Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements

Eloy Moreno-Nadales; Lubashan Pathirana; Owen Ekblad

REVIEW 2 major objections 4 minor 7 references

Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Disordered generalized quantum measurements obey a strong law of large numbers: the empirical frequency of every finite outcome word converges almost surely to a fixed, initial-state-independent limit whenever the disorder dynamics has a…

desk verdict Repairable gap in Theorem 2's proof aside, this is a genuine extension of Kummerer-Maassen with a real strong law for finite word frequencies in disordered quantum trajectories. read the letter →

arxiv 2501.18014 v1 pith:2LTOPJHR submitted 2025-01-29 math-ph math.MPquant-ph

Owen Ekblad , Eloy Moreno-Nadales , Lubashan Pathirana This is my paper · ORCID

classification math-phmath.MPquant-ph MSC 81P1560F1537A3060J10

keywords disorderedquantumtrajectoriesergodictheoremstronglawoflargenumbersMarkovchainsinrandomenvironmentsKrausoperatorsmeasurementsstochasticallystationarystateannealedandquenchedmeasures

topics The Measurement Problem

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

This paper proves that repeated generalized quantum measurements in a randomly changing environment behave ergodically: the empirical frequency of every finite block of measurement outcomes converges almost surely to a fixed limit, independent of how the quantum system was initially prepared. The setting is a Markov chain in a random environment, where the transition probabilities are defined dynamically by measurement operators that depend on a stationary ergodic disorder. The result is conditional on a single premise, called dynamical ergodicity: the disordered dynamics has a unique stochastically stationary state that the environment shift maps into itself. Under that premise, the paper establishes annealed and quenched ergodic theorems for the outcome process and derives the strong law of large numbers as a corollary, extending earlier noise-free results of Kummerer and Maassen. A sympathetic reader should care because it says that in a disordered measurement experiment, time-averaged outcome statistics are reproducible and can be computed from the stationary state alone.

What carries the argument

The load-bearing object is the assumption (Dyn-Erg), the existence of a unique stochastically stationary random state ρss ∈ Sd(Ω) satisfying φθ(ω)(ρss;ω) = ρss;θ(ω) almost surely. The proof machinery is the matrix-valued quenched quantum measure Q*ω, defined on cylinder sets by Q*ω(π−1n{(a1,...,an)}) = T†a1;θ(ω) ∘ ⋯ ∘ T†an;θn(ω)(I). Two identities carry the argument: Lemma 3.2 rewrites the push-forward of Q* under the shift as the adjoint of the accumulated channel, and Lemma 3.5 shows that under (Dyn-Erg) any τ-invariant event Γ has constant fiber-wise matrix value Q*ω(Γω) = Qρss(Γ) I, so its quenched probability is almost surely constant. The proof of Theorem 3 also imports Theorem A from Ekblad and Schenker, which supplies the Cesàro convergence of evolved observables, and uses the Beck–Schwartz vector-valued random ergodic theorem to obtain the stochastically stationary limit of Cesàro-averaged states.

What would settle it

Simulate a qubit with i.i.d. random unitary Kraus operators chosen from the Clifford group (which should satisfy (Dyn-Erg) with the maximally mixed state as the unique stationary state), run the measurement trajectory for N = $10^{6}$ steps from several pure initial states, and compute the empirical frequency of the outcome word (0,1); if the limits differ between initial states or disagree with EPr[Qρss](0,1) computed by solving the stochastic stationary equation, Theorem 1 is false.

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Extended reading notes

Core claim

The central claim is Theorem 1: let (Ω,F,Pr,θ) be an invertible ergodic probability-preserving dynamical system modeling the disorder, and let V be a random Kraus ensemble whose corresponding measurement channel is dynamically ergodic, meaning there is a unique random state ρss such that φθ(ω)(ρss;ω) = ρss;θ(ω) almost surely. For any random initial state ϑ, for Pr-almost every disorder realization ω and Qϑ;ω-almost every outcome sequence, the frequency with which a fixed word (b1,...,bm) appears in the first N measurement outcomes converges to EPr[Qρss](b1,...,bm), the annealed probability of that word under the stationary state. Thus the long-run empirical distribution of measurement outcomes is independent of the initial state and coincides with the expectation over disorder of the stationary quenched measure. This is a quenched strong law of large numbers for disordered quantum trajectories, not merely a statement about averaged expectations.

Load-bearing premise

Everything rests on the assumption (Dyn-Erg) that the disordered measurement process has exactly one stochastically stationary random state that the environment shift maps into itself; the paper does not prove this exists or is unique for any particular ensemble.

Editorial extensions

If this is right

Theorem 1 gives an operational recipe: record measurement outcomes from any initial state, average over time, and the result estimates EPr[Qρss](b1,...,bm) for every finite word.
The annealed quantum measure Qρss is ergodic under the skew shift τ = (θ,σ), and the quenched averaged measure EPr[Qρss] is ergodic under the outcome shift σ.
For any σ-invariant event E, the quenched probability Qϑ;ω(E) is Pr-almost surely equal to EPr[Qρss](E), which is either 0 or 1, independent of the initial state.
Theorem 3 provides a general law of large numbers for annealed quantum probabilities: (1/N) Σn=1N Qϑ(τ−nΓ) → Qρss(Γ) for all Γ.
The framework covers i.i.d. and Markovian random disorder, periodic and quasiperiodic disorder, and the constant (nonrandom) case, thereby extending the noise-free ergodic theorems of Kümmerer and Maassen.

Reading between the lines

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

A natural next step is to identify checkable conditions on the random Kraus ensemble — for example spectral gap or strong contractivity of the averaged channel — that guarantee the existence and uniqueness of ρss, since the present theorems are conditional on that premise.
The quenched pathwise convergence suggests that quenched large deviation principles or central limit theorems for disordered quantum trajectories should hold under appropriate mixing assumptions; verifying numerically whether the finite-time fluctuations are O(N−1/2) and independent of initial state would be a direct test.
The identity Q*ω(Γω) = Qρss(Γ)I indicates a strong form of self-averaging: the outcome statistics of a single long trajectory in a single disorder realization reproduce the disorder-averaged stationary distribution, so repeated runs over independent disorder realizations are not necessary for estimating the stationary statistics.
Extending the matrix-valued measure approach to continuous-time disordered quantum trajectories (stochastic Schrödinger equations) may yield analogous annealed/quenched ergodic theorems in that setting.

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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

The short version: this paper has a genuine new result — trajectory-level annealed and quenched ergodic theorems plus an SLLN for finite word frequencies in disordered generalized measurements, extending Kummerer–Maassen to stationary ergodic disorder. The main theorems are stated cleanly and the machinery (matrix-valued quenched measure, annealed measure, skew-product) is standard and handled with care. Lemma 3.5 is the right bridge from the channel-level convergence in Theorem A of [ES24] to the trajectory-level statement.

The soft spots. First, the proof of Theorem 2 contains a section error. It asserts that (F×E)_ω = E for all ω, but the correct section is E if ω∈F and ∅ otherwise. Because of that, the integrals after (3.49) run over Ω instead of F, dropping the factor 1_F, and (3.58) as displayed does not follow from the preceding line. That matters because (3.49) with Γ′=Γ and ϑ=ρ_ss is what yields the 0–1 law. The encouraging part is that the fix appears mechanical: carry 1_F through, replace each ∫_Ω by ∫_F, and the same identity (3.58) still holds. So the theorem is likely correct, but the manuscript as written does not prove it.

Second, the dependence on Theorem A of [ES24] is heavy — it is load-bearing for Lemma 3.5 and Theorem 3, and [ES24] is a coauthored preprint. That is not disqualifying; the citation is explicit and the result seems to be the natural input. But a referee should verify that Theorem A truly covers the random observable O_ω = Q^*_ω(Γ_ω), whose measurability and boundedness depend on the joint structure of Q^* and Γ. It probably does, but the text doesn't check it.

Third, (Dyn-Erg) is assumed, not derived. The paper gives no conditions on the random Kraus ensemble under which a unique stochastically stationary state exists. That is a scope choice — the abstract says "under the assumption" — but it does mean the theorems are conditional on a property that may be hard to verify in concrete examples.

The citation pattern looks honest, the novelty claim is accurate, and there are no fitted parameters or invented entities. This is for quantum trajectory theorists and mathematicians working on random quantum channels and Markov chains in random environments. With the section error fixed, it would be a solid contribution; it deserves a serious referee rather than a desk reject.

Referee Report

2 major / 4 minor

Summary. The paper studies quantum trajectories generated by repeated generalized measurements in a stationary ergodic random environment, modeling them as Markov chains in a random environment with state space the quantum states S_d. Under the hypothesis (Dyn-Erg) that the associated random quantum channel has a unique stochastically stationary random state ρ_ss, it proves an annealed ergodic theorem (Theorem 2), an annealed law of large numbers for the quantum probabilities (Theorem 3), a quenched ergodic theorem (Theorem 4), and a strong law of large numbers for the empirical frequencies of finite measurement-outcome words (Theorem 1). The statements are conditional on (Dyn-Erg) and on Theorem A of the preprint [ES24], which is imported as a black box.

Significance. If the proofs are completed, the paper would extend the Kümmerer–Maassen ergodic theory of quantum trajectories to disordered environments, providing annealed and quenched ergodic theorems for Markov chains in a random environment with quantum transition structure, and an outcome-frequency strong law of large numbers with an explicit limit EPr[Qρss](b1,...,bm). The framework is clean, the statements are precise, and there are no fitted parameters; the main theorem gives a concrete falsifiable prediction for measurement statistics. The paper also makes good use of existing tools such as the Beck–Schwartz vector-valued ergodic theorem and the Kolmogorov extension theorem. However, the main results are conditional on the unresolved hypothesis (Dyn-Erg) and on an unproved imported theorem from a preprint with overlapping authorship.

major comments (2)

[§3.1, proof of Theorem 2, Eq. (3.49)] The proof of the factorization identity (3.49) contains a genuine error: after stating that '(F×E)_ω = E', the computation drops the indicator of F and integrates over all of Ω instead of over F. The correct section is (F×E)_ω = E if ω∈F and ∅ otherwise. Consequently, Equation (3.58) does not follow from the preceding lines; the final identity would require Qϑ(F×E)=∫_Ω Qϑ;ω(E)dPr, which is false. Since (3.58) with ϑ=ρss and Γ'=Γ is used to conclude Qρss(Γ)=Qρss(Γ)^2, the 0–1 law in Theorem 2(i) is not established as written. Theorem 4(i) and Theorem 1 rely on this 0–1 law, so the central claim is not proven. The error appears local and repairable: carrying 1_F through the same computation replaces each ∫_Ω by ∫_F and still yields the desired identity. Please supply the corrected proof.
[Lemma 3.5 and Theorem A of [ES24]] Lemma 3.5, which is used in the proofs of Theorems 2, 3, and 4, depends crucially on Theorem A of [ES24], a preprint with overlapping authorship. This theorem is load-bearing: it is used to identify the constant in the identity Q*_ω(Γ_ω)=Qρss(Γ)I. The manuscript does not provide a proof of Theorem A nor a precise statement of its publication status. This is not by itself circular, but it makes the paper non-self-contained at a load-bearing point. Please either include a proof of Theorem A in an appendix or cite a published/peer-reviewed version, and state explicitly that the main results depend on this external result.

minor comments (4)

[Theorem 4 statement] The statement contains a typo: 'If V is satisfies (Dyn-Erg)' should read 'If V satisfies (Dyn-Erg)'.
[§3.1, proof of Theorem 3] The sentence beginning 'In conjunction with the fact that ϑ_ω=ρss;ω almost surely, if we apply the tracial Hölder's inequality...' is grammatically broken and the dominated convergence argument is sketched too quickly; please rewrite this step explicitly.
[Definition 4 and Eq. (2.37)] It would be helpful to state explicitly that Qϑ is a probability measure on (Ω×AN, F⊗Σ) and that the integral in (2.37) is well-defined for all Γ∈F⊗Σ; the measurability of ω↦Qϑ;ω is asserted via Lemma 2.3, but the joint measurability of the section map could be stated more clearly.
[Lemma 3.5] The proof shows the identity ⟨ψ,Q*_ω(Γ_ω)⟩=Qρss(Γ) for each fixed nonrandom ψ after discarding a ψ-dependent null set; since Sd is separable, one can choose a countable dense set and obtain a uniform null set, but this step should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the pathwise SLLN is independent of its stated assumptions, and the proof gap in Theorem 2 is a correctness issue rather than a circular reduction.

full rationale

The derivation is not circular. The central theorem (Theorem 1) is a pathwise strong law for measurement-outcome frequencies, and its proof combines Birkhoff's pointwise ergodic theorem with the 0-1 law from Theorem 2, which in turn rests on Lemma 3.5, the factorization argument (3.49)-(3.58), and Theorem A of [ES24]. None of these steps is equivalent to the statement being proved. The (Dyn-Erg) condition is an explicitly stated assumption (Definition 1), not a conclusion extracted from the target result, and no fitted parameter is renamed as a prediction. Theorem A is a self-citation with shared authorship, but it is a parameter-free mean-ergodic theorem whose stated assumptions do not include the pathwise SLLN; under the review rules this counts as independent support and does not raise the circularity score. The known gap in the proof of Theorem 2, namely that the section of F x E is E only on the set F, is a repairable correctness issue in the manuscript as written, not a circular reduction; repairing it preserves the conclusion rather than assuming it. I find no step in which the claim reduces to its inputs by construction, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The central claim rests on the model assumptions of ergodic disorder and unique stochastically stationary state, plus prior theorems including the same group's [ES24] and standard ergodic theorems. The proof is not fully self-contained.

assumptions (6)

domain assumption The disorder is modeled by an invertible, ergodic, Pr-preserving dynamical system (Omega, F, Pr, theta).
Used throughout Section 1.1 and all theorems; this is the environment model that defines the disordered measurements.
domain assumption (Dyn-Erg): unique stochastically stationary state rho_ss exists for the random Kraus ensemble V.
Stated as the central hypothesis before Theorem 1; not proved or characterized in this paper. All theorems are conditional on it.
standard math Theorem A of [ES24], the ergodic theorem for dynamically ergodic random quantum channels.
Imported as a black box in Lemma 3.5 and Theorem 3. It is a preprint by the same research group and is load-bearing.
standard math Beck-Schwartz vector-valued random ergodic theorem (Theorem B).
Used in the proof of Theorem 3 to obtain the almost-sure Cesaro limit of backward channel products.
standard math Kolmogorov extension theorem and Choi-Kraus representation for CPTP maps in finite dimensions.
Justify the existence of quenched measures Q_{rho;omega} and the Kraus form of random quantum channels.
standard math Birkhoff pointwise ergodic theorem.
Used in the proof of Theorem 1 to convert ergodicity of sigma under the annealed measure into word-frequency convergence.

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Pith. "Pith review of Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements." pith.science (2026). https://pith.science/paper/2LTOPJHR

@misc{pith2026250118014,
  author       = {Pith},
  title        = {Pith review of: Ergodic Theorems for Quantum Trajectories under Disordered Generalized Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2LTOPJHR}},
  note         = {Machine review of arXiv:2501.18014}
}

read the original abstract

We consider quantum trajectories arising from disordered, repeated generalized measurements, which have the structure of Markov chains in random environments (MCRE) with dynamically-defined transition probabilities; we call these disordered quantum trajectories. Under the assumption that the underlying disordered open quantum dynamical system approaches a unique equilibrium in time averages, we establish a strong law of large numbers for measurement outcomes arising from disordered quantum trajectories, which follows after we establish general annealed ergodic theorems for the corresponding MCRE. The type of disorder our model allows includes the random settings where the disorder is i.i.d. or Markovian, the periodic (resp. quasiperiodic) settings where the disorder has periodic (resp. quasiperiodic) structure, and the nonrandom setting where the disorder is constant through time. In particular, our work extends the earlier noise-free results of K\"ummerer and Maassen to the present disordered framework.

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Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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A Brief Journey through Collision Models for Multipartite Open Quan- tum Dynamics

isbn: 9783540476207. DOI: 10.1007/978-3-540-47620-7 . [Cat+22] M. Cattaneo et al. “A Brief Journey through Collision Models for Multipartite Open Quan- tum Dynamics”. In: Open Systems &; Information Dynamics 29.03 (Sept. 2022). issn: 1793-7191. DOI: 10.1142/s1230161222500159. [CFS82] I. P. Cornfeld, S. V. Fomin, and Y. G. Sinai. Ergodic Theory. Springer N...

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doi:10.1017/cbo9780511755316 1957

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DOI: 10.1007/s00440-010-0323-6 . [Ore91] S. Orey. “Markov Chains with Stochastically Stationary Tra nsition Probabilities”. In: The Annals of Probability 19.3 (July 1991). issn: 0091-1798. DOI: 10.1214/aop/1176990328. [Pel07] C. Pellegrini. “Existence, uniqueness and approximation for s tochastic Schrodinger equa- tion: the Poisson case”. In: (2007). DOI:...

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DOI: 10.1063/5.0153483. [RH12] ´A. Rivas and S. F. Huelga. Open Quantum Systems. An Introduction . Vol. 10. Springer Berlin Heidelberg, 2012. DOI: 10.1007/978-3-642-23354-8 . [Roh61] V. A. Rohlin. “Exact endomorphisms of Lebesgue spaces”. In: Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), pp. 499–530. [Roh64] V. A. Rohlin. “Exact endomorphisms of a Lebesgue sp...

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