{"id":"3483ddae-197e-4ed8-941c-2d1c4528de26","arxiv_id":"2608.12519","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In chiral continuous-time quantum walks with a single-site dichotomic measurement, all two-time probabilities, measurement back-action, and Leggett-Garg correlators are exactly determined by the rooted return amplitude, enabling flux-engineering of the L3=3/2 bound.","lead":"A quantum walker on a graph with magnetic flux phases: the paper shows that all two-time measurement statistics at the starting vertex are fixed by a single return amplitude, and that flux can push temporal quantum correlations to a known ceiling. The result gives experimentalists a local, single-site probe of synthetic gauge fields and a design rule for optimal Leggett-Garg violation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the central derivations are internally consistent, and the localized-initial-state/Lüders-measurement idealization is explicit and correctly scoped.","rationale":"The reader's verdict of ACCEPT with moderate confidence is appropriate. The paper's central claim is an exact statement under explicitly stated ideal conditions: a perfectly localized initial state and an ideal dichotomic Lüders measurement. Within those conditions, the derivations are internally consistent. I independently checked the main algebraic steps, including the two-time joint probabilities, the back-action identity, the L3-K relation, the two-level Krylov saturation formula, the diamond-graph examples, and the parity-dependent cycle expansion. No circular reasoning, numerical fitting, or claim-without-derivation was found. The only genuinely fragile point is the idealization itself: if the initial state has any weight outside the measured vertex, or if the measurement is not the ideal projective Lüders instrument, the exact return-amplitude reduction and the bound L3 <= 1 + 2K no longer follow. This is precisely the reader's weakest-assumption identification, and it is a legitimate limitation for practical implementations, but it is explicitly acknowledged and does not undermine the theoretical claims as scoped. A robustness check with imperfect preparation and measurement would be a useful complement, but its absence does not warrant changing the verdict.","tokens_in":22309,"tokens_out":33511,"duration_ms":287925,"concrete_test":"Recompute the two-time joint probabilities from the explicit Kraus operators for an initial state rho0 = (1-epsilon)|nu><nu| + epsilon sigma and for a noisy dichotomic POVM with efficiency eta < 1, then check whether the identities L3 = 1 + 2delta - 4P-+ and L3 <= 1 + 2K still hold to leading order in epsilon and (1-eta). If either fails, it quantifies exactly how much of the central claim is tied to the idealized localized-preparation and projective-measurement limit; if they survive, the reduction is robust beyond the stated assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I checked the derivation chain supporting the central claims and did not find an internal inconsistency or a hidden unsupported step. Proposition 1 follows correctly from the Kraus/effect algebra of the dichotomic Lüders instrument; Eqs. (29)-(31) and the back-action identity (30) match a direct recomputation; Eqs. (54)-(58) correctly yield L3 = 1 + 2δ - 4P-+ and hence L3 <= 1 + 2K for the equally spaced protocol; and the two-dimensional rooted-Krylov formula (68) with maximum 1 + 2w(1-w) is correct. I also re-derived the diamond-graph quartic coefficients (44)-(50) and the cycle winding-onset formulas (92)-(93); the signs and powers are consistent with direct small-time expansion of the exact spectral return amplitude. The weakest premise is therefore the same one the reader identified: the exact reduction to A_nu requires the walker to start perfectly localized on the measured vertex and the measurement to be the ideal dichotomic Lüders projection. The paper states this as a setup condition, uses it in every derived identity, and never claims robustness to delocalized preparations or inefficient measurements. That is a limitation of scope, not a flaw in the argument. The absence of shipped code is mitigated by the fact that the numerical results are evaluations of the exact closed-form return amplitude (84) in Eq. (54); no fitting or approximation is involved.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies continuous-time quantum walks on finite graphs with complex hopping amplitudes (chiral CTQWs) and analyzes sequential measurements of the dichotomic return observable Qν = 2|ν⟩⟨ν| − I at a single vertex ν, for a walker initially localized at ν. The central result (Proposition 1, Sec. III) states that the complete two-time joint distribution, and hence the correlators entering the Leggett–Garg functional, are determined exactly by the rooted return amplitude Aν(t;Φ); the derivation is given in Appendix A. From this exact reduction the authors obtain an expression for the Kolmogorov inconsistency Kν,Φ(s,t) (Eq. (30)), an exact identity relating the equally spaced Leggett–Garg functional L3 to the signed measurement disturbance δν,Φ and to the probability P−+ of returning after an intermediate 'complement' outcome (Eq. (56)), and the bound L3 ≤ 1 + 2Kν,Φ (Eq. (58)). Short-time expansions show that the leading disturbance is O(t²) and independent of the Peierls phases, while flux sensitivity generically enters at O(t⁴) (Sec. IV). Section V develops a graph-independent operator criterion (Eqs. (71)–(74)) for balanced two-dimensional rooted Krylov dynamics, under which the Lüders bound L3 = 3/2 is saturated at finite time; this is realized exactly on the diamond graph for the relative-flux configuration (Φ1,Φ2) = (0,π). For flux-threaded cycles, the winding-number expansion (Eq.","tokens_in":22525,"tokens_out":23548,"duration_ms":194148,"significance":"Assuming the results hold, this is a valuable contribution to chiral quantum walks and to the theory of temporal quantum correlations. The paper's main strength is that the central chain is self-contained, exact, and checkable: Proposition 1 follows from the Lüders instrument algebra, Eqs. (29)–(31) and (54)–(58) are consistent with direct recomputation, the balanced-Krylov criterion in Eq. (71) is a clean operator condition, and the diamond and C4 saturation examples are exact spectral statements rather than numerical fits. The parity-dependent onset in Proposition 4 is a falsifiable, graph-specific prediction. The numerical results are evaluations of the exact closed-form return amplitude in Eq. (84), so the absence of a code deposit is not a correctness issue. The explicit scope—localized initial preparation and ideal dichotomic Lüders instrument—is stated as a setup condition and used consistently; the paper does not claim robustness to delocalized preparations or inefficient measurements. The main limitation is therefore one of scope, not of internal consistency.","major_comments":[],"minor_comments":[{"comment":"The combination of the ℓ and −ℓ winding sectors in Eq. (86) uses J−n(z) = (−1)^n Jn(z); stating this identity explicitly immediately before Eq. (86) would remove a potential point of confusion for readers who verify the derivation.","section":"Sec. VI.A, Eq. (86)"},{"comment":"The paper correctly states the localized-initial-state and ideal-Lüders assumptions in Sec. II.B, but a one-sentence reminder in the Conclusions that all central identities, including L3 ≤ 1 + 2Kν, would acquire additional terms for delocalized preparations or inefficient measurements would help prevent overgeneralization.","section":"Sec. VII"},{"comment":"The observation windows and grid resolutions for the numerical survey (for example, T = 40 for C10 and the 501 × 4001 grid in Fig. 3) are given in the text and Appendix E; including the key values directly in the figure captions would improve readability and reproducibility.","section":"Sec. VI.D and Figs. 3–4"},{"comment":"There are minor typographical inconsistencies in the rendering of the name 'Lüders' (for example, 'L\\\"uders' in the abstract) and in the use of 'Lüders bound' versus 'Lüders value'; these should be harmonized in the final version.","section":"Abstract and Secs. V–VI"}],"recommendation":"accept","confidential_remarks":"I concur with the reader's positive assessment. The derivations are self-contained and checkable, the numerical results are exact evaluations of the closed-form return amplitude rather than fits, and the localized-initial-state/ideal-Lüders idealization is explicitly scoped and does not undermine the claims. The paper is a good fit for the journal, and I have no unresolvable concerns."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nI've read the paper. Short version: it deserves a serious referee. The core claim—that for a walker initialized on the measured vertex, the full two-time statistics reduce exactly to the rooted return amplitude Aν(t)—is correct and cleanly proved. The derived bound L3 ≤ 1 + 2Kν is exact, and the positive statement that any Leggett-Garg violation implies nonzero back-action follows without any hidden step. The genuinely new contribution is the flux-engineering criterion in Eq. (71): balanced two-dimensional rooted Krylov dynamics is the necessary and sufficient route to saturating the Lüders bound 3/2 within that class, and the diamond graph realizes it by making extra rooted modes dark. The parity-dependent onset t^N vs t^{2N} on cycles is also a nice concrete signature, and the winding-number derivation in App. D is consistent.\n\nI rechecked the stress-test derivations. The expansions, the quartic coefficients, and the C4 saturation example all reproduce from the exact spectral return amplitude. The numerics are exact evaluations, not fits; no circularity, no p-hacking. The self-citation to [36] is used appropriately, as protocol comparison.\n\nThe soft spot is exactly what the authors state: the result requires the walker to start perfectly localized on the measured vertex and the measurement to be the ideal dichotomic Lüders instrument. The paper explicitly scopes this as a setup condition, so it's a limitation of scope rather than a flaw, but it is the fragile premise for any experimental realization. A short robustness discussion would help. A second, smaller caveat: the temporal speedup numbers (S10 ≈ 3.1) are finite-window numerical observations on specific graphs, not general theorems. They are honestly computed, but should be presented as such.\n\nWho is this for? People working on quantum walks with synthetic gauge fields, sequential measurement protocols, and Leggett-Garg tests. They get an exact reduction tool and a constructive design principle. It is incremental relative to [36] and the known LG bound, but the reduction and the criterion are new enough to matter. I recommend engaging with it in peer review; I would accept it after minor revision focused on the robustness caveat and on distinguishing theorems from numerical observations.","headline":"Exact two-time return statistics and a flux-engineering criterion for the Lüders bound; clean derivations, with the localized-initial-state idealization the only real caveat.","tokens_in":23105,"tokens_out":2978,"would_cite":true,"duration_ms":27199,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"One return amplitude fixes sequential measurement statistics in chiral quantum walks.","keywords":["chiral continuous-time quantum walks","gauge-invariant flux","measurement back-action","Leggett-Garg inequalities","Lueders bound","rooted return amplitude","Krylov subspace","winding number expansion"],"falsifier":"On a four-site chiral cycle with flux $\\phi=\\pi$, implement the equally spaced protocol and search over $\\tau$; the paper predicts the maximum of $L_3$ is exactly $3/2$ at $\\tau=\\pi/(6\\sqrt{2})$. A second decisive test: prepare the walker in a superposition with a small weight on a neighboring vertex and check whether the two-time probabilities still depend only on $A_\\nu(t)$; any visible dependence on other transition amplitudes would refute the exact reduction.","tokens_in":22058,"feed_emoji":"🌀","tokens_out":8399,"duration_ms":73149,"temperature":0.7,"pith_summary":"This paper argues that in a chiral continuous-time quantum walk, if the walker starts exactly on one vertex $\\nu$ and one measures only whether it has returned to $\\nu$, the full two-time statistics are captured by a single gauge-invariant quantity, the rooted return amplitude $A_\\nu(t;\\Phi)$. Because that amplitude is built from interference among closed walks, the graph's fluxes become tunable controls on measurement back-action and on temporal correlations of the Leggett\\textendash Garg type. The paper proves the exact bound $L_3 \\le 1 + 2K_{\\nu,\\Phi}$, so any Leggett\\textendash Garg violation in this protocol certifies that the intermediate measurement disturbed the walk. It also proves a graph-independent criterion under which flux engineering saturates the L\\\"uders bound $L_3 = 3/2$, and exhibits two exact realizations: a two-flux diamond graph and the four-cycle at half flux. A winding-number expansion gives a parity-dependent onset of flux effects, at order $t^N$ on even cycles and $t^{2N}$ on odd cycles.","feed_headline":"One return amplitude fixes a walker's measurement statistics","feed_subtitle":"Gauge flux then tunes back-action and can push Leggett-Garg correlations to the 3/2 Lueders bound.","key_machinery":"The carrying object is the rooted return amplitude $A_\\nu(t;\\Phi)=\\langle \\nu|e^{-iH_\\chi(\\Phi)t}|\\nu\\rangle$ together with its rooted spectral measure $\\mu_{\\nu,\\Phi}=\\sum_r w_r\\delta_{\\lambda_r}$. Every two-time return probability, the back-action $K_{\\nu,\\Phi}$, and the Leggett\\textendash Garg functional $L_3$ reduce to exact algebraic combinations of this single function. The saturation mechanism is the operator condition $(H_\\Phi-a_\\nu I)^2|\\nu\\rangle=\\gamma_\\nu|\\nu\\rangle$, which makes the rooted Krylov subspace two-dimensional with equal spectral weights; the cycle analysis uses the Jacobi\\textendash Anger expansion to reorganize $A_N(t;\\phi)$ into winding sectors $J_{\\ell N}(2t)\\cos(\\ell\\phi)$ whose short-time orders control the parity-dependent flux onset.","core_discovery":"On the paper's own terms, the central discovery is that the joint probabilities of the dichotomic return measurement are exactly $P_{++}=p_\\nu(t_1)p_\\nu(\\tau)$, $P_{+-}=p_\\nu(t_1)[1-p_\\nu(\\tau)]$, $P_{-+}=|A_\\nu(t_2)-A_\\nu(\\tau)A_\\nu(t_1)|^2$, and $P_{--}=1-p_\\nu(t_1)-P_{-+}$, for any finite graph, any hopping phases, and any time-independent Hamiltonian. Thus the two-time correlators, the Kolmogorov inconsistency $K_{\\nu,\\Phi}$, and the equally spaced Leggett\\textendash Garg functional $L_3(\\tau)$ are exact functionals of $A_\\nu(t;\\Phi)$ only. From these identities the paper derives that $L_3 \\le 1+2K_{\\nu,\\Phi}$, that the leading short-time disturbance is flux-independent with flux sensitivity first entering through rooted closed-walk sums at quartic or higher order, and that balanced two-dimensional rooted Krylov dynamics, enforced by the operator condition $(H_\\Phi-a_\\nu I)^2|\\nu\\rangle=\\gamma_\\nu|\\nu\\rangle$, saturates the L\\\"uders bound. On flux-threaded cycles the same framework yields a winding-number expansion in which the leading flux contrast is of order $t^N$ for even $N$ and $t^{2N}$ for odd $N$, with half flux on the four-cycle reaching $L_3=3/2$.","pith_inferences":["Inference: the same reduction is recursive. Adding a third intermediate measurement should express the three-time joint distribution through products of return amplitudes $A_\\nu(t_i)A_\\nu(t_j-t_i)\\cdots$, so flux-engineering criteria for higher-order Leggett\\textendash Garg functionals $L_n$ would follow from the same Krylov argument.","Inference: the exact dependence of $L_3$ on $A_\\nu$ suggests a flux-metrology protocol. The quantum Fisher information of the dichotomic return statistics is likely an explicit functional of $A_\\nu$ and its derivatives, so single-vertex return measurements could estimate a flux with precision governed by the local spectral measure; the paper raises the question but does not compute this.","Inference: because odd-cycle return statistics are $\\pi$-periodic in the flux, a single odd cycle cannot distinguish $\\phi$ from $\\phi+\\pi$. A two-flux graph like the diamond, where the protocol separates relative flux configurations, would be needed to lift that ambiguity."],"forward_implications":["Any Leggett\\textendash Garg violation measured with this protocol implies nonzero measurement back-action at the intermediate time, since $L_3 \\le 1+2K_{\\nu,\\Phi}$.","Local single-vertex measurements become a complete probe of the rooted spectral measure: optimizing $L_3$ over $\\tau$ reveals the spectral weights $w_r$ and gaps $\\lambda_r$ seen from $\\nu$.","Flux can be designed to make rooted modes dark: on the diamond graph with $(\\Phi_1,\\Phi_2)=(0,\\pi)$ the outer vertex saturates $L_3=3/2$ at $\\tau=\\pi/(6\\sqrt{2})$, a gain of about $0.0410$ over zero flux.","On cycles the leading flux contrast in back-action is order $t^N$ for even $N$ with sign $(-1)^{N/2}$, and order $t^{2N}$ for odd $N$.","On larger cycles flux mainly advances temporal accessibility: for $C_{10}$ at half flux the earliest time to reach 99% of the zero-flux optimized violation drops from about $31.4$ to about $10.2$, a speedup factor around $3.1$."],"supporting_citations":[{"why":"Supplies the continuous-time quantum walk Hamiltonian $D-A$ used throughout.","marker":"[1]"},{"why":"Establishes the graph-based CTQW model and its transport diagnostics.","marker":"[2]"},{"why":"Introduces chiral quantum walks and the gauge-invariant flux parametrization of edge phases.","marker":"[8]"},{"why":"Defines the Leggett-Garg inequality and macrorealism assumptions the functional $L_3$ tests.","marker":"[28]"},{"why":"Provides the standard three-time Leggett-Garg functional and its quantum bound.","marker":"[29]"},{"why":"Previous position-resolved sequential-measurement study whose single-vertex dichotomic protocol is contrasted here.","marker":"[36]"},{"why":"Carries the L\\\"uders bound $L_3=3/2$ that the balanced rooted-Krylov mechanism saturates.","marker":"[45]"}],"fun_headline_variants":["Back-action and Leggett-Garg correlations set by return amplitude","Flux tunes back-action in chiral quantum walks","Flux engineering saturates the Lüders bound in quantum walks","Half flux reaches Lueders bound on four-cycle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the walker starts exactly localized on the measured vertex and that the measurement is an ideal, perfectly efficient projection onto return-versus-complement; any weight on other vertices or any measurement inefficiency would introduce extra terms beyond the return amplitude and break the bound and the parity onsets.","fun_headline_variants_meta":{"raw":{"variants":["Back-action and Leggett-Garg correlations set by return amplitude","Flux tunes back-action in chiral quantum walks","Flux engineering saturates the Lüders bound in quantum walks","Half flux reaches Lueders bound on four-cycle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000981,"raw_usage":{"total_tokens":4286,"prompt_tokens":1186,"completion_tokens":3100,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":802,"completion_tokens_details":{"reasoning_tokens":3033}},"tokens_in":802,"tokens_out":3100,"duration_ms":20302,"temperature":1.0,"reasoning_tokens":3033,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:07:51.123277+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a four-site chiral cycle with flux $\\phi=\\pi$, implement the equally spaced protocol and search over $\\tau$; the paper predicts the maximum of $L_3$ is exactly $3/2$ at $\\tau=\\pi/(6\\sqrt{2})$. A second decisive test: prepare the walker in a superposition with a small weight on a neighboring vertex and check whether the two-time probabilities still depend only on $A_\\nu(t)$; any visible dependence on other transition amplitudes would refute the exact reduction.","supporting_citations":[{"cited_title":"ForC6 and C10, the optimized violation varies by less than one percent over the flux range in the observation windows considered below","cited_arxiv_id":null,"evidence_quote":"Supplies the continuous-time quantum walk Hamiltonian $D-A$ used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the graph-based CTQW model and its transport diagnostics."},{"cited_title":"Smirne, D","cited_arxiv_id":null,"evidence_quote":"Provides the standard three-time Leggett-Garg functional and its quantum bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous position-resolved sequential-measurement study whose single-vertex dichotomic protocol is contrasted here."},{"cited_title":"Fritz, Quantum correlations in the temporal Clauser– Horne–Shimony–Holt (CHSH) scenario, New J","cited_arxiv_id":null,"evidence_quote":"Carries the L\\\"uders bound $L_3=3/2$ that the balanced rooted-Krylov mechanism saturates."}],"review_version":1}