{"id":"6df448bd-d1e8-4bc7-9208-5e16e1c135cf","arxiv_id":"2412.05456","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Local weak values in any quantum circuit evolve only at local gates and obey simple classical oscillator equations in an exchange interaction, suggesting a localized all-at-once underpinning for quantum circuits.","lead":"This paper analyzes the local weak values of individual qubits in quantum circuits and finds they stay constant between gates and evolve through gates according to simple local equations, even when the qubits are part of a massively entangled state. The authors argue this supports a localized, all-at-once hidden-variable account of quantum circuits that scales linearly with the number of qubits.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The dynamic-locality results (Eqs. 13-22) are algebraic and credible, but the leap to a localized reality rests entirely on the Section V template, which is overconstrained (14 constraints on 12 variables), future-input-dependent, and has no general probability rule; the universal claim is…","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the Section V model is overconstrained, future-input-dependent, and only a template. The dynamic-locality theorems are solid and independent of Section V; they show weak values are local in the Heisenberg sense. However, the paper's advertised conclusion is about a localized reality underpinning all circuits. That conclusion requires an actual hidden-variable model. The only model offered is explicitly incomplete: overconstrained, no derivation of Eq. (29), no unified probability rule, and only one partially-entangled example. Because the authors themselves list these as open problems in Section V.D and VI.B, the correct verdict is conditional, not accept. My proposed test directly checks the model's generality: rotate a measurement basis or extend the circuit, and see whether solutions and probabilities survive. If they do not, the template does not generalize; if they do, the strongest objection is weakened. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":21640,"tokens_out":5236,"duration_ms":59703,"concrete_test":"Implement the Section V constraint system numerically: for the √SWAP circuit of Fig. 4, keep |i⟩=x⊗y and outcomes measured in the z basis, then rotate the final measurement basis of qubit a by an angle θ (e.g., measure σz cosθ + σx sinθ) and rerun the gradient-descent search for solutions of Eqs. (27)-(31) plus boundary constraints. If for any θ≠0 the solution count either vanishes or no longer matches the three Born probabilities, the model is tuned to one circuit and cannot serve as a template for universal circuits. A complementary check: append a second √SWAP gate to the circuit and count solutions for the same outcomes; no well-defined solution count that yields Born probabilities would be strong evidence against a general modular construction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The strongest parts of the paper are the algebraic demonstrations in Sections III-IV: local weak values are constant on noninteracting wires, rotate through one-qubit gates, and satisfy Eq. (22) during an exchange interaction. Appendix A derives Eq. (22) directly from Eq. (1), so I do not question those results. The load-bearing step is the inference from dynamically local weak values to a localized reality underpinning arbitrary circuits. That inference is carried entirely by the Section V model. That model is not a derivation: Eq. (29), sa·sb=1, is imposed so that the nonlinear coefficient in Eqs. (32)-(33) collapses to 2; no mechanism produces this pre-interaction correlation. Section V.D counts 14 real constraints on 12 real initial variables, so solutions exist only in special cases. For the single √SWAP example the authors find 2/2/4 solutions matching outcome probabilities, but even there Im(w) is recovered only after discarding half the solutions, and the solution-counting rule is incompatible with the single-qubit 1/Re[w]^2 rule of Eq. (11). The paper openly calls Section V a template. Thus the central claim that any quantum circuit can plausibly be underpinned by localized variables is not yet supported: there is no demonstration that the constraints are satisfiable for a generic circuit, nor a probability rule that reproduces Born probabilities outside the single example. The future-input dependence of Eq. (29) and the measurement constraints is acknowledged, but that does not by itself supply the missing dynamics.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies local (single-qubit) weak values in complete quantum circuits. It proves (Sections III and IV, with Appendix A) that these weak values are dynamically local: constant on non-interacting wires, rotating through single-qubit gates, and obeying the coupled oscillator equation d²w/dτ² = (π²/2)(w_b − w_a) during an exchange interaction, regardless of entanglement elsewhere. On this basis the authors argue that any quantum circuit can plausibly be underpinned by a localized, all-at-once hidden-variable model. Section V presents a template model in which each qubit carries a complex 3-vector s(t), constrained by s·s = 1 and by preparation/measurement constraints, and shows that for a single √SWAP circuit the number of solutions matches the Born probabilities while the average Re[s] reproduces Re[w]. The authors themselves identify serious open problems with this model: it is overconstrained (14 real constraints on 12 real variables) and lacks a general probability rule.","tokens_in":21984,"tokens_out":4017,"duration_ms":41867,"significance":"If the program were completed, the result would be significant: it would suggest a spacetime-localized, linearly-scaling account of arbitrary quantum circuits, with retrodictive constraints replacing nonlocal causation. The algebraic core of the paper is strong and reproducible: the dynamic-locality results in Sections III–IV follow directly from Eq. (1), and Appendix A gives a clean derivation of Eq. (22). The authors are also commendably explicit about the limitations of the Section V template. However, the central claim of the paper—that a localized reality 'appears to underpin' any quantum circuit—is not established by the evidence presented. The dynamic-locality results concern weak values, not a complete hidden-variable model, and the only concrete model offered (Section V) is overconstrained, future-input-dependent, and fitted to a single example. The paper is therefore best read as a strong motivation and a preliminary template, not as a demonstration of the universal claim.","major_comments":[{"comment":"The overconstraint problem is load-bearing for the universality claim. The authors count 14 real constraints on the 12 real initial variables of the two-qubit model (three complex equations (27)–(29) give six real constraints, and the four preparation/measurement constraints (36)–(39) give eight more). They then show that the √SWAP example happens to have solutions, but no argument is given that generic circuits, or even generic two-qubit states, admit solutions. Since the paper's conclusion that 'any quantum circuit can plausibly be underpinned by localized variables' depends on the Section V model being a template, the overconstraint means the central claim is currently unsupported beyond the single example.","section":"§V.D"},{"comment":"The model uses two incompatible probability rules. In the single-qubit sector, Eq. (11) assigns probability P(w) ∝ 1/Re(w)², whereas the √SWAP analysis in Section V.C assigns equal weight to each solution of the constraint equations, with the correct Born probabilities emerging from the number of solutions. The authors acknowledge in §V.D that no unified rule is known ('It is not clear what probability rule would work for both this example and for the single-qubit case'). Without a general probability rule that reproduces Born probabilities for arbitrary circuits, the Section V model cannot underpin quantum circuits, and the claim that weak values 'could generate weak values without using state vectors' remains confined to the worked example.","section":"§II.C and §V.C"},{"comment":"The model's constraints are future-input-dependent in a way that is not derived but imposed. Eq. (29), s_a·s_b = 1, requires the two qubits' hidden vectors to be correlated before they interact, and the measurement constraints (38)–(39) depend on the future outcome. The authors honestly label this 'counterintuitive' and 'evidently retrocausal'. However, the paper presents this future-input dependence as a feature that explains Bell violations, not as a hypothesis to be tested. Because no mechanism or independent motivation is given for why these particular constraints should hold, the empirical match in Section V.C is essentially a reverse-engineering of the constraints from the known weak values. This weakens the inference from 'the model fits one example' to 'localized reality underpins circuits.'","section":"§V.A, Eqs. (29), (38)–(39)"},{"comment":"The leap from dynamic locality of weak values to a localized underlying reality is not supported by the algebraic results alone. Sections III–IV show that weak values, as computed from Eq. (1), evolve locally; but weak values are averages over an ensemble (or best estimates), and the paper itself introduces s(t) in Section V precisely because w(t) does not satisfy the model constraints in all cases (e.g., the |10⟩ outcome). Thus the dynamic-locality results are necessary but not sufficient for the conclusion. The only bridge is the Section V template, whose limitations are acknowledged. To make the universal claim credible, the paper would need either a more complete model or a clearly weakened conclusion stating that the evidence supports only the possibility, not the plausibility, of such an underpinning.","section":"§VI.A"}],"minor_comments":[{"comment":"The denominator in both equations is written as ⟨f|U_ex[α]|ψ⟩, but it should be ⟨f|U_ex[α]|i⟩, consistent with Eq. (1).","section":"Eqs. (20)–(21)"},{"comment":"In the expression for Im(dw_a/dτ), the second cross product is written as Im(w_b)×Im(w_a); from Eq. (23) it should be Im(w_b)×Re(w_a) (with the first term Re(w_b)×Im(w_a)).","section":"§IV.C"},{"comment":"The text refers to a 'BRST algorithm' for gradient descent; this is presumably a typo for a standard optimizer such as BFGS. Please clarify the algorithm used.","section":"§V.C"},{"comment":"The axis labels and legend are difficult to read in the reproduced figure; please increase the font size and ensure all curve labels are legible.","section":"Figure 3"}],"recommendation":"major_revision","confidential_remarks":"The authors are transparent about the Section V model's limitations, which is commendable, but the manuscript's title and abstract promise more than the evidence delivers. The algebraic results in Sections III–IV are solid and publishable as a standalone contribution; the interpretive claim needs either a substantially developed model or a careful restatement as a conjecture. The paper fits the journal's scope, but the editors may want to weigh whether the current 'template' level of evidence meets the bar for a claim of 'a localized reality appears to underpin quantum circuits.'"},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague—\n\nThe thing worth knowing about this paper is that the algebraic core is real and the interpretive superstructure is not. Sections III and IV, plus Appendix A, show that local weak values in any quantum circuit are dynamically local: constant on noninteracting wires no matter what the rest of the entangled state is doing, rotating through one-qubit gates, and obeying a clean oscillator equation d²w_a/dτ² = (π²/2)(w_b − w_a) during an exchange interaction. That last result is new, and Appendix A derives it directly from the weak-value definition. I checked the logic; it is straightforward and sound. Anyone working with weak values or retrocausal models will want this.\n\nThe soft spot is the leap from those results to the claim that a localized reality underpins any quantum circuit. That leap is carried by the Section V model, and the model is explicitly a template. It is overconstrained: 14 real constraints on 12 real initial variables. The probability rule—counting solutions—does not generalize, and the authors admit it does not square with the single-qubit 1/Re[w]^2 rule. They recover Im(w) only by discarding half the solutions. The future-input dependence (Eq. 29 and the measurement constraints) is an assumption of the research program, not a derivation. So the universal claim is not supported yet. But the authors say this themselves in Section V.D. The paper is honest about its own gaps, which is more than many papers in this area manage.\n\nThe dynamic-locality results stand on their own. They are formal, reproducible algebra: no free parameters, no invented entities beyond the s(t) vectors in the template model. The citation pattern is appropriate—Wharton and Argaman's Colloquium, Sutherland's work, the weak-value experimental literature. Self-citation is not a problem here; the cited work is relevant.\n\nWho should read this: quantum foundations people interested in weak values, retrocausality, and all-at-once accounts. It won't convert a skeptic, but it gives the program a concrete, checkable handle on circuit dynamics. As a referee, I would send it out. The algebraic results deserve scrutiny, and the Section V model deserves a referee's push on whether the constraints can be relaxed or the probability rule improved. The paper should survive as a contribution if the claims are scaled back to “this is a promising template, not a demonstration.”\n\nRecommendation: accept for review. If I were the editor, I'd want a referee who can check the algebra and who is sympathetic but not credulous.","headline":"Clean, checkable algebra on dynamically local weak values (the oscillator equation is genuinely new) under a universal claim that outruns the evidence; the Section V model is an honest template, not a demonstration.","tokens_in":22473,"tokens_out":2645,"would_cite":true,"duration_ms":25682,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.Ta","03.67.Lx"],"model":"deepseek-v4-flash","headline":"This paper argues that the local weak values of individual qubits in a quantum circuit remain localized and evolve by simple local rules even when the global state is entangled, providing evidence that a spacetime-localized, all-at-once…","keywords":["weak values","quantum circuits","local hidden variables","all-at-once models","retrocausality","exchange interaction","sqrt(SWAP) gate","Bell's theorem"],"falsifier":"Perform a weak-measurement experiment on the $\\sqrt{\\mathrm{SWAP}}$ circuit of Figure 4, preparing $|x\\rangle|y\\rangle$ and post-selecting on $|10\\rangle$, and compare the measured $\\mathrm{Re}[w_a(\\tau)]$ and $\\mathrm{Re}[w_b(\\tau)]$ with the analytic forms in Appendix B. If the trajectories deviate from the coupled-oscillator equation, or if the pre-interaction weak values are found not to depend on the choice of the final measurement basis while the preparation and post-selected outcome are held fixed, the central locality claim fails.","tokens_in":21439,"feed_emoji":"⚛️","tokens_out":7168,"duration_ms":66454,"temperature":0.7,"pith_summary":"This paper argues that the local weak values of individual qubits in any complete quantum circuit behave as localized, real properties, even when the overall quantum state is massively entangled. On circuit wires the weak values stay constant; through single-qubit gates they rotate exactly as the gate rotates a Bloch vector; and during an exchange interaction they evolve by a simple second-order differential equation of coupled oscillators. The paper takes these regularities as evidence that a spacetime-localized, all-at-once account could underpin any quantum circuit, with resources scaling linearly in the number of qubits rather than exponentially. It also presents a toy model using hidden complex 3-vectors whose solutions reproduce the weak values on average and whose counting reproduces outcome probabilities for a sqrt(SWAP) circuit.","feed_headline":"Weak values stay local even in a massively entangled quantum circuit","feed_subtitle":"Single-qubit weak values obey a simple local equation through every gate, hinting that entanglement needs no nonlocal link.","key_machinery":"The main object is the local weak value vector $w$, a complex 3-vector assigned to each wire of a quantum circuit, defined by Eq. (1) with single-qubit Pauli operators and identity on the remaining qubits. The load-bearing mechanical results are: constancy of $w$ on free wires (Section III.B), the rotation rule through single-qubit gates (Section III.C), and the oscillator equation inside the exchange interaction, derived analytically in Appendix A. The authors also propose a template hidden-variable model (Section V) in which each qubit carries a complex 3-vector $s$ constrained by $s \\cdot s = 1$ and, during exchange, by $s_a \\cdot s_b = 1$ and first-order torque equations; these constraints imply the oscillator dynamics, and counting the solutions of a $\\sqrt{\\mathrm{SWAP}}$ circuit reproduces the Born-rule probabilities of the three allowed outcomes.","core_discovery":"The central discovery is that the complex weak-value vector $w = (W[\\sigma_x], W[\\sigma_y], W[\\sigma_z])$ for each qubit, computed from the usual post-selected weak-value formula, obeys dynamic locality in any circuit built from single-qubit rotations and the exchange interaction. If a qubit is acted on by no gate, its $w$ is constant even when other qubits evolve in the same entangled state; if it passes through a single-qubit gate, its $w$ rotates exactly as the corresponding Bloch vector rotates, independent of the rest of the state; and inside a $\\mathrm{SWAP}_\\alpha$ exchange gate its components obey $\\frac{d^2 w_a}{d\\tau^2} = \\frac{\\pi^2}{2}(w_b - w_a)$, a classical coupled-oscillator equation. The authors take these results as evidence that entangled quantum circuits can be underpinned by localized variables that are future-input-dependent: the hidden variables must be solved all-at-once, using both past preparation and future measurement constraints, rather than by a forward-causal dynamical law.","pith_inferences":["If the weak-value locality carries over to larger circuits, an experimental test could track weak values through a controlled $\\sqrt{\\mathrm{SWAP}}$ interaction and check whether the measured trajectories satisfy the oscillator equation for each outcome; this would distinguish the all-at-once picture from forward-causal hidden-variable models.","The future-input dependence required by the model is a retrocausal element: the final measurement basis is a boundary condition on the past. A reader who rejects retrocausal constraints will see the exchange-interaction constraint $s_a \\cdot s_b = 1$ as the point where a purely dynamical account breaks down.","The paper's approach connects naturally to the \"second-order qubit\" program mentioned in the text; expanding the per-qubit hidden space from 6 to 7 or 8 real variables would relieve the over-constraint problem and may yield a general local model for universal circuits.","If the solution-counting rule generalizes, it would provide a probabilistic interpretation in terms of hidden-state counting rather than wavefunction amplitudes, offering a route to quantum probabilities that does not invoke a collapse postulate."],"forward_implications":["If the central claim is right, no instantaneous nonlocal connection is needed to explain Bell-inequality violations in circuits; the locality is carried by weak values that only depend on past and future boundary conditions, not on distant simultaneous events.","A generic quantum circuit can be analyzed \"all-at-once\", replacing the exponentially large state vector with a set of $N$ local hidden vectors, provided the future measurement setting is allowed to constrain earlier hidden states.","Weak values can be generated without state vectors as an intermediate step, at least in the $\\sqrt{\\mathrm{SWAP}}$ example, because averaging the hidden $s$ solutions reproduces $\\mathrm{Re}(w)$ throughout the gate.","The outcome probabilities of a partially entangled two-qubit circuit emerge from the number of allowed hidden-variable solutions, matching the Born-rule weights for the $|00\\rangle$, $|10\\rangle$, $|11\\rangle$ outcomes, and vanishing for the impossible $|01\\rangle$ outcome.","The same modular strategy applied to a universal gate set (single-qubit rotations plus $\\sqrt{\\mathrm{SWAP}}$) suggests an infinite family of entanglement geometries could be treated by local rules, one gate at a time."],"supporting_citations":[{"why":"Supplies the concept of future-input-dependent locally mediated reformulations and the Bell-theorem motivation that the paper's all-at-once models build on.","marker":"[1]"},{"why":"Provides the derivation of post-selected weak values that the paper uses as its working definition in Eq. (1).","marker":"[3]"},{"why":"Gives the characterization of weak values as the best estimate given boundary data, allowing the paper to interpret $w$ without an actual weak measurement.","marker":"[10]"},{"why":"Establishes the Roberts/Sutherland view that weak values can be objective local elements of reality, which the paper adopts and extends.","marker":"[4]"},{"why":"Represents the state of the art in future-input-dependent models, previously limited to maximally entangled states, which this paper's partially entangled $\\sqrt{\\mathrm{SWAP}}$ model extends.","marker":"[27]"},{"why":"Provides the benchmark of explaining a zero-probability entangled case with classical electromagnetism, which the Section V model surpasses by covering all non-zero probability outcomes.","marker":"[28]"}],"fun_headline_variants":["Weak values stay put in entangled circuits","Entanglement's local twist: weak values","Quantum circuits may be local after all","Hidden variables in quantum circuits are local","Entangled qubits, local weak values: a surprise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The model requires that the hidden vectors of two qubits entering an exchange interaction already satisfy $s_a \\cdot s_b = 1$ before the interaction begins, which means the past states are constrained by future measurement settings; if retrocausal or future-boundary constraints are disallowed, the model cannot reproduce entangled weak values or outcome statistics.","fun_headline_variants_meta":{"raw":{"variants":["Weak values stay put in entangled circuits","Entanglement's local twist: weak values","Quantum circuits may be local after all","Hidden variables in quantum circuits are local","Entangled qubits, local weak values: a surprise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000246,"raw_usage":{"total_tokens":1566,"prompt_tokens":997,"completion_tokens":569,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":613,"completion_tokens_details":{"reasoning_tokens":503}},"tokens_in":613,"tokens_out":569,"duration_ms":6167,"temperature":1.0,"reasoning_tokens":503,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T20:41:33.937924+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform a weak-measurement experiment on the $\\sqrt{\\mathrm{SWAP}}$ circuit of Figure 4, preparing $|x\\rangle|y\\rangle$ and post-selecting on $|10\\rangle$, and compare the measured $\\mathrm{Re}[w_a(\\tau)]$ and $\\mathrm{Re}[w_b(\\tau)]$ with the analytic forms in Appendix B. If the trajectories deviate from the coupled-oscillator equation, or if the pre-interaction weak values are found not to depend on the choice of the final measurement basis while the preparation and post-selected outcome are held fixed, the central locality claim fails.","supporting_citations":[{"cited_title":"Wharton and N","cited_arxiv_id":null,"evidence_quote":"Supplies the concept of future-input-dependent locally mediated reformulations and the Bell-theorem motivation that the paper's all-at-once models build on."},{"cited_title":"Aharonov, D","cited_arxiv_id":null,"evidence_quote":"Provides the derivation of post-selected weak values that the paper uses as its working definition in Eq. (1)."},{"cited_title":"Dressel, Weak values as interference phenomena, Phys- ical Review A 91, 032116 (2015)","cited_arxiv_id":null,"evidence_quote":"Gives the characterization of weak values as the best estimate given boundary data, allowing the paper to interpret $w$ without an actual weak measurement."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the Roberts/Sutherland view that weak values can be objective local elements of reality, which the paper adopts and extends."},{"cited_title":"Neder and N","cited_arxiv_id":null,"evidence_quote":"Represents the state of the art in future-input-dependent models, previously limited to maximally entangled states, which this paper's partially entangled $\\sqrt{\\mathrm{SWAP}}$ model extends."},{"cited_title":"Wharton and E","cited_arxiv_id":null,"evidence_quote":"Provides the benchmark of explaining a zero-probability entangled case with classical electromagnetism, which the Section V model surpasses by covering all non-zero probability outcomes."}],"review_version":1}