Different monitoring schemes applied to the same Markovian open quantum system generate inequivalent stochastic quantum trajectories — distinct "unravelings" of one Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation — while every linear functional of the trajectories is fixed by the unconditional density matrix. Piñol et al. (Phys. Rev. Research 6, L032057 (2024)) showed that a nonlinear functional, the quantum-trajectory-averaged variance (QTAV) of the inversion in resonance fluorescence, differs between direct-photodetection (Poisson) and homodyne/heterodyne (Wiener) unravelings, and proposed it as an operational discriminator, computable through a truncated adjoint-Lindbladian moment hierarchy. The open problem is whether this proposed operational discrimination survives a published challenge to its accessibility: Gaona-Reyes, Altamura and Bassi (Phys. Rev. Research 7, 043295 (2025)) argue that nonlinear trajectory averages cannot be accessed without prior knowledge of the measurement scheme, and that assumed accessibility would enable superluminal signaling, rendering any operational protocol to distinguish unravelings fundamentally unfeasible — a conclusion that the original program has so far neither conceded nor rebutted in print, bypassing it only in a digital setting (arXiv:2601.19889) where trajectory labels are available by construction. The quantitative underpinnings of the proposal remain correspondingly incomplete: its moment hierarchy does not close and is truncated without convergence guarantees, Wiener-type QTAV values are known only numerically, and robustness at realistic detector efficiency and thermal occupation is documented only through illustrative simulations. Settling the accessibility dispute determines whether nonlinear trajectory statistics are experimentally meaningful physical quantities or artifacts of a chosen measurement model.

Background

An open quantum system — a system coupled to an environment — is commonly described, under Markovian assumptions, by the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) master equation for its density matrix: a deterministic linear evolution generated by a Hamiltonian and by so-called Lindblad or jump operators that encode the dissipative channels. When the environment is continuously monitored (e.g., the light emitted by an atom is detected), the system state conditioned on the measurement record becomes a stochastic process, a quantum trajectory; the ensemble of such trajectories, whose average reproduces the master equation, is called an unraveling of that master equation. One and the same GKSL master equation admits infinitely many unravelings, corresponding to different monitoring schemes. In resonance fluorescence — a coherently driven two-level atom with Rabi frequency Ω decaying at rate γ — the two paradigmatic cases are the Poisson-type unraveling induced by direct photodetection of the scattered photons (piecewise-deterministic evolution interrupted by quantum jumps) and the Wiener-type unraveling induced by homodyne or heterodyne detection of the emitted field (homodyne and heterodyne detection measure one, respectively both, quadratures of the emitted field, and the conditioned state undergoes diffusive stochastic evolution).

By construction, the average of any quantity that is linear in the conditional state (such as the mean inversion ⟨σ_z(t)⟩) is identical for all unravelings, since it is fixed by the unconditional density matrix. Piñol et al. (Phys. Rev. Research 6, L032057 (2024)) showed that this is no longer true for nonlinear functionals of the trajectories. Their central quantity is the quantum-trajectory-averaged variance (QTAV) of the inversion, ⟨σz(t)⟩2‾−⟨σz(t)⟩‾2\overline{\langle\sigma_z(t)\rangle^2} - \overline{\langle\sigma_z(t)\rangle}^2, where the overbar denotes the ensemble average over trajectories (over realizations of the measurement record). For direct photodetection they obtained an exact expression from the waiting-time distribution between photon emissions and a long-time asymptotic formula (their Eq. (3)), exhibiting damped oscillations at frequency 4Ω; for homodyne and heterodyne detection the QTAV shows dephasing behavior that they compute only numerically — by Monte-Carlo trajectory simulation and by a moment-hierarchy method. That method evolves the ensemble moments of products of conditional expectations (e.g., Eω[xixjxk]E_\omega[x_i x_j x_k], where xi=⟨Xi⟩tx_i=\langle X_i\rangle_t are conditional expectations of basis observables and ω indexes the noise) through a linear system y˙=My\dot{y}=My built from the adjoint Lindbladian (the Heisenberg-picture generator acting on observables rather than on the state) and the Itô calculus of stochastic differentials; because the equation for each moment couples to higher moments, the hierarchy does not close and must be truncated (they use tenth order), with no convergence theory or error bounds.

The paper also analyzes realistic imperfections: with detector efficiency η < 1 and thermal occupation n̄ of the bath, the conditional state is no longer pure and the QTAV degrades rapidly (Monte-Carlo simulations indicate values of order 10⁻³ for η ≲ 0.05). A measurement protocol is sketched in which two copies of the system are prepared identically and trajectory pairs are post-selected on having the same photocounting record, so that the trajectory-to-trajectory variance can be estimated from pairs of measurements on identically conditioned copies; the accompanying experimental analysis of dark counts, signal-to-noise scaling, and efficiency budgets is, however, developed only for the linear intensity-correlation observable g⁽²⁾ (the normalized second-order photon-correlation function), leaving the nonlinear case unspecified.

Two later developments define the current frontier. First, Keys and Wehr (arXiv:2402.05352) generalized the framework to arbitrary finite-dimensional systems and arbitrary Lindblad operators, computing the Itô corrections for variance-type and entropy-type nonlinear functionals of both Poisson and Wiener unravelings. Second, Gaona-Reyes, Altamura and Bassi (Phys. Rev. Research 7, 043295 (2025)) challenged the operational reading itself: nonlinear unraveling-dependent quantities can only be computed by filtering the measurement record through the stochastic master equation (the evolution equation of the measurement-conditioned state) of the very monitoring scheme one seeks to identify, and if they were accessible without prior knowledge of that scheme, a Bell-type argument shows that superluminal signaling would be possible; they conclude that any operational protocol to distinguish unravelings is fundamentally unfeasible. In response to the experimental demands of the optical setting, Piñol et al. (arXiv:2601.19889 (2026)) implemented synthetic unravelings — a projective-measurement scheme and a random-unitary "kick" scheme with identical ensemble-averaged evolution — on one and two IBM superconducting qubits, where the measurement records (trajectory labels) are available by construction, and distinguished them through trajectory-resolved variances and trajectory-averaged entropies. Whether such label-based digital demonstrations answer or merely bypass the inaccessibility argument is not settled in the literature. Related trajectory-level physics — the dependence of measurement-induced entanglement transitions on the chosen unraveling, and precision bounds for trajectory observables — is an active adjacent field, but none of it settles the accessibility dispute at the center of this problem.

Problem Statement

Decide whether different unravelings of one and the same GKSL master equation — in the setting instantiated by resonance fluorescence of a driven two-level emitter (Rabi frequency Ω, decay rate γ, jump operator σ₋) — can be operationally distinguished by nonlinear averages of quantum trajectories, such as the quantum-trajectory-averaged variance QTAV(O)=Eω[⟨O⟩t2]−(Eω[⟨O⟩t])2\mathrm{QTAV}(O)=E_\omega[\langle O\rangle_t^2]-(E_\omega[\langle O\rangle_t])^2, where ω indexes the measurement record.

Concretely, the task is to resolve the published dispute over this question. Piñol et al. (Phys. Rev. Research 6, L032057 (2024)) propose to distinguish unravelings by the QTAV of the inversion, which differs between direct-photodetection (Poisson) and homodyne/heterodyne (Wiener) unravelings while every linear functional of the trajectories is fixed by the unconditional density matrix, and they sketch a measurement scheme in which two copies of the system are prepared identically and trajectory pairs are post-selected on having the same photocounting record. Gaona-Reyes, Altamura and Bassi (Phys. Rev. Research 7, 043295 (2025)) counter that such nonlinear quantities can only be computed by filtering the measurement record through the stochastic master equation of the very monitoring scheme one seeks to identify, that they therefore cannot serve as an independent discriminator, and that assuming access to them without prior knowledge of the unraveling would enable superluminal signaling.

A resolution is either:

  • a rigorous no-go argument formalizing the position of Gaona-Reyes, Altamura and Bassi, covering protocols that lack prior knowledge of the monitoring scheme; or
  • a concrete protocol — which may build on the two-copy/post-selection scheme of Piñol et al. or on a digital setting in which trajectory labels are available by construction — that evades the stated assumptions of that argument and demonstrates discrimination among a stated hypothesis class of monitoring schemes with a stated statistical decision procedure.

The scope of the answer must be explicit: a no-go must state the class of protocols it excludes, and a positive protocol must state the hypothesis class of monitoring schemes it discriminates among and whether prior knowledge of the scheme is assumed, since discrimination of an unknown monitoring scheme and verification of an implemented, known scheme against alternatives are different tasks with different accessibility assumptions. A positive protocol must also make explicit how it avoids the Bell-type superluminal-signaling consequence without violating relativistic causality.

The quantitative state of the art frames, but does not replace, this question: the adjoint-Lindbladian moment hierarchy used to compute the relevant averages does not close and must be truncated without convergence guarantees or error bounds, Wiener-type QTAV values are known only numerically, and the signal degrades rapidly under realistic detector efficiency η and thermal occupation n̄. Any concrete protocol must confront these limitations, and extensions of the diagnostics beyond a single emitter — where a two-qubit digital demonstration exists — remain open directions noted in the source literature rather than required deliverables here.

References

  1. E. Piñol, Th. K. Mavrogordatos, D. Keys, R. Veyron, P. Sierant, M. A. García-March, S. Grandi, M. W. Mitchell, J. Wehr, and M. Lewenstein, “Telling different unravelings apart via nonlinear quantum-trajectory averages”, Physical Review Research 6, L032057 (2024). DOI: 10.1103/PhysRevResearch.6.L032057; arXiv:2312.03452; https://arxiv.org/abs/2312.03452
  2. D. Keys and J. Wehr, “Nonlinear functionals of master equation unravelings”, arXiv:2402.05352 (2024). https://arxiv.org/abs/2402.05352
  3. J. L. Gaona-Reyes, D. G. A. Altamura, and A. Bassi, “Theoretical limits of protocols for distinguishing different unravelings”, Physical Review Research 7, 043295 (2025). DOI: 10.1103/6qnt-t3wl; arXiv:2502.19268; https://arxiv.org/abs/2502.19268
  4. E. Piñol, P. Sierant, D. Keys, R. Veyron, M. A. García-March, T. Reese, M. W. Mitchell, J. Wehr, and M. Lewenstein, “Distinguishing synthetic unravelings on quantum computers”, arXiv:2601.19889 (2026). https://arxiv.org/abs/2601.19889
  5. J. Richter, M. Haque, and L. Sá, “Localization and Delocalization of Quantum Trajectories in the Liouvillian Spectrum”, arXiv:2511.19700 (2025). https://arxiv.org/abs/2511.19700
  6. J. M. Ryan, S. Gorbaty, S. W. Teitsworth, and C. Noel, “Quantum-classical crossover in noisy monitored oscillators”, arXiv:2608.08267 (2026). https://arxiv.org/abs/2608.08267
  7. T. Lamich, L. Zarraoa, S. Elsehimy, M. W. Mitchell, and R. Veyron, “A single atom emitting resonance fluorescence into a coherent beam, and its use for non-destructive atom thermometry”, arXiv:2607.29515 (2026). https://arxiv.org/abs/2607.29515
  8. R. Nongthombam and A. K. Sarma, “Homodyne measurement of a non-Hermitian qubit undergoing spontaneous emission”, Physical Review A 113, 063701 (2026). DOI: 10.1103/gv74-cktg; arXiv:2510.13345; https://arxiv.org/abs/2510.13345
  9. R. Nongthombam and A. K. Sarma, “Entanglement dynamics in a two transmon qubit system under continuous measurement and postselection”, arXiv:2604.25311 (2026). https://arxiv.org/abs/2604.25311
  10. M. Eissler, I. Lesanovsky, and F. Carollo, “Unraveling-induced entanglement phase transition in diffusive trajectories of continuously monitored noninteracting fermionic systems”, Physical Review A 111, 022205 (2025). DOI: 10.1103/PhysRevA.111.022205; arXiv:2406.04869; https://arxiv.org/abs/2406.04869
  11. T. Van Vu, “Fundamental bounds on precision and response for quantum trajectory observables”, PRX Quantum 6, 010343 (2025). DOI: 10.1103/PRXQuantum.6.010343; https://doi.org/10.1103/PRXQuantum.6.010343