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arxiv: 2606.31321 · v1 · pith:QYHBCX5Mnew · submitted 2026-06-30 · 🪐 quant-ph · cs.SY· eess.SY· math-ph· math.MP· math.OC

Projection Operator Stochastic Equations for Non-Markovian Quantum Systems Under Continuous Measurement-Based Feedback

H. I. Nurdin This is my paper

Pith reviewed 2026-07-01 05:36 UTC · model grok-4.3

classification 🪐 quant-ph cs.SYeess.SYmath-phmath.MPmath.OC
keywords non-Markovianprojection operatorstochastic equationsquantum feedbackcontinuous measurementhomodynephoton countingquantum optics
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The pith

For non-Markovian quantum systems with measurement feedback, the projection operator stochastic equations keep their form but gain stochastic terms from the measurement record.

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

This paper generalizes the projection operator technique to non-Markovian quantum systems under continuous measurement-based feedback. It derives stochastic differential equations for the projected state that match the structure of the open-loop case. The key difference is that terms previously treated as deterministic now depend stochastically on the measurement outcomes. This covers generalized measurements such as homodyne detection and photon counting. The result matters because it provides a way to model feedback control in systems that violate the Markov assumption common in quantum optics and related fields.

Core claim

The paper establishes that the stochastic equations obtained for the projected state take the same form as the open-loop case, but previously deterministic terms become stochastic ones which depend on the measurement record. This generalization applies when feedback is based on the continuous-measurement record for a class of measurements including continuous possibly adaptive homodyne and photon counting.

What carries the argument

Projection operator applied to the Markovian embedding of the non-Markovian system, extended to include stochastic feedback from the measurement record.

If this is right

  • The equations for the projected state maintain the same structural form under feedback.
  • Deterministic terms in the open-loop equations become stochastic and measurement-dependent.
  • The method works for adaptive homodyne and photon counting measurements.
  • It enables analysis of feedback control in non-Markovian settings via the embedding.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Feedback control laws for non-Markovian systems can be designed using the same projection framework as open-loop.
  • This may simplify numerical simulations by allowing use of Markovian tools with adjusted stochastic inputs.
  • Extensions could include deriving optimal feedback strategies based on these equations.

Load-bearing premise

A non-Markovian quantum system can be embedded into a larger Markovian quantum system in a way that allows the projection operator method to apply even after feedback is added via the measurement record.

What would settle it

Comparison of the derived stochastic equations against direct simulation of the embedded Markovian system with feedback applied to a specific non-Markovian model, checking if the projected dynamics match.

Figures

Figures reproduced from arXiv: 2606.31321 by H. I. Nurdin.

Figure 1
Figure 1. Figure 1: Feedback connection of the ρ p sa,· and ρ q sa,· , with ρ q sa,· acting as the state of a memory subsystem (denoted by the dotted box). This opens an approach to approximate the memory term, a problem posed in [12], by developing a suitable method to approximate z. 6 Conclusion and future work In this paper, we have generalized the non-Markovian stochastic SDEs for the evolution of a non-Markovian quantum … view at source ↗
read the original abstract

Quantum Markov models have been successfully used to accurately model various physical quantum systems in fields such as quantum optics, optomechanics and superconducting circuits and they provide the basis for (measurement-based) quantum feedback control. However, the quantum Markov assumption is a strong one and it is not expected to hold for general quantum systems of interest. The projection operator approach is one approach that has been developed to model non-Markovian quantum systems by considering its embedding in a larger Markovian quantum system, but mainly in the context of quantum master equations for the dynamics of the unmonitored reduced quantum state of a quantum system. This approach was recently adapted for continuously measured non-Markovian quantum systems, which enables open-loop control but did not yet consider the presence of feedback of the stochastic measurement record, deriving non-Markovian SDEs for the evolution of the projected state of the Markovian embedding. This paper generalizes these stochastic equations to the setting of stochastic feedback based on the continuous-measurement record and shows that the equations take the same form but that previously deterministic terms become stochastic ones which depend on the measurement record, as would be intuitively expected. The stochastic equations are obtained for a generalized class of measurements that includes continuous (possibly adaptive) homodyne and photon counting measurements.

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.

Referee Report

2 major / 2 minor

Summary. The paper generalizes the projection operator technique, previously applied to open-loop continuously measured non-Markovian quantum systems, to the case of stochastic feedback control based on the continuous measurement record. It derives stochastic differential equations for the projected state of the Markovian embedding and asserts that these retain the same structural form as the open-loop equations, with previously deterministic coefficients replaced by stochastic, record-dependent terms. The result is stated for a generalized class of measurements that includes continuous (possibly adaptive) homodyne detection and photon counting.

Significance. If the central derivation is valid, the result would allow projection methods to be used for closed-loop non-Markovian models, extending their utility in quantum optics, optomechanics, and superconducting-circuit feedback control. The manuscript does not report machine-checked proofs, reproducible code, or parameter-free derivations.

major comments (2)
  1. [Section 3 (derivation of the stochastic equations)] The central claim that the projected SDEs retain the same form under feedback rests on the assumption that the projection superoperator commutes with the feedback-modified Liouvillian in the same manner as with the open-loop case. No explicit commutation relations or additional averaging conditions for arbitrary adaptive measurements are stated; this step is load-bearing for the generalization and requires a concrete derivation or counter-example check.
  2. [Section 4 (generalized measurements)] The abstract and introduction claim the result holds for a generalized class of measurements, yet the manuscript supplies no error analysis, numerical verification against the full Markovian embedding, or test cases with adaptive feedback that would confirm the projection remains valid when the control Hamiltonian depends on the stochastic record.
minor comments (2)
  1. [Section 2] Notation for the stochastic measurement record and its dependence in the feedback terms should be introduced earlier and used consistently throughout the derivations.
  2. [Introduction] The manuscript would benefit from an explicit statement of the precise conditions under which the projection technique was previously validated for open-loop cases, to allow direct comparison with the feedback extension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. Below we respond point-by-point to the major comments. We agree that additional clarification on commutation is warranted and will revise accordingly, while maintaining that the formal derivation stands for the stated class of measurements.

read point-by-point responses

  1. Referee: [Section 3 (derivation of the stochastic equations)] The central claim that the projected SDEs retain the same form under feedback rests on the assumption that the projection superoperator commutes with the feedback-modified Liouvillian in the same manner as with the open-loop case. No explicit commutation relations or additional averaging conditions for arbitrary adaptive measurements are stated; this step is load-bearing for the generalization and requires a concrete derivation or counter-example check.

    Authors: We agree that the commutation step merits explicit treatment in the feedback setting. The projection superoperator acts on the system degrees of freedom while the feedback-modified Liouvillian includes a stochastic control term that is a function of the measurement record; because this term remains a bounded operator on the full embedding space and does not mix system and environment subspaces, the same commutation relations derived in the open-loop case continue to hold. In the revised manuscript we will insert a short subsection that states these relations explicitly for both homodyne and photon-counting feedback and notes the absence of additional averaging conditions beyond those already required for the open-loop projection. revision: yes

  2. Referee: [Section 4 (generalized measurements)] The abstract and introduction claim the result holds for a generalized class of measurements, yet the manuscript supplies no error analysis, numerical verification against the full Markovian embedding, or test cases with adaptive feedback that would confirm the projection remains valid when the control Hamiltonian depends on the stochastic record.

    Authors: The manuscript presents a formal derivation rather than a numerical study. The generalized measurement class is defined precisely so that the stochastic feedback enters only through the already-projected record-dependent coefficients; the projection validity therefore follows from the same embedding construction used in the open-loop case. We acknowledge that explicit error bounds or adaptive-feedback simulations would be useful for applications, but they lie outside the scope of the present theoretical generalization. In revision we will add a brief paragraph in Section 4 clarifying this scope and referencing existing numerical methods for the underlying Markovian embedding. revision: partial

Circularity Check

0 steps flagged

No significant circularity; generalization of prior projection technique stands on independent assumptions

full rationale

The paper extends an existing projection-operator method for non-Markovian systems under continuous measurement to the feedback setting, asserting that the resulting SDEs retain the same structural form with deterministic coefficients replaced by record-dependent stochastic ones. No quoted derivation step reduces the claimed result to a fitted parameter, a self-definitional identity, or a load-bearing self-citation whose content is itself unverified; the central step is the (unproven here) assumption that the projection superoperator continues to commute appropriately once the control Hamiltonian depends on the stochastic record. This is an external modeling assumption rather than a circular reduction, consistent with the reader's assessment of score 2 and with the absence of any renaming, ansatz smuggling, or uniqueness theorem imported from the same authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the projection-operator embedding of a non-Markovian system into a Markovian one; no free parameters, invented entities, or additional axioms are visible from the abstract alone.

axioms (1)
  • domain assumption A non-Markovian quantum system can be embedded in a larger Markovian quantum system so that the projection operator recovers the reduced dynamics under continuous measurement and feedback.
    This embedding is the foundational premise of the projection-operator approach referenced throughout the abstract.

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Reference graph

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