{"id":"b6961d30-3548-4772-ab6c-1dfd21392812","arxiv_id":"2412.07123","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A nonlinear-equation framework that claims full amplitude reconstruction from single-qubit measurements fails because single-qubit marginals contain too little information to identify a multi-qubit state.","lead":"This paper proposes to estimate every amplitude of an n-qubit state by measuring only one qubit at a time, then solving a nonlinear equation system. The proposal cannot work because single-qubit measurement statistics depend only on the measured qubit's reduced state, so they cannot determine a full multi-qubit state.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Single-qubit outcome probabilities depend only on each qubit's reduced density matrix, so states like (|00>+|11>)/√2 and (|00>-|11>)/√2 give identical equations in Eq. (71); amplitude reconstruction is information-theoretically impossible.","rationale":"The reader's weakest assumption identifies the same load-bearing flaw: the measurement probabilities in Eq. (71) depend only on single-qubit reduced density matrices, so the equation system cannot uniquely determine all complex amplitudes. My analysis confirms this and sharpens it with an explicit two-qubit counterexample where the right-hand sides are exactly identical for distinct physical states. The conclusion's own caveat about the unproven Jacobian norm is corroborated, and in the counterexample the Jacobian is singular rather than merely large. The error-analysis theorem rests on the same unsupported invertibility assumption, so the claimed sample complexities O(4^n/δ^4), O(6^n/δ^4), and O(2^n/δ^4) are premature. The reader's verdict of REJECT stands; no adjustment is needed.","tokens_in":18566,"tokens_out":5382,"duration_ms":59655,"concrete_test":"Analytical check: instantiate Eq. (71) for n=2 with the normalized states |ψ_+>=(|00>+|11>)/√2 and |ψ_->=(|00>-|11>)/√2, using any finite set of angles θ. For both states, each single-qubit reduced density matrix is I/2, so every entry of the right-hand side — including p_0^(θ) and p_1^(θ) for every θ — is exactly 1/2. The two resulting nonlinear systems are therefore identical. Run any numerical solver, e.g., Newton's method, on the common system: it will either fail due to a singular Jacobian or return a solution consistent with both states. This directly demonstrates non-identifiability, since a successful estimator would have to output different amplitudes (e.g., a_11=1/√2 versus a_11=-1/√2) from identical data.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the right-hand side of Eq. (71) — all single-qubit outcome probabilities — uniquely determines the 2^{n+1}-2 real parameters of an n-qubit pure state. This fails for a structural reason. For any measurement on qubit k in basis {|θ_0>, |θ_1>}, the outcome probability is Tr((|θ_s><θ_s| ⊗ I_{≠k}) |ψ><ψ|) = <θ_s| ρ_k |θ_s>, where ρ_k is the single-qubit reduced density matrix of qubit k. Therefore every probability appearing in Eq. (71) is a function only of the n reduced density matrices, i.e., at most 3n real parameters, not the 2^{n+1}-2 parameters of the full pure state. Concretely, for n=2, the states |ψ_+>=(|00>+|11>)/√2 and |ψ_->=(|00>-|11>)/√2 both have ρ_1=ρ_2=I/2, so every single-qubit measurement outcome probability equals 1/2 for every basis and every angle θ. The Z-basis equations and the θ-dependent equations in Eq. (71) are identical for the two states; any solution to one system is a solution to the other. No estimator based solely on these outcomes can return different amplitudes. Theorem 1 additionally presupposes an invertible Jacobian at the true solution, but for these states the Jacobian is singular because the equations do not distinguish the two solutions. The paper's own closing admission that |J^{-1}| is not proven to be O(1) is thus not merely a technical gap: in a central regime the Jacobian is not invertible at all.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a framework to estimate all amplitudes of an n-qubit pure state from measurements of a single qubit at a time, in the computational basis and in M = ceil(2^n - n) arbitrary rotated bases. The measurement probabilities are assembled into a system of nonlinear algebraic equations (Eq. 71), and the author argues that solving this system yields approximations to all real and imaginary parts of the amplitudes. The paper derives sample complexity bounds for three error metrics: O(4^n/δ^4) for maximum absolute norm error, O(6^n/δ^4) for total variation, and O(2^n/δ^4) for average L1 error, all under the assumption that the inverse Jacobian norm |J^{-1}| is O(1).","tokens_in":18991,"tokens_out":5585,"duration_ms":49796,"significance":"If the central claim were true, it would imply that full pure-state tomography is possible from single-qubit marginals, which would be a striking result. The manuscript is clearly written in parts and explicitly states its main assumption about the Jacobian. However, the claim fails on information-theoretic grounds: every probability obtained from a single-qubit measurement is a function only of the measured qubit's reduced density matrix, so the equations in Eq. (71) cannot distinguish states with identical single-qubit reduced states. The numerical complexity claims are therefore not established, and the proposed method cannot work as stated.","major_comments":[{"comment":"The central claim is information-theoretically impossible. For any single-qubit measurement on qubit k in basis {|θ_0>,|θ_1>}, the outcome probability is Tr((|θ_s><θ_s| ⊗ I_{≠k})ρ) = <θ_s|ρ_k|θ_s>, where ρ_k is the reduced density matrix of qubit k. Hence every right-hand side in Eq. (71) depends only on the n single-qubit reduced density matrices, i.e., on at most 3n real parameters, whereas a pure n-qubit state has 2^{n+1}-2 real parameters. Concretely, for n=2 the states (|00>+|11>)/√2 and (|00>-|11>)/√2 have identical reduced density matrices (each equal to I/2), so every single-qubit measurement outcome probability is 1/2 in every basis; the equation systems (Eq. 7 or Eq. 71) are identical for the two states and cannot lead to different amplitude estimates. This contradicts the abstract's claim that arbitrary-basis single-qubit outcomes 'can be used to assist the finding of amplitudes.' It also implies that the Jacobian of the system at these states is singular, so the invertibility premise of Theorem 1 fails exactly in this regime.","section":"II.A, Eq. (71); Algorithm 1"},{"comment":"The equation count is off by a factor of two. Each single-qubit measurement basis yields two probabilities p0 and p1 with p0+p1=1, so only one independent equation per basis. With n Z-basis measurements and M = ceil(2^n - n) rotated-basis measurements, Algorithm 1 produces at most n + M = 2^n independent equations, not the 2^{n+1} equations needed to determine the 2^{n+1} real variables. The displayed system (Eq. 71) lists 2n + 2M equations, but half are redundant. For the n=2 example, Eq. (7) has four rows but only three independent rows (since p0^1+p1^1=1 and p0^2+p1^2=1), and the claim of a 'square linear system' in the 3-qubit case (Eq. 26) similarly includes redundant rows. The system is therefore underdetermined by a factor of two, and no unique solution can be guaranteed.","section":"II.B, Eq. (71); Algorithm 1"},{"comment":"The perturbation bound in Theorem 1 is not valid as stated. The proof replaces f_i(˜x) by its first-order Taylor expansion around x and then treats J(˜x - x) = ˜b - b as an exact identity. Taylor's theorem only guarantees f_i(˜x) = f_i(x) + ∇f_i(x)·(˜x-x) + O(|˜x-x|^2), so the omitted remainder can be arbitrarily large unless a uniform bound on the second derivatives is established. No such bound is given, and the statement |˜x - x| ≤ |J^{-1}| |˜b - b| does not follow. Because all three sample-complexity claims (Eqs. (122), the total-variation bound in II.B.2, and the average-L1 bound in II.B.3) rely on Eq. (90), this is a load-bearing gap in the derivation.","section":"II.B, Theorem 1 and Eq. (90)"},{"comment":"The manuscript explicitly acknowledges in the conclusion that 'we are not able to prove' that |J^{-1}| = O(1) and that the Jacobian behavior 'can only be revealed numerically.' Since the headline measurement counts O(4^n/δ^4), O(6^n/δ^4), and O(2^n/δ^4) are all derived after dropping the |J^{-1}| factor, these complexity claims are conditional on an unproven and, as shown in the first major comment, sometimes false assumption. For states with all single-qubit marginals equal to I/2, the Jacobian is singular, so |J^{-1}| is infinite and the bounds provide no guarantee. The complexity results are therefore not established.","section":"III. Conclusion"}],"minor_comments":[{"comment":"The text uses 'complex variables' where 'real variables' is meant: a 2^n-dimensional complex amplitude vector has 2^{n+1} real degrees of freedom, not 2^{n+1} complex variables; this should be corrected.","section":"Throughout (e.g., after Eq. (70) and II.B)"},{"comment":"Algorithm 1 says to 'measure the first qubit O(2^n - n) times'; this should read 'measure the first qubit in O(2^n - n) different bases' or 'for O(2^n - n) values of θ', because copying the same measurement setting is not what is intended.","section":"Algorithm 1"},{"comment":"There are several typographical errors, e.g., 'Algerbraic' in the Theorem 1 heading and 'the back equation 7, 14' in Section II.B; these should be corrected.","section":"II.B (Theorem 1 heading) and II.B (after Eq. (7))"},{"comment":"The notation for probabilities is inconsistent: p0/p1 is reused for the first and second qubit in Section II.A (leading to the superscript notation introduced there), but the superscript is omitted in Eq. (71) for the Z-basis probabilities. Clarifying this would improve readability.","section":"II.A and Eq. (71)"}],"recommendation":"reject","confidential_remarks":"This is a clear case of a central claim that cannot be true for information-theoretic reasons. The author's framework would, if correct, constitute a major breakthrough, but the analysis does not overcome the basic fact that single-qubit marginals do not determine a pure state. I see no sign of bad faith; the errors are honest. However, the manuscript cannot be fixed by a revision within its current scope unless the author restricts the claims to quantities determined by single-qubit marginals (e.g., single-qubit reduced density matrices) or uses additional correlated measurements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Candidly: this paper is not close to correct. The central claim—that arbitrary single-qubit measurements can determine all amplitudes of an n-qubit pure state—is false, and the paper's own equations show why. Every probability in Eq. (71) is of the form Tr(|θ><θ|⊗I_{≠k} |ψ><ψ|), which depends only on the one-qubit reduced density matrix of the measured qubit. For the two-qubit states |ψ_+>=(|00>+|11>)/√2 and |ψ_->=(|00>-|11>)/√2, all single-qubit reduced states are I/2, so every single-qubit outcome probability is 1/2 for every basis and every angle. The full equation system is identical for these two states, so no solution procedure can tell them apart. The paper claims to recover the complex amplitudes in the Z basis; that is information-theoretically impossible from these marginals.\n\nWhat is genuinely new here is modest: the explicit angle-dependent equations for single-qubit probabilities are not in the cited tomography papers, and the idea of building a nonlinear system from them is at least clearly presented. The author also deserves credit for stating up front that the Jacobian norm |J^{-1}| is unproven, and for noting that the approach would need it to be O(1). That honesty does not rescue the argument; it just identifies one more unsupported step.\n\nThe soft spots beyond the central impossibility are numerous and serious. The equation count ignores that p0+p1=1 makes half of the probabilities redundant, so the system has far fewer independent equations than variables. Theorem 1's proof is a linearization that drops higher-order terms and then states a bound as if it were exact; it also assumes the Jacobian is invertible at the solution, which fails for states with degenerate marginals. The sample-complexity claims all carry the unproven |J^{-1}| factor, so the displayed O(4^n/δ^4), O(6^n/δ^4), and O(2^n/δ^4) scalings are conditional on the very assumption that is most in doubt.\n\nThis is not a paper for a serious referee. It is a worked example of why single-qubit marginals are insufficient for full state reconstruction, though the author does not draw that conclusion. A desk reject is appropriate; if the author wants to pursue the idea, the correction is to acknowledge that only the reduced density matrices can be estimated, and the nonlinear system is overdetermined for that smaller goal.","headline":"The central claim fails: single-qubit measurement probabilities depend only on each qubit's reduced density matrix, so states like (|00>+|11>)/√2 and (|00>-|11>)/√2 are indistinguishable no matter how many angles are used.","tokens_in":19439,"tokens_out":3514,"would_cite":false,"duration_ms":36063,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P68","81P15"],"pacs":["03.65.Wj","03.67.-a"],"model":"deepseek-v4-flash","headline":"The paper claims that measuring one qubit in arbitrary bases recovers every complex amplitude of an n-qubit pure state via a nonlinear equation system.","keywords":["quantum state tomography","single-qubit measurement","amplitude estimation","nonlinear algebraic equations","Born rule","sample complexity","measurement basis","pure state recovery"],"falsifier":"Measure the two-qubit states $(|00\\rangle + |11\\rangle)/\\sqrt{2}$ and $(|00\\rangle - |11\\rangle)/\\sqrt{2}$ through the paper's procedure: for either qubit and in every basis, the two outcome probabilities are $1/2$ and $1/2$ for both states, so the right-hand sides of the equation system coincide exactly even though the amplitude $a_{11}$ differs in sign. Writing out the equations for these two states and observing that the systems are identical would settle whether single-qubit outcomes can separate them.","tokens_in":18354,"feed_emoji":"⚛️","tokens_out":13572,"duration_ms":124075,"temperature":0.7,"pith_summary":"The paper proposes that all complex amplitudes of an unknown $n$-qubit pure state $|\\psi\\rangle = \\sum_{i=0}^{2^n-1} a_i |i\\rangle$ can be recovered by measuring only one qubit at a time, as long as the measurement basis can be chosen freely. The outcomes are turned, through the Born rule, into a system of nonlinear algebraic equations whose unknowns are the real and imaginary parts of every amplitude, and solving that system classically yields the full state. The author derives sample counts for three error criteria: $O(4^n/\\delta^4)$ single-qubit measurements for all amplitude norms to additive error $\\delta$, $O(6^n/\\delta^4)$ for total-variation distance $\\delta$, and $O(2^n/\\delta^4)$ for average $L_1$ error. If the framework holds, an arbitrary pure state would be fully characterizable from its single-qubit marginals alone, without ever measuring qubits jointly.","feed_headline":"One qubit's outcomes claimed to reveal every amplitude","feed_subtitle":"Full tomography from one qubit's outcomes, with sample counts scaling as 4^n, 6^n, or 2^n.","key_machinery":"The object that carries the argument is the system of nonlinear algebraic equations built from Born-rule probabilities of single-qubit measurements. A measurement of one qubit in the basis $\\{|\\theta\\rangle_0, |\\theta\\rangle_1\\}$, with $|\\theta\\rangle_0 = \\cos\\theta|0\\rangle + \\sin\\theta|1\\rangle$ and $|\\theta\\rangle_1 = \\sin\\theta|0\\rangle - \\cos\\theta|1\\rangle$, yields one equation per outcome expressing a sum of rotated-amplitude moduli squared as an estimable probability; the Z-basis measurements supply the marginal constraints. Together these give $2^{n+1}$ equations in the $2^{n+1}$ variables $\\mathrm{Re}(a_i), \\mathrm{Im}(a_i)$. The second load-bearing mechanism is Theorem 1, which bounds the shift of the solution when the right-hand sides are noisy, converting the per-probability estimation error $\\epsilon$ into the sample-complexity claims through the Jacobian norm $\\|J^{-1}\\|$.","core_discovery":"On its own terms, the paper's central claim is that measurement outcomes from a single qubit, taken in the computational (Z) basis on every qubit and then in $M = \\lceil 2^n - n \\rceil$ additional rotated bases on the first qubit, determine every amplitude $a_i$, real and imaginary parts included. Each rotated-basis outcome contributes equations of the form $\\sum_{i_1,...,i_{n-1}} |\\cos\\theta\\, a_{0i_1...i_{n-1}} \\pm \\sin\\theta\\, a_{1i_1...i_{n-1}}|^2 = p^{(\\theta)}_{0/1}$, and the Z-basis outcomes contribute marginal equations; together they form a square system of $2^{n+1}$ equations in the $2^{n+1}$ real unknowns. The paper proves an error-propagation bound (Theorem 1): when the right-hand sides are estimated from finitely many measurements, the solution error satisfies $|\\tilde{x} - x| \\le \\|J^{-1}\\|\\,|\\tilde{b} - b|$, with $J$ the Jacobian of the system. Assuming $\\|J^{-1}\\| = O(1)$, it then derives the three stated measurement counts, and concludes that joint multi-qubit measurement is not asymptotically stronger than single-qubit measurement for this task.","pith_inferences":["The uniqueness of the solution is an implicit assumption the paper does not examine: two two-qubit states with identical single-qubit marginals, such as $(|00\\rangle + |11\\rangle)/\\sqrt{2}$ and $(|00\\rangle - |11\\rangle)/\\sqrt{2}$, produce identical right-hand sides for every single-qubit basis, so any recovery claim must confront this degeneracy.","All three sample-complexity formulas carry a hidden factor $\\|J^{-1}\\|^2$; the paper concedes it cannot prove $\\|J^{-1}\\| = O(1)$ and expects the behavior to be revealed numerically, so the advertised counts are conditional on that norm staying bounded.","A natural extension would be to count classical operations as well: solving $2^{n+1}$ coupled nonlinear equations may dominate the quantum measurement cost, and choosing the angles $\\theta$ to minimize the Jacobian condition number is an unexplored optimization problem.","The framework raises an information-theoretic question it does not answer: which discrete sets of single-qubit bases are sufficient, and minimal, for a pure state to be determined by its marginals, since the choice $M = \\lceil 2^n - n \\rceil$ is proposed but not proven minimal."],"forward_implications":["If the framework is sound, a pure state can be characterized without ever measuring two qubits jointly, relaxing the hardware requirements of conventional multi-qubit tomography.","The claimed counts set the trade-offs: additive amplitude error costs $O(4^n/\\delta^4)$, total-variation error $O(6^n/\\delta^4)$, and average $L_1$ error $O(2^n/\\delta^4)$, which for the last metric matches the naive all-qubit baseline under the $\\|J^{-1}\\| = O(1)$ condition stated in the paper.","The construction makes the choice of measurement basis the source of extra information, so the framework frames basis selection, not joint measurement, as the resource that yields full state recovery.","The paper draws the corollary that joint measurements are not asymptotically stronger than single-qubit ones for amplitude estimation, since any joint measurement would merely generate a different set of equations of the same kind."],"supporting_citations":[{"why":"Provides the prior single-qubit Pauli-measurement tomography protocol whose O(10^n/δ^2) trace-distance complexity is the main comparison for the total-variation result.","marker":"[9]"},{"why":"The sample-optimal full tomography protocol that supplies the multi-qubit measurement baseline for the total-variation comparison.","marker":"[13]"},{"why":"The earlier study of collective versus local measurement in qubit estimation, cited alongside [9] as prior single-qubit Pauli-measurement work.","marker":"[10]"},{"why":"The learning-based method for estimating linear properties of a density matrix that motivates framing measurement outcomes as statistical estimates.","marker":"[7]"}],"fun_headline_variants":["Single-qubit measurements enough to estimate all amplitudes","Nonlinear equations from one qubit give all amplitude estimates","Tomography with only one qubit measured, then classical solving","All amplitudes from single-qubit bases via algebraic system","Estimating n-qubit amplitudes with a single measured qubit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework assumes that the probabilities obtained by measuring a single qubit in arbitrary bases carry enough information to uniquely determine all complex amplitudes, so that the nonlinear system has the true amplitudes as its only solution.","fun_headline_variants_meta":{"raw":{"variants":["Single-qubit measurements enough to estimate all amplitudes","Nonlinear equations from one qubit give all amplitude estimates","Tomography with only one qubit measured, then classical solving","All amplitudes from single-qubit bases via algebraic system","Estimating n-qubit amplitudes with a single measured qubit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000399,"raw_usage":{"total_tokens":2217,"prompt_tokens":1205,"completion_tokens":1012,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":821,"completion_tokens_details":{"reasoning_tokens":932}},"tokens_in":821,"tokens_out":1012,"duration_ms":10334,"temperature":1.0,"reasoning_tokens":932,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T19:06:04.015449+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the two-qubit states $(|00\\rangle + |11\\rangle)/\\sqrt{2}$ and $(|00\\rangle - |11\\rangle)/\\sqrt{2}$ through the paper's procedure: for either qubit and in every basis, the two outcome probabilities are $1/2$ and $1/2$ for both states, so the right-hand sides of the equation system coincide exactly even though the amplitude $a_{11}$ differs in sign. Writing out the equations for these two states and observing that the systems are identical would settle whether single-qubit outcomes can separate them.","supporting_citations":[{"cited_title":"Efficient estimation of pauli observables by derandomization.Physical review letters, 127(3):030503, 2021","cited_arxiv_id":null,"evidence_quote":"The sample-optimal full tomography protocol that supplies the multi-qubit measurement baseline for the total-variation comparison."}],"review_version":1}