{"id":"1615ea54-2625-4034-aa43-45bf66db5ffa","arxiv_id":"2603.22686","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A deterministic master equation for non-Markovian quantum feedback follows from rewriting finite-memory signal rules as higher-dimensional Markovian vector signals (Eq. 2), with momentum and T-step embeddings worked out.","lead":"This paper derives a deterministic master equation for quantum feedback control when the control signal depends on the history of past measurements, not only the latest one. Memory is encoded in extra bookkeeping variables, so that non-Markovian feedback becomes a higher-dimensional but memoryless problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Continuum-limit recovery of Refs. [22,29] is asserted, and Eq. (12) is inconsistent with Eq. (11) by a factor γ, leaving a key advertised application unverified.","rationale":"The reader correctly identified the finite-T, model-dependent scope as the main limitation, and I agree that Eq. (2) is an exact rewriting for known deterministic finite-memory feedback. I do not see a flaw in the central embedding: Eq. (13) is a standard shift-register construction and the derivation in Appendix A is sound modulo a typo in the dummy index. However, the paper makes an additional load-bearing claim, namely that the framework recovers the filtered quantum Fokker-Planck master equations [22,29] in the Gaussian/linear continuum limit. That claim is not derived, and the one explicit continuum expression, Eq. (12), is inconsistent with Eq. (11) by a factor γ. Since this recovery is a major advertised application and is partly self-cited, it deserves a concrete check before being accepted. The finite-T overstatement is real but already acknowledged by the authors, so it does not change the conditional verdict. The factor-γ discrepancy may be a typo, but as written it prevents verification of the recovery claim. Hence I recommend keeping the reader's CONDITIONAL verdict unchanged.","tokens_in":6737,"tokens_out":12311,"duration_ms":117620,"concrete_test":"Recompute the continuum limit of Eq. (10) for a linear feedback rule, e.g. g_{n+1}(x,s)=κ x_{n+1}, taking δt→0 and β=1-γδt. Derive s(t) from the s_{n+1}=s_n+m_{n+1} update and compare the resulting integral kernel with Eq. (12). Then compare the prefactor and kernel shape with the corresponding continuum kernel in Ref. [29]. If the extra γ disappears under correct integration, Eq. (12) is a typo and the recovery claim may still hold; if it persists, Sec. III.B's recovery statement is quantitatively incorrect.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The core rewriting in Eq. (2) is exact, and the finite-T Markovian embedding in Eq. (13) is a correct bookkeeping construction. The load-bearing weakness is the paper's advertised connection to existing filtered master equations (Sec. III.B last paragraph and Sec. IV). This recovery is asserted without derivation, and the only explicit continuum calculation is internally inconsistent: substituting Eq. (11) into s(t)=∫m ds gives s(t)=∫ ds (1-e^{-γ(t-s)})g(x_s), not Eq. (12)'s γ∫ ds (1-e^{-γ(t-s)})g(x_s). The extra prefactor γ changes the kernel's low-frequency gain, so the claimed quantitative match to Refs. [22,29] cannot be checked as written. This is not an attack on Eq. (2) itself, but it undermines the paper's concrete claim that the new non-Markovian formulation recovers general filtering results in the continuum limit.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a deterministic master equation for quantum feedback with non-Markovian signal processing, Eq. (2). The main idea is to promote the scalar feedback signal to a high-dimensional vector y that stores the relevant past via auxiliary variables, so that a feedback rule depending on T past signals (Eq. (1)) becomes a Markovian update in the enlarged space. The derivation in Appendix A follows the delta-function method of Ref. [23]. The paper then gives two explicit embeddings: a 'momentum' rule (Eqs. (8)-(10)) and a general T-step shift register (Eq. (13)). It claims that, for Gaussian Kraus operators and linear feedback maps, the continuum limit recovers the filtered quantum Fokker-Planck master equations of Refs. [22,29].","tokens_in":6962,"tokens_out":3302,"duration_ms":34234,"significance":"The core formal identity in Eq. (2) is an exact rewriting of the conditioned dynamics once a finite-dimensional signal map is fixed; it is close to a tautology but provides a useful bookkeeping device. The explicit T-step Markovian embedding in Eq. (13) is a concrete and correct construction for finite-memory signal processing, and the momentum example connects to a known optimization heuristic. These are useful contributions. However, the advertised continuum-limit recovery of existing filtered master equations is asserted rather than demonstrated, and the only explicit continuum calculation contains a factor-γ error. If the continuum connections were supplied, the paper would substantiate its main practical claim; as it stands, the contribution is a framework plus embeddings, with an unverified key application.","major_comments":[{"comment":"Equation (12) does not follow from Eq. (11). Substituting m(t)=γ∫ds e^{-γ(t-s)}g(x_s) into s(t)=∫ds m(s) gives s(t)=∫ds (1-e^{-γ(t-s)})g(x_s), without the overall prefactor γ. The extra γ changes the low-frequency gain of the kernel and therefore the quantitative behavior of the feedback filter. Since this calculation is the only explicit continuum limit shown, the claimed recovery of Refs. [22,29] is not verifiable as written.","section":"Sec. III.A, Eqs. (11)-(12)"},{"comment":"The statement that 'in the limit δt→0, one can use Gaussian Kraus operators... and linear transformation for g_n to recover the results in Refs. [22,29]' is asserted without derivation. No explicit mapping from the finite-dimensional discrete embedding (Eq. (13)) to the continuum filtered master equation is provided, and the only continuum example (Sec. III.A) has the factor-γ error above. This is a load-bearing advertised application; the authors should either supply the derivation or temper the claim.","section":"Sec. III.B, last paragraph"},{"comment":"The paper states that Eq. (2) 'can reproduce all previous results in the field for appropriate choices of feedback rule and quantum instruments.' This is broad, but the paper itself acknowledges that the construction is model-dependent (Sec. IV). The claim is acceptable if the intended scope is finite-memory deterministic feedback rules. However, the title's 'non-Markovian signal processing' should be qualified as 'finite-memory deterministic feedback,' since unbounded memory or unknown functional forms cannot be embedded by this construction.","section":"Sec. IV, first paragraph"}],"minor_comments":[{"comment":"There are notation slips: in the first line after applying E_{n+1|1:n}, the argument of f_{n+1} should be x' rather than x_n, and in the dummy-variable step the second delta should be δ_{y, f_{n+1}(x',y')} (with x' consistent). The derivation is correct in substance, but these typos should be fixed.","section":"Appendix A, Eq. (A4)"},{"comment":"The notation m_{n+1}=β m_n + (1-β)g_{n+1}(x_{n+1},s_n) is fine, but the text preceding Eq. (11) says 'we can write Eq. (10) as' and then displays two equations; it may help to label them (11a) and (11b) for clarity.","section":"Eq. (10)"},{"comment":"The construction assumes the feedback function g_{n+1} is known and deterministic. This is stated in Eq. (1), but the paper could emphasize that the auxiliary variables are exact bookkeeping only when this assumption holds, not for stochastic or unknown signal processing.","section":"Sec. III.B, Eq. (13)"}],"recommendation":"major_revision","confidential_remarks":"The paper's central identity (Eq. (2)) is correct as a reformulation, and the embedding recipes are useful. The main weakness is the unverified continuum-limit recovery, compounded by the explicit γ error in Eq. (12). I would encourage the authors to provide a careful continuum derivation or to explicitly restrict their claims to the discrete-time framework. The paper is within scope for a quantum-information journal, but it is not yet ready for acceptance in its present form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper does what it says in a narrow sense — it writes down a deterministic master equation for feedback that depends on finitely many past signals, by embedding the signal history into a higher-dimensional vector. The main equation, Eq. (2), is an exact rewriting of the conditionally evolved ensemble state once the signal map is fixed. If you accept that starting point, the result is closed and deterministic. The genuinely new piece compared with Rosal et al. is the explicit embedding recipes: the momentum rule in Eq. (10) and the general T-step shift register in Eq. (13). These are spelled out, checkable, and immediately usable; the (T+1)-dimensionality statement is clearly made. Credit where due: the construction is transparent, the model-dependence is acknowledged in Sec. IV, and the appendix follows the known delta-function derivation cleanly apart from a minor index typo in Eq. (A4).\n\nThe soft spots are real but not fatal. First, Eq. (12) does not follow from Eq. (11). Integrating m(t) as given gives s(t) = ∫ ds (1 − e^{−γ(t−s)}) g(x_s), without the leading γ. The extra γ changes the DC gain of the equivalent filter, so the advertised quantitative connection to Refs. [22, 29] cannot be verified as written. This looks like a simple error in the text, but because that recovery is the paper's concrete demonstration of practical relevance, it has to be fixed before the claim is usable. Second, the recoveries themselves are asserted rather than derived. The last paragraph of Sec. III.B says Gaussian Kraus operators plus linear f reproduce Refs. [22, 29]; no calculation is shown. Given that one of those references is the first author's own, an independent derivation or at least a sketch is needed. Third, the title's 'general, possibly non-Markovian' overstates the demonstrated scope: the construction requires a known deterministic feedback rule with finite memory T. That is a meaningful but restricted class, and the paper does state this caveat in Sec. IV.\n\nNone of this undermines Eq. (2) itself. The paper is a formulation, close to true by construction, and its value is in the explicit embeddings and the dimensionality trade-off. For people doing feedback control with finite-bandwidth electronics, delayed feedback, or digital filtering, this is a handy formal tool.\n\nMy recommendation: send it to peer review. A referee should ask for the Eq. (12) fix and for the recovery calculation to be made explicit. After those revisions, it's a solid contribution.","headline":"A useful formal extension of deterministic feedback master equations to finite-memory signal processing; the core rewriting is sound, but the continuum-limit example has a factor-γ inconsistency and the recoveries are asserted.","tokens_in":7526,"tokens_out":2512,"would_cite":true,"duration_ms":21986,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that any feedback rule depending on finitely many past signals can be rewritten exactly as a Markovian update on an enlarged signal vector, yielding a deterministic quantum master equation for non-Markovian feedback.","keywords":["quantum feedback control","non-Markovian feedback","deterministic master equation","Markovian embedding","signal processing","measurement-based feedback","memory effects","quantum trajectories"],"falsifier":"A concrete check: pick a qubit, a two-step memory feedback rule such as s_{n+1} = s_n + s_{n-1} + α x_{n+1}, implement the shift-register embedding, and compute the feedback-resolved state two ways — directly by Monte Carlo sampling of measurement trajectories and by iterating Eq. (2). Any difference in the first or second moments of the signal and the average quantum state at the same finite step n would falsify the claim.","tokens_in":6555,"feed_emoji":"🔄","tokens_out":5579,"duration_ms":44856,"temperature":0.7,"pith_summary":"The paper sets out to close a gap in deterministic quantum feedback theory: ensemble-averaged master equations exist for Markovian feedback, but non-Markovian feedback has generally required stochastic trajectory sampling. Its central claim is that any feedback rule depending on finitely many past signal values can be turned into a Markovian update by promoting the signal to a vector of auxiliary memory variables. In that lifted space, the ensemble-resolved quantum state obeys a closed, deterministic master equation, so memory effects can be modeled without averaging trajectories. Two explicit embeddings are given: a momentum rule that accumulates past feedback, and a shift-register construction that stores T past signal values in T extra components.","feed_headline":"Non-Markovian feedback becomes a Markovian master equation","feed_subtitle":"Past signals become auxiliary variables, so delayed feedback obeys one closed master equation.","key_machinery":"The load-bearing object is the Markovian embedding of the feedback signal. The scalar signal s_n is replaced by a vector y_n = (s_n, m_n^{(1)}, ..., m_n^{(T)}) whose components are momentum-like differences m_n^{(k)} = s_{n+1-k} - s_{n-k}; because y_{n+1} is a function only of x_{n+1} and y_n, the feedback loop becomes a one-step Markovian map. The deterministic master equation then sums over measurement outcomes x′ and previous signal states y′, using the instrument M_{x′}(y′) and a delta-function that enforces the deterministic update y = f_{n+1}(x′, y′). This machinery converts non-Markovian signal processing into an enlarged but memoryless state space.","core_discovery":"The central discovery is Eq. (2): for feedback determined by s_{n+1} = g_{n+1}(x_{n+1}, s_n, ..., s_{n-T}), the feedback-resolved state evolves as ϱ_{n+1}(y) = Σ_{x′,y′} δ_{y, f_{n+1}(x′,y′)} M_{x′}(y′) ϱ_n(y′). This is not an approximation; it is an exact rewrite of the conditional dynamics once the scalar signal is embedded in a vector whose update is one-step Markovian. The practical content is the embedding recipe: momentum variables m_n^{(k)} = s_{n+1-k} - s_{n-k} form a shift register that makes all past values explicitly available. Consequently, the dimension of the signal vector, T+1, directly counts the memory depth needed, and the ensemble-averaged dynamics is deterministic and clo","pith_inferences":["Because Eq. (2) is exact for any known finite-memory deterministic feedback rule, a natural practical diagnostic follows: one can fit experimental feedback data by progressively increasing T until the deterministic master equation closes, thereby measuring the effective memory depth of the electronics.","An extension the authors do not pursue: adaptive or learned feedback rules, in which the feedback function itself changes with data, would break the closure and need a mixture or hierarchy of embeddings; the equation presented here is for fixed, known rules only.","The dimensionality trade-off — memory depth costs one extra signal component per past step — suggests a compression problem: for specific feedback functions, smarter coordinates than the shift register may yield lower-dimensional embeddings, so finding the minimal T for a given kernel becomes a natural optimization target."],"forward_implications":["If the central claim is right, every finite-memory feedback protocol — delayed feedback, finite-bandwidth electronics, digital filters — can be simulated by a deterministic master equation rather than by averaging stochastic trajectories.","The dimension (T+1) of the signal vector gives an operational measure of a protocol's non-Markovianity: the number of past steps one must keep to make the dynamics closed.","The momentum example implies that a two-dimensional embedding already produces a memory kernel s(t) = γ ∫ ds (1 − e^{-γ(t−s)}) g(x_s) in the continuum limit, connecting discrete feedback rules to physically common exponential filtering.","In the continuum limit with Gaussian measurement operators and linear feedback, the equation reproduces existing quantum Fokker-Planck master equations with general filtering, making the framework a common parent of several earlier results."],"fun_headline_variants":["Exact Markovian master equation from non-Markovian feedback","Memory variables make non-Markovian feedback exactly Markovian","Deterministic quantum master equation for delayed feedback","Non-Markovian feedback, rewritten exactly as Markovian"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The feedback rule must be a known deterministic function of at most finitely many past signal values; if the memory is unbounded, or the functional form of the rule is unknown, the finite (T+1)-dimensional embedding breaks and Eq. (2) loses its closed deterministic form.","fun_headline_variants_meta":{"raw":{"variants":["Exact Markovian master equation from non-Markovian feedback","Memory variables make non-Markovian feedback exactly Markovian","Deterministic quantum master equation for delayed feedback","Non-Markovian feedback, rewritten exactly as Markovian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000471,"raw_usage":{"total_tokens":2133,"prompt_tokens":649,"completion_tokens":1484,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":393,"completion_tokens_details":{"reasoning_tokens":1415}},"tokens_in":393,"tokens_out":1484,"duration_ms":11583,"temperature":1.0,"reasoning_tokens":1415,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T17:35:05.225921+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: pick a qubit, a two-step memory feedback rule such as s_{n+1} = s_n + s_{n-1} + α x_{n+1}, implement the shift-register embedding, and compute the feedback-resolved state two ways — directly by Monte Carlo sampling of measurement trajectories and by iterating Eq. (2). Any difference in the first or second moments of the signal and the average quantum state at the same finite step n would falsify the claim.","supporting_citations":[],"review_version":1}