How to design quantum-jump trajectories via distinct master equation representations

Andrea Smirne; Dariusz Chru\'sci\'nski; Jyrki Piilo; Kimmo Luoma

REVIEW 3 major objections 5 minor 49 references

How to design quantum-jump trajectories via distinct master equation representations

T0 review · 3 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read A family of quantum-jump unravelings, called R-ROQJ, redivides any master equation into deterministic drift and jumps, and whenever the associated rate operator is positive the averaged trajectories reproduce the original open-system…

desk verdict The R-ROQJ construction is sound and genuinely useful, but the paper asserts—without proof—positivity of the driven rate operator R1' in Sec. 5.2; the gap is real but repairable, and the central contribution deserves referee time. read the letter →

arxiv 2009.11312 v3 pith:YHQ3MQ46 submitted 2020-09-23 quant-ph

Dariusz Chruściński , Kimmo Luoma , Jyrki Piilo , Andrea Smirne This is my paper · ORCID

classification quant-ph MSC 81S2281P15

keywords quantumjumpsunravelingsopensystemsmasterequationrateoperatorsdissipativitynon-MarkoviandynamicsMonteCarlowavefunction

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

Quantum trajectories for one open-system master equation are not unique: the paper constructs a family of 'rate-operator quantum jumps' (R-ROQJ) that reshuffles the generator's terms between deterministic drift and sudden jumps. Whenever the chosen rate operator $R_\psi(t)=J'_t(|\psi\rangle\langle\psi|)$ is positive, the averaged trajectories exactly reproduce the original master equation, so each valid choice is also a continuous-measurement interpretation. The paper proves that a condition called dissipativity—the Kadison–Schwarz inequality applied to all operators—guarantees such a positive rate operator exists, going beyond completely positive divisibility. Concrete qubit examples show the split can be engineered so that post-jump states are fixed in time or so that coherent driving is absorbed into the jump part, which simplifies simulations and exposes new features such as phase jumps in rational multiples of $\pi/4$.

What carries the argument

The rate operator $R_\psi(t)=J'_t(|\psi\rangle\langle\psi|)$, the image of the current pure state under the chosen jump map, fixes both post-jump states (its eigenvectors) and jump probabilities (its eigenvalues). The gauge freedom $C(t)$ in Eqs. (2)–(4) parametrizes how the master equation is split, and positivity of $R_\psi$ is what makes the split a valid unraveling with a continuous-measurement reading. The load-bearing existence result is Lindblad's dissipativity condition (39), $L^\dagger_t(X^\dagger X)\ge L^\dagger_t(X^\dagger)X+X^\dagger L^\dagger_t(X)$ for all $X$, which by averaging over the unitary group yields a positive map $J_t$ (proof in Appendix B); for qubits, Prop. 1's special form of $C(t)$ (Eq. (33)) forces the eigenvectors of $J'_t$ onto the two fixed states $|\phi_\pm\rangle=(|\phi_1\rangle\pm|\phi_2\rangle)/\sqrt 2$.

What would settle it

For a qubit generator with P-divisible but non-dissipative rates (e.g., $\gamma_1=\gamma_2=1$, $\gamma_3=-a\tanh t$ with $a>1/2$), search all allowed $C(t)$ matrices over all times and initial states, and compute the smallest eigenvalue of $R_\psi(t)$: a case where every choice leaves a negative eigenvalue would prove that dissipativity or something equally strong is strictly necessary for R-ROQJ positivity, while failure of Prop. 1 whenever the deterministic evolution introduces a relative phase between jumps would show condition (35) is load-bearing.

Watch

Extended reading notes

Core claim

The central claim is that, for any time-local master equation of GKLS form, one may pick an arbitrary linear operator $C(t)$ and move part of the dissipative map into the Hamiltonian and decay terms: $J'_t(\rho)=J_t(\rho)+\frac12(C(t)\rho+\rho C^\dagger(t))$, $H'=H+\frac12 B(t)$, $\Gamma'=\Gamma+A(t)$. If the rate operator built from the redefined map, $R_\psi(t)=J'_t(|\psi\rangle\langle\psi|)$, is positive, then the pure-state trajectories with deterministic generator $K'(t)=H'(t)-\frac i2\Gamma'(t)$ and jumps into the eigenstates of $R_\psi$ at rate equal to the corresponding eigenvalue average precisely to the original master equation. Positivity of $R_\psi$ is not automatic for P-divisible dynamics; the paper proves (Prop. 3, via Lindblad's averaging argument) that if the propagators satisfy the Kadison–Schwarz inequality for all operators—the dissipativity condition (39)—one can represent the generator with a positive $J_t$, hence a positive rate operator and a continuous-measurement interpretation. For qubits, Prop. 1 gives conditions (real Choi matrix plus no relative phase under the deterministic evolution) under which the post-jump eigenstates are fixed, so trajectories live on three deterministic families. The eternal non-Markovian qubit model is used to exhibit qualitatively different unravelings, including one whose post-measurement states are time-independent and one that removes external driving from the deterministic evolution.

Load-bearing premise

The general claim that every dynamics covered by the paper admits a positive rate operator rests on dissipativity—the Kadison–Schwarz inequality applied to all operators, not just Hermitian ones—which is strictly stronger than P-divisibility and already fails for the eternally non-Markovian example, so the paper's existence guarantee is narrower than the P-divisible class where the numerical examples live.

Editorial extensions

If this is right

R-ROQJ extends continuous-measurement interpretations beyond CP-divisible dynamics to any dynamics whose propagators obey Kadison–Schwarz for all operators; within that class every unraveling is a genuine measurement scheme.
For qubits satisfying Prop. 1, simulations need only track three deterministically evolving state families instead of an infinite ensemble, cutting numerical cost and simplifying experimental jump implementation.
A chosen time-independent non-Hermitian Hamiltonian can be imposed as the deterministic generator, and external time-dependent driving can be shifted entirely into the jump part, as shown by $R'_1$ in the driven example.
The same open-system dynamics admits inequivalent continuous-measurement schemes (different instruments, probabilities, and post-jump states); for instance, $R_2$ gives time-independent post-measurement states while $W$ gives time-dependent ones.
The freedom in splitting the generator is distinct from the standard unitary freedom in choosing Lindblad operators, so it provides a new handle for tailoring simulations to specific experimental platforms.

Reading between the lines

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

The paper leaves open whether dissipativity is also necessary for the existence of a positive R-ROQJ; a systematic scan of qubit generators with $\gamma_1,\gamma_2\ge|\gamma_3|$ but $\gamma_{1,2}<2|\gamma_3|$ could map the true positivity region beyond the sufficient condition proved here.
The $b=1$ driven case, where phase jumps are rational multiples of $\pi/4$, suggests using R-ROQJ to engineer discrete phase-space lattices for the ensemble—an application the authors do not pursue.
Because $C(t)$ can be chosen continuously (e.g., the parameter $y(t)$ in Appendix D), one could optimize among unravelings for a given task, such as minimizing total jump rate or maximizing the weight of rare jumps, though the paper does not formulate such an optimization.
The fixed-basis idea might generalize to higher-dimensional 'no-phase' manifolds instead of only qubit $|\phi_\pm\rangle$ states; the paper's proof is explicitly two-dimensional, so an extension would need a new argument.

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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 take: the core construction is sound, and the paper deserves serious referee time, but it gets sloppy in exactly one load-bearing place—the driven example in Sec. 5.2 asserts positivity of R1' without proof. The reader's CONDITIONAL verdict is about right.

What is actually new: the R-ROQJ family exploits the known gauge freedom in splitting a master equation into deterministic and jump parts, but the specific rate-operator formulation (Eqs. (20)–(24)), the fixed-post-jump-state result (Prop. 1), and the trick of absorbing a time-dependent drive into a modified rate operator (Sec. 5.2) are not in the prior ROQJ literature. The proof that any positive rate operator of the form J'(|ψ><ψ|) yields a valid unraveling is clean and self-contained; the ensemble-average computation (Eqs. (25)–(30)) checks out. The paper is also honest about the scope of the general positivity guarantee: dissipativity (Eq. 39) is strictly stronger than P-divisibility, and the eternally non-Markovian example is not dissipative, so positivity there is proven case-by-case in Appendix C. That is the right way to handle it.

Soft spots, in proportion. First, Sec. 5.2's R1'. The paper defines it in Eq. (47) and says the simulation works "provided that the rate operator R1' is positive," with no proof. That is a real gap in a key illustrative example. The stress-test note's direct calculation shows the assertion is actually true for the stated parameters: with γ1=γ2=1, γ3=−a, a = ½ tanh t, one gets eigenvalues 1−a/2 ± ½√[(1+b²)(x²+y²)+a²z²], whose minimum for a≤1/2, b≤1 is 3/4 − √2/2 > 0. So the result survives, but the missing argument should be added. Second, the dissipativity condition limits the general theorem; the paper says so, but casual readers should not conclude every P-divisible dynamics admits a positive R-ROQJ. Third, there is no code or data deposit, so the numerics are not independently reproducible as shipped; given the simple qubit model, that is minor. The citation pattern is fine—[30] and Lindblad's dissipativity are both properly credited.

Who this is for: people doing quantum-trajectory simulations or working on non-Markovian open systems. The simulations agree with the analytics, and the R2 example (three fixed states) is a genuinely nice simplification. I'd send this to peer review, with referees asked to check the Sec. 5.2 positivity and the exact scope of Prop. 3. With that repaired, it is a citable, useful paper.

Referee Report

3 major / 5 minor

Summary. The paper proposes a class of quantum-jump unravelings, called R-ROQJ, based on the freedom to rewrite the dissipative part of a time-local master equation via an arbitrary operator C(t). The central construction is: for any representation whose rate operator R_ψ(t)=J'_t(|ψ(t)⟩⟨ψ(t)|) is positive, the deterministic evolution of Eq. (7), the jumps of Eq. (21), and the probabilities of Eq. (23) define a valid unraveling of the original master equation. The paper proves this averaging identity, gives a general existence condition for positive rate operators under a dissipativity assumption (Prop. 3), and derives a condition for fixed post-jump states in two-level systems (Prop. 1). The theoretical results are illustrated on the eternally non-Markovian qubit master equation, including numerical simulations with several rate operators, and a driven version where a rate operator R1' is designed so that the deterministic evolution is driving-independent.

Significance. If the claims hold, the paper provides a useful and nontrivial extension of quantum-jump methods: it shows how the decomposition of a master equation into deterministic and jump parts can be engineered, which is of practical value for numerical simulation and for the design of continuous-measurement interpretations. The core averaging proof is clean and self-contained, and the explicit positivity proofs in Appendix C for the rate operators R1, R2, and R3 are a concrete asset. The connection to dissipativity is mathematically precise and correctly identifies a sufficient condition for the existence of a positive rate operator. However, the central advertised application to driven systems relies on an unproven positivity assertion for R1' in Sec. 5.2, and the general dissipativity theorem does not cover the eternally non-Markovian example used in the paper; these points limit the completeness of the manuscript as it stands.

major comments (3)

[Sec. 5.2, Eq. (47)] The definition of R1'_ψ(t)=R1_ψ(t)+ i b(t)/2 [σ_z, |ψ(t)⟩⟨ψ(t)|] is followed by the statement that the deterministic evolution can be made independent of the driving 'provided that the rate operator R1' is positive', but no proof of this positivity is given. This is load-bearing: the jump probabilities in Eq. (23) are nonnegative only if R1' is positive, and the simulations in Figs. 3 and 5 therefore depend on an unproved condition. The gap is repairable: for the stated parameters γ1=γ2=1, γ3=-1/2 tanh t, and 0≤b(t)≤1, a direct Bloch-sphere calculation gives eigenvalues 1-a/2 ± 1/2 sqrt[(1+b^2)(x^2+y^2)+a^2 z^2] with a=1/2 tanh t, whose minimum is 3/4 - sqrt(2)/2 > 0. I recommend adding this proof, or a citation to it, in Appendix C or in Sec. 5.2 itself.
[Sec. 4, Prop. 3 and Eq. (39)] The general existence theorem for positive rate operators is conditional on dissipativity, i.e., the Kadison-Schwarz inequality for all operators X, not just Hermitian ones. This is strictly stronger than P-divisibility and is not satisfied by the eternally non-Markovian master equation used in Sec. 5, as the paper itself notes. This is not an error, but it means the general theorem does not cover the main example; the example works because positivity of R1, R2, and R3 is proved directly in Appendix C. I recommend that the text explicitly state, in the introduction and the conclusions, that the dissipativity-based guarantee and the example-based positivity proofs are separate results, so that readers do not infer that the eternally non-Markovian example is covered by Prop. 3.
[Sec. 3.2, Prop. 1] The statement of Prop. 1 would benefit from a more precise formulation of the condition 'the deterministic evolution in-between the jumps does not introduce a relative phase'. As written, Eq. (35) is stated as an implication involving a(t,dt), but it is not tied to a concrete condition on the operators H'(t) and Γ'(t) or on the propagator D(t,s). The proof in Appendix A makes the intended restriction clear (states must remain of the form ρ12=ρ21 in the chosen basis), but the proposition itself would be easier to verify if condition (35) were replaced by an explicit algebraic condition on D(t,s), such as preservation of the subspace of real-coefficient states.

minor comments (5)

[Throughout] The paper uses both 'unraveling' and 'unravelling'; please standardize the spelling.
[Fig. 5 caption] There is a typo in 'paneld c)'; it should read 'panel (c)'.
[Sec. 6, Conclusions] The word 'determinisitic' should be 'deterministic'.
[Sec. 5.2] In the sentence after Eq. (47), 'provided that the rate operator R1' is positive' should include a pointer to the proof, once it is added, since the positivity is not self-evident.
[Abstract] The phrase 'ease their numerical simulations' would read better as 'simplify their numerical simulations'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: the R-ROQJ averaging theorem and Proposition 3 are forward derivations from stated assumptions.

full rationale

The central construction in Section 3 is not circular: the paper defines R_psi(t)=J_t(|psi><psi|), the jump operators in Eq. (22), and the probabilities in Eq. (23), then explicitly computes the conditioned average in Eq. (30), showing that it equals |psi><psi| - i[H,|psi><psi|]dt - (1/2){Gamma(t),|psi><psi|}dt + J_t(|psi><psi|)dt, which is exactly the first-order expansion of the master-equation generator in Eq. (1). No parameter is fitted, and the target master equation is not assumed in the definition of the trajectories; positivity of R_psi is an input condition, and the averaging theorem is proved directly from the definitions. Proposition 3 is likewise constructive: Appendix B starts from the dissipativity condition in Eq. (39), defines K(t) by Eq. (B.1), constructs J^dagger_t(X)=L^dagger_t(X)-(K(t)X+XK^dagger(t)) in Eq. (B.7), averages the Kadison-Schwarz inequality over U(N), and concludes J^dagger_t(X^dagger X)>=0, hence the constructed J_t is positive. This is a derivation from Lindblad's condition, not an importation of the conclusion. The examples in Section 5 are consistency checks rather than fitted predictions: the rate operators R1, R2, and R3 are chosen with positivity proved in Appendix C, and the simulations are verified against the analytical master equation, which is the input of the construction. The paper does cite previous work by overlapping authors (e.g., Refs. [30,40] for the W-ROQJ and its P-divisibility characterization), but those citations are used for background and for the side statement in Proposition 2, not for the central R-ROQJ averaging theorem or for Proposition 3, so they are not load-bearing circularity. The genuine gap is in Section 5.2: the paper says the driving can be absorbed 'provided that the rate operator R1' is positive' (Eq. (47)), but no proof of that positivity is given; for the stated parameters the claim is true but unproved in the paper. That is a completeness gap, not a circular reduction, and it does not change the circularity score.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim introduces a new mathematical object, the rate operator R_ψ, but no new physical entities. The only genuine free parameter in the construction is the gauge operator C(t), represented by y(t) in the qubit example; this is a freedom, not a fit. The example model parameters (γ1=γ2=1, γ3=-tanh t, μ=1, σ=0.25) are illustrative choices and do not affect the central theorem.

free parameters (1)

y(t) = 0 in plotted example; allowed range 0 ≤ y(t) ≤ γ1(t)+0.5γ2(t)-0.5γ3(t)
Defines a continuous family of fixed-post-jump unravelings (Appendix D). Not fitted to data; chosen by hand within a positivity-preserving range.

assumptions (4)

domain assumption Finite-dimensional Hilbert space (N-dimensional open system).
Used throughout; the spectral resolutions and matrix representations rely on finite dimension.
domain assumption Master equation in time-local form (1) with real coefficients c_α(t).
Assumed to hold for the open system dynamics; this is the standard structure for trace- and Hermiticity-preserving generators.
domain assumption Dissipativity condition (39) for all X in B(H_S).
This is the key premise for the existence of a positive J_t in Prop. 3; it is strictly stronger than P-divisibility.
domain assumption Existence of a basis and operator C(t) satisfying the reality condition (32) and the no-phase condition (35).
These conditions are assumed for the qubit fixed-post-jump construction (Prop. 1).

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Cite this review

Pith. "Pith review of How to design quantum-jump trajectories via distinct master equation representations." pith.science (2026). https://pith.science/paper/YHQ3MQ46

@misc{pith2026200911312,
  author       = {Pith},
  title        = {Pith review of: How to design quantum-jump trajectories via distinct master equation representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YHQ3MQ46}},
  note         = {Machine review of arXiv:2009.11312}
}

read the original abstract

Every open-system dynamics can be associated to infinitely many stochastic pictures, called unravelings, which have proved to be extremely useful in several contexts, both from the conceptual and the practical point of view. Here, focusing on quantum-jump unravelings, we demonstrate that there exists inherent freedom in how to assign the terms of the underlying master equation to the deterministic and jump parts of the stochastic description, which leads to a number of qualitatively different unravelings. As relevant examples, we show that a fixed basis of post-jump states can be selected under some definite conditions, or that the deterministic evolution can be set by a chosen time-independent non-Hermitian Hamiltonian, even in the presence of external driving. Our approach relies on the definition of rate operators, whose positivity equips each unraveling with a continuous-measurement scheme and is related to a long known but so far not widely used property to classify quantum dynamics, known as dissipativity. Starting from formal mathematical concepts, our results allow us to get fundamental insights into open quantum system dynamics and to enrich their numerical simulations.

Figures

Figures reproduced from arXiv: 2009.11312 by the authors.

**Figure 1.** Simulation results using rate operator R1 [(a), (b)] and R2 [(c), (d)] for model (41) without driving. Example trajectories are displayed in (a) and (c). Final steady state distributions for Bloch vector X and Z components are shown in (b) and (d). The insets in (b) and (d) demonstrate the agreement between the analytical results (solid line) and simulation results (dots) with 104 trajectories. The error bars are si… view at source ↗

**Figure 2.** Simulation results using rate operator R3 [(a), (b)] and W [(c), (d)]. Seven example pure state trajectories are displayed in (a) and (c) by plotting the probability of state |1i. Final steady state distributions for Bloch vector X and Z components, with 500 trajectories, are shown in (b) and (d). Here, each dot corresponds to a single trajectory. In (d), all the 500 dots have the value Z = 0 and X = −1/2 or X = +1/… view at source ↗

**Figure 4.** The phase evolution (upper panel) and phase [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: Time independent driving for b(t) = 1. In panel a) we plot the dynamics using rate operator R1 and the driving term is clearly visible in the real part of the coherences. In panel b) we verify in the inset the validity with the simulations and plot the distribution of …

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[49]

Now, sinceAandB arepositive, themap Jt ispositiveandhence R1ψ(t)≥0

conversely, ifγ2≥γ1, then (C.8) is equivalent toγ1 +γ3≥0, whichendstheproofofpositivityof A. Now, sinceAandB arepositive, themap Jt ispositiveandhence R1ψ(t)≥0. Positivity of R2ψ(t) is evident due to the relation R2ψ(t) = R1ψ(t)−γ3(t)|ψ(t)⟩⟨ψ(t)|, (C.9) and γ3(t)< 0. Finally, ...

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

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