Randomly measured quantum particles and thermal noise

Victor Gurarie

REVIEW 2 major objections 5 minor 16 references

Randomly measured quantum particles and thermal noise

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Random position measurements and thermal noise look identical in linear observables but separate cleanly in the square of the average position.

desk verdict Random measurements and thermal noise are indistinguishable at the linear level but the nonlinear observable <x>^2 separates them (t vs t^3); careful and honest calculation, with one verification gap in the replica normalization. read the letter →

arxiv 2507.06382 v1 pith:GVAOCYWY submitted 2025-07-08 cond-mat.dis-nn quant-ph

Victor Gurarie This is my paper

classification cond-mat.dis-nnquant-ph

keywords randommeasurementsthermalnoisenonlinearobservablesKeldyshreplicameasurementbackactionquantumdiffusionopensystemsposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Random position measurements and random (e.g. thermal) noise are shown to drive the averaged density matrix of a free quantum particle through the same evolution equation, so every linear observable agrees between the two settings. The paper then calculates a nonlinear observable — the square of the quantum average position, averaged over measurement outcomes or noise realizations — and finds that the two settings separate sharply at short times: measurements give a growth linear in time, while noise gives a growth cubic in time. Establishing this difference matters because any experiment that wants to attribute decoherence or diffusion to measurement backaction rather than to an ordinary fluctuating environment needs an observable that can actually tell the two apart.

What carries the argument

The argument is carried by the replicated Keldysh density matrix in the replica limit $n \to 1$. Each of $n$ copies of the particle is written with forward and backward trajectories (Keldysh contours), and averaging over the Gaussian random field $V(x,t)$ produces an effective $n$-replica Schrödinger-like equation, Eqs. (21) and (22). The only difference between random measurements and random noise is the sign of the terms coupling different replicas within the same Keldysh sector; this sign is invisible for $n = 1$ but controls the nonlinear observable once the system is replicated and the limit $n \to 1$ is taken. The calculation is completed perturbatively in the rate $\lambda$, with Gaussian measurement/noise correlation $W(x) = (1/\sqrt{2\pi}\,\ell)\,e^{-x^2/(2\ell^2)}$, and the replica normalization is handled by dividing by the trace of the density matrix (Appendix A.2). The square of the quantum average position is extracted by differentiating the replicated density matrix with respect to two classical Keldysh momentum components and integrating out the quantum components.

What would settle it

Simulate many individual quantum trajectories of a free particle under random Gaussian position measurements (Kraus operators with rate $\lambda$) and, separately, under a random Gaussian potential with the same correlation function; from each trajectory record the single-shot mean position $x$ at several short times, square the averages over trajectories, and fit the short-time growth. The paper's claim predicts a linear-in-$t$ growth for measurements and a $t^3$ growth for noise; observing the same time dependence in both settings, or a different exponent, would refute Eqs. (26) and (28).

Watch

Extended reading notes

Core claim

The paper's central claim is that random measurements and random noise are indistinguishable at the level of linear expectation values yet distinguishable through nonlinear observables. Averaging the density matrix over the random measurement field yields equation (11), and averaging over a random noisy potential yields exactly the same equation, so quantities like the mean squared position follow the identical law (17) in both cases. Once two copies of the system are coupled through the square of a quantum average, the replicated evolution equations (21) and (22) differ by the sign of the same-replica interaction terms, and a perturbative solution in the measurement/noise rate $\lambda$ gives different short-time behavior: $\overline{\langle x\rangle^2} \sim \lambda t$ for random measurements and $\overline{\langle x\rangle^2} \sim \lambda t^3$ for random noise (Eqs. (26) and (28)). The paper thereby offers an operational criterion: measure the square of the average position, and its time scaling identifies which process generated it.

Load-bearing premise

The distinguishing result depends on a technical normalization recipe for the replicated density matrix in the $n \to 1$ limit (dividing by the trace and by extra factors of $2$ and $2^{n/2}$, keeping only the $\alpha=1$, $\beta=2$ replica pair); if that recipe is wrong, the predicted $t$ versus $t^3$ scaling difference would not follow.

Editorial extensions

If this is right

Every linear observable of a randomly measured particle — mean position, mean squared position, momentum — matches the same quantity under random noise; the linear law (17) holds for both, including the $t^3$ diffusion term with no $\hbar$.
The nonlinear observable $\overline{\langle x\rangle^2}$ distinguishes the two: at short times it grows as $\lambda t$ under random measurements and as $\lambda t^3$ under random noise, with distinct dependence on the initial width $\Delta$ and measurement length $\ell$.
The difference $\langle x^2\rangle - \overline{\langle x\rangle^2}$ initially decreases in time under random measurements before eventually growing, an early suppression of quantum spreading that does not occur under random noise (Eq. (30)).
For large times the two settings also separate: $\overline{\langle x\rangle^2}$ grows as $t^2 \ln t$ for measurements and as $t^2$ for noise (Appendix A), so the distinction is not limited to short times.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The scaling contrast ($t$ vs $t^3$) suggests a practical protocol: in a cold-atom or trapped-ion experiment that can apply either random projective-type measurements or engineered noisy potentials, accumulating enough shots to estimate the squared mean position should reveal which mechanism is active; the paper does not propose such an experiment.
Because linear observables are provably blind, any future experiment claiming to detect measurement-induced backaction should use a nonlinear functional of the density matrix; this paper's observable is one example, and other nonlinear functionals (for example higher moments or fidelity-type measures) may show similar or sharper signatures.
The fact that the short-time measurement signal $\sim \lambda t$ is quantum (contains $\hbar$) while the noise signal $\sim \lambda t^3$ is classical suggests a thermodynamic reading: measurement backaction injects quantum fluctuations that noise does not, an interpretation the paper states only implicitly.
A direct numerical check of Eqs. (26) and (28) using Monte Carlo sampling of Kraus operators versus realizations of the noisy potential would independently test the replica normalization, since the paper's analytic derivation relies entirely on a perturbative replica calculation without such a check.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick read of arXiv:2507.06382. The headline result: for a single particle, random measurements and random noise are indistinguishable in all observables linear in the density matrix, but the nonlinear observable ^2 separates them cleanly — linear in t for measurements (Eq. 26), cubic in t for noise (Eq. 28). That is a useful, concrete distinction, and the paper shows it explicitly rather than leaving it at the level of replica symmetries.

What is actually new: the equivalence of the single-replica density-matrix equations was known, and the x^2 ~ t^3 linear result goes back to refs. 11 and 13. The new content is the explicit first-order-in-lambda computation of ^2 for both processes, the observation that the replica-coupling signs differ between the two (Eqs. 21 vs 22), and the resulting t vs t^3 distinction. The appendix is algebraically careful; Appendix A.5 rederives (17), which is a good internal check.

Soft spots, in proportion: the load-bearing step is the two-replica normalization in A.2. The stress-test worry about a missing factor is not borne out — I checked the -1/(2*2^{n/2}) prescription against the Gaussian integral (A32) and it gives the unnormalized numerator correctly. But the paper provides no numerical simulation or independent derivation of the final coefficients in (26) and (28), so an algebraic slip in that normalization cannot be entirely excluded. That is a real gap, though not a demonstrated error. Also, the results are first order in lambda and only valid at short times t << m*Delta^2, which the paper states plainly. The large-time expressions (A34)-(A37) are less emphasized, and those also deserve scrutiny.

The central argument holds. The distinction is real as scoped, and the paper is honest about its limits. The replica trick is transparent enough that a referee can verify the algebra.

Who it's for: anyone working on measurement-induced transitions, quantum trajectories, or open quantum systems who needs an operational way to tell measurement from thermal noise. I would send it to a serious referee. My main advice to the editor: make sure the referee actually rechecks the normalization in A.2 and the coefficients in (26)/(28), because that's where the result lives.

Referee Report

2 major / 5 minor

Summary. This paper contrasts the dynamics of a quantum particle subject to random position measurements with that of a particle in a random time-dependent potential (thermal noise). For each case the author writes down the Kraus/time-dependent Schrödinger description, averages over the random realizations, and shows that the averaged density matrix obeys the same master equation, so all linear observables coincide. The known result ⟨x²⟩ = Δ² + ℏ²t²/(4m²Δ²) + λt³/(3√2πm²ℓ³) is reproduced. To find a distinction, the paper computes the nonlinear observable ⟨x⟩² (the noise-averaged square of the quantum expectation value of position) using a replica trick with n→1. For random measurements it obtains ⟨x⟩² ≈ 2√2Δ⁴λt/(ℏ²√π(2Δ²+ℓ²)^{3/2}) at short times, while for random noise it obtains ⟨x⟩² ≈ 7λt³/(3√2πm²(2Δ²+ℓ²)^{3/2}). The derivation is perturbative in λ, and the appendices provide the algebraic details.

Significance. If correct, the paper offers an operational distinction between measurement-induced dynamics and thermal noise: linear observables are blind to the difference, but the short-time growth of ⟨x⟩² scales as t for random measurements and as t³ for noise. This is a conceptually interesting and potentially testable result. The manuscript is self-contained, and the internal algebra is largely consistent: the n=1 linear correlator is rederived in Appendix A.5 and matches the known result, and the normalization factor in Appendix A.2 can be checked to be correct. The main weakness is that the central quantitative formulas (26) and (28) are not verified by any numerical simulation or independent derivation, and the replica-trick normalization is presented in a terse way that leaves room for doubt.

major comments (2)

[Appendix A.3] The derivation of the two-derivative, two-replica normalization is incomplete. The check in Eq. (A31) establishes the prefactor for a single derivative only, and the statement that non-(1,2) replica pairs contribute zero because 'the derivative over k3,cl followed by setting it to zero brings down k3,q' does not match the differentiation procedure actually used, since only kcl,1 and kcl,2 are differentiated. The conclusion may be correct (the vanishing follows from oddness of the integrand in the quantum momentum of the background replicas), but this is not shown. Because this step determines the coefficients in Eqs. (26) and (28), a complete derivation of the selection rule and of the factor -1/(2·2^{n/2}) is needed.
[Section 3, Eqs. (26) and (28)] The quantitative predictions for the nonlinear observable rest entirely on the replica calculation and are not checked numerically or by an independent method. Given that replica normalizations are a common source of algebraic prefactor errors, the authors should provide either a numerical simulation of the stochastic Schrödinger equation (for example, sampling the Kraus operators for small λ) or an independent analytic derivation to confirm the coefficients and the t versus t³ scalings. This is a verification gap, not a demonstrated error, but it is load-bearing for the paper's central quantitative claim.

minor comments (5)

[Appendix A.5, Eq. (A40)] The final equality in Eq. (A40) drops the free-spreading terms Δ² and t²/(4m²Δ²); as written, the right-hand side is only the λ-dependent part of ⟨x²⟩. Please correct this to avoid confusion.
[Appendix A.2] There is a typo: 'Diving by N' should be 'Dividing by N'.
[Appendix A.1, Eq. (A1)] The notation 'i ∂ρ/dt' should be 'i ∂ρ/∂t' for consistency with the rest of the paper.
[Introduction, Eq. (18)] The notation ⟨x⟩² could be defined more explicitly: it denotes the average over measurement outcomes or noise realizations of the squared quantum expectation value of the position, not the square of the averaged expectation value.
[General] The restoration of ℏ in Eqs. (26) and (28) is physically meaningful (the measurement result is quantum, the noise result is classical), but a brief explanatory sentence would help readers understand why ℏ appears in one formula and not the other.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the random-measurement/noise distinction follows from an explicit replica perturbation calculation with no fitted parameters; the only self-citation (ref. 12) is backed by an independent rederivation in Appendix A.5.

full rationale

The paper's central results, Eqs. (26) and (28), are obtained by a parameter-free perturbative solution of the replicated master equations (21) and (22), with no constants fitted to the quantities being predicted. The equality of linear observables, Eq. (11), is derived directly from Gaussian averaging of the two different evolution problems, not assumed from the target result. The linear result (17) is attributed to prior independent work (refs. 11,13) and is also rederived self-containedly in Appendix A.5, so the self-citation to ref. 12 is not load-bearing. The replica normalization prescription in Appendix A.2, including the factors of 2 and 2^(n/2) and the restriction to the alpha=1, beta=2 replica pair, is a technical assumption that could affect the coefficients if wrong, but it does not presuppose Eqs. (26) and (28), and no step in the derivation reduces to its own output by construction. The absence of a numerical check of the final nonlinear formulas is a verification gap and a correctness risk, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted: lambda, l, Delta, and m are physical inputs of the model. The central results are derived, not fit to data. The axioms are the standard weak-measurement, Gaussian-noise, Kraus, and replica-limit assumptions, plus the explicit first-order perturbative truncation. No new entities are introduced.

assumptions (5)

domain assumption Weak position measurements can be represented by Kraus operators K[V]=integral dx e^{V(x) delta t} |x><x| with Gaussian random V, and the continuous-time limit yields the non-unitary Schrodinger equation (3).
Stated in Eqs. (1)-(6); this is the standard weak-measurement formalism the paper builds on.
domain assumption The Gaussian random potential V(x,t) is white in time with correlation lambda delta(t1-t2) W(x1-x2), Eq. (4).
This Markovian Gaussian noise model is the basis for all path-integral averaging; it is a modeling choice, not derived.
domain assumption The n to 1 replica limit, with normalization by N and factors 2^(n/2) and 2, correctly evaluates nonlinear observables such as <x>^2.
Used in Eqs. (18)-(23) and Appendix A.2; the validity of the replica trick is assumed rather than proved.
domain assumption The weak-measurement operators K[V] are Hermitian, so K_j x K_j in Eq. (18) equals K_j dagger x K_j.
For real V the operator is Hermitian, but the paper does not state this explicitly.
ad hoc to paper First-order perturbation theory in lambda correctly captures the short-time scaling of <x>^2.
The paper says 'we have no reasons to believe that the perturbation theory would be exact' (after Eq. 22), yet uses it for the central quantitative results.

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Cite this review

Pith. "Pith review of Randomly measured quantum particles and thermal noise." pith.science (2026). https://pith.science/paper/GVAOCYWY

@misc{pith2026250706382,
  author       = {Pith},
  title        = {Pith review of: Randomly measured quantum particles and thermal noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GVAOCYWY}},
  note         = {Machine review of arXiv:2507.06382}
}

read the original abstract

We consider the motion of a quantum particle whose position is measured in random places at random moments in time. We contrast this motion with the motion of a quantum particle in a potential which varies randomly in space and in time, which could also be thought of as (possibly thermal) noise. We calculate expectations of observables both linear and nonlinear in the density matrix. We demonstrate explicitly that while linear observables cannot distinguish between random measurements and random noise, measurable distinctions can be seen in nonlinear observables.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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author author B. Skinner , author J. Ruhman , \ and\ author A. Nahum ,\ 10.1103/PhysRevX.9.031009 journal journal Phys. Rev. X \ volume 9 ,\ pages 031009 ( year 2019 ) NoStop

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doi:10.1103/physrevx.13.041046 2023

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Guo , author M

author author H. Guo , author M. S. \ Foster , author C.-M. \ Jian , \ and\ author A. W. \ Ludwig ,\ https://arxiv.org/abs/2410.07317 journal journal arXiv preprint arXiv:2410.07317 \ ( year 2024 ) NoStop

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doi:10.1088/0034-4885/29/1/306 1966

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author author L. D. \ Landau \ and\ author E. M. \ Lifshitz ,\ @noop title Statistical Physics, Part 1 ,\ edition 3rd \ ed.,\ series Course of Theoretical Physics , Vol. volume 5 \ ( publisher Pergamon Press ,\ address Oxford ,\ year 1980 ) NoStop

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