for stochastic quantum trajectories will get value from the distribution they derive. It is worth sending to peer review because the methods are established and the claim is testable; a referee can check the reduction step and see whether the numerics cover the general case or only special states. Overall, the work is solid enough to deserve that step rather than a desk rejection.

Referee Report

2 major / 3 minor

Summary. The paper investigates the purification of a monitored qubit using a collisional model that leads to a stochastic master equation. It claims that for initial mixed states, the dynamics reduce exactly to a multiplicative Langevin equation for the purity parameter, which is then solved exactly using the Onsager-Machlup path-integral formalism to obtain the full probability distribution of the qubit's trajectories. The results show a dynamical crossover from diffusion-dominated to measurement-dominated regimes, with the analytical solutions agreeing strongly with numerical simulations.

Significance. If the central reduction holds, this provides an exact analytical tool for computing trajectory distributions in continuously monitored quantum systems, which is a significant contribution to the field of quantum trajectories and open quantum systems. The path-integral approach allows for closed-form expressions that can serve as benchmarks for numerical methods and could inspire similar treatments in higher-dimensional systems.

major comments (2)

[§3] §3 (Reduction to Langevin equation): The claim that the stochastic master equation reduces to a closed multiplicative Langevin equation solely in the purity p is problematic. For standard continuous monitoring of σ_z, the Itô SDE for p = Tr(ρ²) includes a drift term -2γ(r_x² + r_y²)dt that depends on the separate value of r_z (specifically -2γ(p - r_z² - 1/2)dt), preventing closure on p alone without additional assumptions such as initial r_x = r_y = 0. This issue must be resolved for the subsequent path-integral solution to be valid.
[§4] §4 (Path-integral solution): The application of the Onsager-Machlup formalism assumes the SDE is one-dimensional and closed. If the reduction in §3 does not hold for general initial mixed states, the derived probability distribution does not correspond to the actual qubit dynamics and the agreement with numerics may be limited to special cases.

minor comments (3)

[Abstract] Abstract: Specify the quantitative metric used to claim 'strong agreement' with numerical simulations, such as the L1 distance between distributions or mean-squared error.
[Introduction] Introduction: Provide more context on how the collisional model approximates continuous monitoring, including the limit taken for the interaction strength and frequency.
[Figure 1] Figure 1: Ensure the caption clearly explains the parameters used in the plotted distributions and the initial conditions for the mixed states.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting important points regarding the validity of the reduction to a closed equation for the purity. We address each major comment below and propose revisions to clarify the scope of our results.

read point-by-point responses

Referee: [§3] §3 (Reduction to Langevin equation): The claim that the stochastic master equation reduces to a closed multiplicative Langevin equation solely in the purity p is problematic. For standard continuous monitoring of σ_z, the Itô SDE for p = Tr(ρ²) includes a drift term -2γ(r_x² + r_y²)dt that depends on the separate value of r_z (specifically -2γ(p - r_z² - 1/2)dt), preventing closure on p alone without additional assumptions such as initial r_x = r_y = 0. This issue must be resolved for the subsequent path-integral solution to be valid.

Authors: We agree that in general the equation for the purity does not close without additional assumptions. In our work, we restrict to initial mixed states that are diagonal in the measurement basis, i.e., with vanishing transverse Bloch components r_x = r_y = 0. In this case, r_z² = 2p - 1, and the drift term becomes a function of p only, yielding the multiplicative Langevin equation. We will revise §3 and the abstract to explicitly state this initial condition assumption. revision: yes
Referee: [§4] §4 (Path-integral solution): The application of the Onsager-Machlup formalism assumes the SDE is one-dimensional and closed. If the reduction in §3 does not hold for general initial mixed states, the derived probability distribution does not correspond to the actual qubit dynamics and the agreement with numerics may be limited to special cases.

Authors: With the clarification that our results apply to initial states with r_x = r_y = 0, the SDE is indeed closed and one-dimensional. The numerical simulations presented in the manuscript use the same class of initial states, and the excellent agreement supports the analytical solution. We will update the manuscript to emphasize that the results are for this specific class of initial mixed states, rather than arbitrary mixed states. revision: yes

Circularity Check

0 steps flagged

Derivation chain from collisional model to Onsager-Machlup path integral is self-contained; only minor non-load-bearing self-citation possible

full rationale

The paper starts from the standard stochastic master equation obtained via the collisional model, performs an exact reduction of the qubit density-matrix dynamics to a multiplicative Langevin equation in the purity scalar for the stated initial conditions, and then applies the established Onsager-Machlup formalism to obtain the probability distribution. No parameter is fitted to data and then re-labeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The central steps are therefore independent of the target result. A score of 2 accounts for the possibility of routine self-citation of the collisional-model framework without that citation carrying the load of the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard assumptions of quantum measurement theory and the validity of the collisional-model approximation for continuous monitoring. No free parameters are introduced or fitted; the solution is presented as exact within the model.

axioms (1)

domain assumption The stochastic master equation for continuous monitoring in the collisional model framework accurately describes the conditioned evolution of the qubit density matrix.
Invoked at the outset to justify the reduction to a single purity parameter.

pith-pipeline@v0.9.0 · 5441 in / 1367 out tokens · 63453 ms · 2026-05-14T19:56:46.389254+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Y. Li, X. Chen, and M. P. A. Fisher,Quantum Zeno effect and the many-body entanglement transition, Phys. Rev. B98, 205136 (2018)

work page 2018
[2]

Skinner, J

B. Skinner, J. Ruhman, and A. Nahum,Measurement- Induced Phase Transitions in the Dynamics of Entangle- ment, Phys. Rev. X9, 031009 (2019)

work page 2019
[3]

Y. Li, X. Chen, and M. P. A. Fisher,Measurement- driven entanglement transition in hybrid quantum cir- cuits, Phys. Rev. B100, 134306 (2019)

work page 2019
[4]

M. J. Gullans and D. A. Huse,Dynamical purifica- tion phase transition induced by quantum measurements, Phys. Rev. X10, 041020 (2020)

work page 2020
[5]

Jian, Y.-Z

C.-M. Jian, Y.-Z. You, R. Vasseur, and A. W. W. Lud- wig,Measurement-induced criticality in random quantum circuits, Phys. Rev. B101, 104302 (2020)

work page 2020
[6]

Schomerus,Noisy monitored quantum dynamics of er- godic multi-qubit systems, J

H. Schomerus,Noisy monitored quantum dynamics of er- godic multi-qubit systems, J. Phys. A: Math. Theor.55, 214001 (2022)

work page 2022
[7]

J. M. Deutsch,Quantum statistical mechanics in a closed system, Phys. Rev. A43, 2046–2049 (1991)

work page 2046
[8]

Srednicki,Chaos and quantum thermalization, Phys

M. Srednicki,Chaos and quantum thermalization, Phys. Rev. E50, 888–901 (1994)

work page 1994
[9]

Rigol, V

M. Rigol, V. Dunjko, and M. Olshanii,Thermalization and its mechanism for generic isolated quantum systems, Nature452, 854–858 (2008)

work page 2008
[10]

Nandkishore and D

R. Nandkishore and D. A. Huse,Many-Body Localization and Thermalization in Quantum Statistical Mechanics, Annu. Rev. Condens. Matter Phys.6, 15–38 (2015)

work page 2015
[11]

Grimmett,Percolation, 2nd ed

G. Grimmett,Percolation, 2nd ed. (Springer-Verlag, Berlin, 1999)

work page 1999
[12]

Vasseur, A

R. Vasseur, A. C. Potter, Y.-Z. You, and A. W. W. Lud- wig,Entanglement transitions from holographic random tensor networks, Phys. Rev. B100, 134203 (2019)

work page 2019
[13]

A. C. Potter and R. Vasseur, inEntanglement in Spin Chains(Springer International Publishing, 2022), pp. 211–249

work page 2022
[14]

Jacobs and D

K. Jacobs and D. A. Steck,A straightforward intro- duction to continuous quantum measurement, Contemp. Phys.47, 279–303 (2006)

work page 2006
[15]

H. M. Wiseman and G. J. Milburn,Quantum Measure- ment and Control(Cambridge University Press, 2010)

work page 2010
[16]

Jacobs,How to project qubits faster using quantum feedback, Phys

K. Jacobs,How to project qubits faster using quantum feedback, Phys. Rev. A67, 030301 (2003)

work page 2003
[17]

A. N. Korotkov,Selective quantum evolution of a qubit state due to continuous measurement, Phys. Rev. B63, 115403 (2001)

work page 2001
[18]

T. A. Brun,A simple model of quantum trajectories, Am. J. Phys.70, 719–737 (2002)

work page 2002
[19]

H. M. Wiseman,Quantum trajectories and quantum mea- surement theory, Quantum Semiclass. Opt.8, 205–222 (1996)

work page 1996
[20]

Ciccarello, S

F. Ciccarello, S. Lorenzo, V. Giovannetti, and G. M. Palma,Quantum collision models: Open system dynam- ics from repeated interactions, Phys. Rep.954, 1–70 (2022)

work page 2022
[21]

Onsager and S

L. Onsager and S. Machlup,Fluctuations and Irreversible Processes, Phys. Rev.91, 1505–1512 (1953)

work page 1953
[22]

Machlup and L

S. Machlup and L. Onsager,Fluctuations and Irreversible Process. II. Systems with Kinetic Energy, Phys. Rev.91, 1512–1515 (1953)

work page 1953
[23]

L. F. Cugliandolo and V. Lecomte,Rules of calculus in the path integral representation of white noise Langevin equations: the Onsager–Machlup approach, J. Phys. A: Math. Theor.50, 345001 (2017)

work page 2017
[24]

de Pirey, L

T. de Pirey, L. F. Cugliandolo, V. Lecomte, and F. van Wijland,Path integrals and stochastic calculus, Adv. Phys.71, 1–85 (2022)

work page 2022
[25]

Maruyama,Continuous Markov processes and stochastic equations, Rend

G. Maruyama,Continuous Markov processes and stochastic equations, Rend. Circ. Mat. Palermo4, 48–90 (1955)

work page 1955
[26]

V. E. Beneˇ s,Exact finite-dimensional filters for certain diffusions with nonlinear drift, Stochastics5(1–2), 65–92 (1981)

work page 1981
[27]

S¨ arkk¨ a and A

S. S¨ arkk¨ a and A. Solin,Applied stochastic differential equations, Cambridge University Press (2019)

work page 2019
[28]

I. V. Girsanov,On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory Probab. Appl.5, 285–301 (1960)

work page 1960
[29]

Risken,The Fokker–Planck Equation: Methods of So- lution and Applications, 2nd ed

H. Risken,The Fokker–Planck Equation: Methods of So- lution and Applications, 2nd ed. (Springer, Berlin, 1989)

work page 1989
[30]

C. W. Gardiner,Handbook of Stochastic Methods (Springer-Verlag, Berlin, 1985)

work page 1985
[31]

N. G. Van Kampen,Stochastic Processes in Physics and Chemistry(Elsevier Science, Amsterdam, 2011)

work page 2011
[32]

Patel and P

A. Patel and P. Kumar,Weak measurements, quantum state collapse, and the Born rule, Phys. Rev. A92, 022115 (2015)

work page 2015