{"id":"609aabd4-3423-4dcd-bb77-db6db1cb874d","arxiv_id":"2608.04916","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Every two-qubit state obeys the complementary quantum correlation inequality, so two mutually unbiased measurement tables provide a tomography-free lower bound on total correlation.","lead":"This paper proves that for any two quantum bits, the correlation seen in two complementary measurements never overestimates the total correlation present before measurement. This turns a decade-old conjecture into a theorem and gives a simple way to certify entanglement and quantum communication rates from just two measurement tables.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified. The concavity certificate for Eq. (2) survives re-derivation, and the oriented pairing plus cross-pinching closes correctly.","rationale":"The reader's weakest-assumption analysis correctly identified the qubit half-information exclusion lemma as the load-bearing piece. I checked that piece in detail: the entropy identities are exact, the Hessian of g is computed with the correct signs, the Schur complement is legitimate because the relevant matrices are positive definite in the open ball, and the proof of Q >= 0 is complete once delta >= 1/3 is established. I also verified the two main summation steps: the oriented inequalities in Eq. (11) both supply the same crossed mutual information I(X_A:Z_B), and the data-processing inequality in Eq. (12) has the right direction to close the budget when added. The boundary extension by continuity is valid because g is continuous on the closed unit ball. No numerical or executable evidence is used as a premise, so the proof-dependency audit is accurate. The paper's scope caveats about trusted qubit dimensions and calibrated measurements are appropriate and do not undermine the theorem. Since the central argument holds under scrutiny, the reader's ACCEPT verdict should stand without modification.","tokens_in":8987,"tokens_out":18824,"duration_ms":153396,"concrete_test":"Use a computer algebra system to symbolically verify Eq. (43) starting from Eq. (39), then evaluate Q numerically over a dense grid of (x,y,z) in the Bloch ball with delta defined as in Eq. (7); if any grid point gives a negative determinant or a negative diagonal entry, the concavity certificate in Eqs. (33)-(43) fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I re-checked the load-bearing premise identified by the reader, the half-information exclusion lemma in Eq. (2), and found it sound. The Hessian reduction is internally consistent: for the open Bloch ball, R and D_2 are positive definite; the Schur complement Q in Eq. (39) has nonnegative diagonal entries; and the exact determinant decomposition in Eq. (43) is nonnegative under delta >= 1/3, with the two cases delta >= 1/2 and 1/3 <= delta < 1/2 both handled correctly. The scalar series bound in Eq. (41) indeed gives delta >= 1/3. Equation (29) is exactly Jensen's gap for g, and the extension from the open ball to the closed ball via (1-epsilon) scaling is valid because g is continuous on the closed ball. The oriented pairing in Eqs. (46)-(48) and the cross-pinching inequality in Eq. (54) have the correct signs; adding them yields Eq. (55). I found no circular step, no unsupported assumption, and no omitted proof in the central derivation. The operational corollaries follow from the proved inequality with standard hashing and channel-coding bounds, and the paper states their trusted-qubit scope. The only residual risk is a possible transcription error in a long algebraic expression, which is a low-probability contingency rather than a demonstrated flaw.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves the complementary-quantum-correlation (CQC) inequality for all two-qubit states: for any pair of mutually unbiased local bases, I(X_A:X_B) + I(Z_A:Z_B) <= I(A:B). The proof introduces a half-information-exclusion lemma for arbitrary ensembles of qubit states, chi(C:Q) - I(C:X) >= (1/2) I(C:Z), and establishes it by proving concavity of g(x,y,z) = h(sqrt(x^2+y^2+z^2)) - h(x) - h(z)/2 on the Bloch ball. The concavity certificate is an exact Hessian computation reduced by a Schur complement to a 2x2 matrix whose positivity follows from the scalar bound delta >= 1/3. Two copies of the lemma, oriented oppositely on the two parties, are combined with a relative-entropy data-processing inequality for local dephasings, yielding the main inequality. The paper then derives operational corollaries: S_XZ is a state-independent lower bound on I(A:B), S_XZ > 1 certifies distillable entanglement and a one-way entanglement-distillation rate at least S_XZ - 1, the Choi-state version lower-bounds the quantum capacity of a qubit channel, and two entropic uncertainty relations follow. Appendices contain full derivations and a proof-dependency audit.","tokens_in":9237,"tokens_out":31344,"duration_ms":256314,"significance":"The result closes the qubit case of the CQC conjecture and identifies the dimension boundary at d = 2, complementing the cited higher-dimensional counterexamples. This is an original, self-contained analytic proof rather than a numerical or conjectural contribution. Strengths include the explicit Hessian/Schur-complement certificate, the continuous extension to the closed Bloch ball, the exact decomposition of the determinant, and the proof-dependency audit stating that executable checks are not premises. The operational corollaries are concrete: two 2x2 correlation tables provide a tomography-free lower bound on total correlation, entanglement certification, and a channel-capacity witness in a trusted-qubit model. The proof technique, combining a curvature-based exclusion bound on one system with data processing, is likely to be reusable. I found no load-bearing errors.","major_comments":[],"minor_comments":[{"comment":"The formula for delta should be typeset as (1 - r/arctanh r)/r^2 so that it is not misread as 1 - (r/arctanh r)/r^2; the Schur-complement and series arguments depend on this definition.","section":"Eq. (7) and Eq. (38)"},{"comment":"The statement that the coefficient 1/2 is 'precisely sufficient' could be sharpened by explaining that two oppositely oriented applications of the half-information lemma account for the full crossed correlation I(X_A:Z_B), while the data-processing bracket closes the remaining budget.","section":"Physical implications"},{"comment":"The companion paper [10] is cited for the d >= 3 counterexamples, but its overcounting mechanism is not described; a sentence summarizing that mechanism would help readers appreciate why the qubit proof is not generic.","section":"References"},{"comment":"The notation h(t) = h_2((1+t)/2) makes h(0) = 1; since this value is used in the Jensen-gap identity and the Hessian at the origin, a brief reminder where h is introduced would improve readability.","section":"Appendix A, Eq. (24)"}],"recommendation":"accept","confidential_remarks":"For the editor: the manuscript is rigorous, self-contained, and the proof-dependency audit addresses reproducibility concerns. The companion paper [10] is not needed for the qubit proof, and its citation is appropriate. The AI-assistance disclosure is explicit and does not affect the technical content. I recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Best read here is the theorem, not the corollaries. The paper closes the two-qubit case of the CQC conjecture, which has been numerically probed for a decade, and it does so with a parameter-free analytic proof. The key move is concavity of the function g on the Bloch ball, established by a Hessian calculation that reduces to a Schur complement with a clean determinant decomposition. I re-derived the main lines (Eqs. 33–43); they check out. The oriented pairing in Eqs. 46–48 and the data-processing inequality in Eq. 54 close the sum correctly. The half-information exclusion lemma in Eq. 2 is the load-bearing piece and it looks sound.\n\nWhat is genuinely new: the universal statement over all two-qubit states, not just pure states or structured families. The proof also explains why d=2 is special: binary spectrum gives a curvature bound δ≥1/3, exactly enough for the factor 1/2. That explanation is satisfying and matches the known d≥3 counterexamples.\n\nThe operational corollaries are decent but secondary. The SXZ lower bound on QMI and SXZ−1 distillable-entanglement rate are straightforward consequences; the capacity lower bound is just coherent information of the Choi state. The paper is honest about scope: trusted qubit spaces, calibrated MUBs, not device-independent. Good.\n\nSoft spots, in proportion. First, no machine-checked proof. The Hessian algebra is long enough that a transcription error is conceivable; the stress-test note offers the same caveat. That is a low-probability risk, not a demonstrated gap. Second, the sharp-boundary claim depends on the companion d≥3 counterexample paper; this is fine, but it is a dependency beyond the present proof. Third, the corollaries are deterministic; finite-sample corrections are left to future work. Minor.\n\nThe citation pattern looks fair. The proof-dependency audit is a useful addition, and the executable checks are explicitly marked as non-premises. No data or fitted constants appear anywhere.\n\nWho is this for: quantum information theorists working on entropic inequalities and dimension-dependent complementarity; experimental groups wanting a two-measurement lower bound on correlation in qubit systems. It deserves a serious referee; the central inequality is important and the proof is detailed enough to verify. I would send it to peer review. My verdict: accept if the algebra is independently checked; I do not expect major conceptual revision.","headline":"A parameter-free analytic proof of the two-qubit CQC conjecture; the concavity certificate survives re-derivation and the corollaries are honest lower bounds.","tokens_in":9741,"tokens_out":2706,"would_cite":true,"duration_ms":21226,"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":"Two complementary tables bound every qubit pair's total correlation.","keywords":["complementary quantum correlations","quantum mutual information","qubits","information exclusion","mutually unbiased bases","entanglement certification","quantum channel capacity","Bloch ball"],"falsifier":"Compute, to high precision, the classical mutual informations $I(X_A:X_B)$ and $I(Z_A:Z_B)$ and the quantum mutual information $I(A:B)$ for any two-qubit state with Pauli measurements; a single state where $S_{XZ}$ exceeds $I(A:B)$ would falsify the central claim. Equivalently, search the Bloch ball for an ensemble of vectors $\\{v_c\\}$ and probabilities for which $g(\\bar{v})-\\sum_c p_c g(v_c)$ is negative, since that would violate the concavity lemma underpinning the proof.","tokens_in":8794,"feed_emoji":"⚛️","tokens_out":8427,"duration_ms":71539,"temperature":0.7,"pith_summary":"The paper claims that for every two-qubit state, the sum of the classical mutual informations obtained from two complementary local measurements never exceeds the quantum mutual information of the unmeasured state, so two two-by-two correlation tables give a tomography-free lower bound on total correlation. This matters because most experiments access only a few measurement settings, and the result says the qubit case is the one dimension where simply adding complementary readouts is safe. The proof traces the protection to a single-qubit half-information exclusion lemma: information lost by reading one Pauli axis pays for at least half of the information visible along the complementary axis. That factor one half comes from concavity of a binary-entropy function on the Bloch ball, and data processing joins the two parties' tradeoffs into the full correlation budget. If correct, the same score $S_{XZ}>1$ certifies entanglement, a one-way entanglement-distillation rate at least $S_{XZ}-1$, and a lower bound on the quantum capacity of a qubit channel.","feed_headline":"Two complementary tables bound every qubit pair's total correlation.","feed_subtitle":"Two two-by-two tables replace tomography for qubit correlations, entanglement, and channel capacity.","key_machinery":"The load-bearing object is the Bloch-ball function $g(x,y,z)=h(\\sqrt{x^2+y^2+z^2})-h(x)-\\frac{1}{2}h(z)$, where $h(t)$ is the binary entropy of a qubit with Bloch coordinate $t$. For any ensemble of Bloch vectors, the half-information exclusion inequality is exactly Jensen's inequality for $g$, because the difference $\\chi(C:Q)-I(C:X)-\\frac{1}{2}I(C:Z)$ equals $g(\\bar{v})-\\sum_c p_c g(v_c)$. The proof that $g$ is concave on the closed unit ball is a Hessian certificate: writing $-\\nabla^2 g = R-D$, a Schur complement reduces positivity to a two-by-two matrix whose nonnegative diagonal and determinant follow from the scalar bound $\\delta \\ge 1/3$. This concavity is what forces the factor one half, and it is the reason qubits are special: in dimension three and higher, the entropy is not controlled by a single radial coordinate, and complementary readouts can double-count a shared classical label.","core_discovery":"The central discovery is the universal validity for qubits of the complementary-quantum-correlation inequality $I(X_A:X_B)+I(Z_A:Z_B) \\le I(A:B)$, for any pair of mutually unbiased local bases, equivalently Pauli $X$ and $Z$ measurements. The paper proves it in full generality, with no rank, purity, symmetry, separability, or marginal assumptions, by establishing a qubit half-information-exclusion lemma: $\\chi(C:Q)-I(C:X) \\ge \\tfrac{1}{2}I(C:Z)$ for every ensemble of qubit states, then applying it in opposite orientations on the two parties. The two resulting deficits share the same crossed table $I(X_A:Z_B)$, so they pay for that table exactly once, while relative-entropy data processing under local dephasing accounts for the remaining quantum information. The theorem is tight in distinct regimes: a perfectly correlated classical bit attains $(I_X,I_Z,I)=(1,0,1)$, and a maximally entangled pair attains $(1,1,2)$.","pith_inferences":["The paper leaves open whether finite-sample and calibration corrections to $S_{XZ}$ can be stated as simple additive penalties; a natural extension is to develop such confidence intervals for the two-table score, since experiments will never have exact unbiasedness or infinite statistics.","The 'orient two deficits so they share the same crossed table' strategy suggests a general recipe: in any state space, find a curvature constant that lets one local readout charge the complementary correlation, then join opposite orientations by data processing. Whether qutrit subsets or restricted measurement families admit an analogous constant is a concrete open question.","Because leakage into local dimension three or higher restores overcounting, the score should in practice be paired with a dimension witness before being trusted; the paper does not develop that protocol, but its dimensional boundary makes such a pairing the natural experimental safeguard.","The same data that certify entanglement also give a one-way secret-key rate after entanglement distillation, so the two-table score could serve as a simple high-level certification tool for quantum key distribution with untrusted sources, assuming the trusted-qubit calibration holds."],"forward_implications":["For any two-qubit source, the experimentally accessible score $S_{XZ}=I(X_A:X_B)+I(Z_A:Z_B)$ lower-bounds $I(A:B)$, so two two-by-two correlation tables replace full tomography whenever a lower bound on total correlation suffices.","A score $S_{XZ}>1$ certifies that the state is entangled, and the coherent information in either direction is at least $S_{XZ}-1$, so a one-way entanglement-distillation rate at least $\\max\\{0,S_{XZ}-1\\}$ is achievable per copy.","For a qubit-input, qubit-output channel, applying the inequality to its normalized Choi state gives $Q(\\mathcal{N}) \\ge \\max\\{0,S_{XZ}(\\omega)-1\\}$, certifying a nonzero asymptotic quantum communication rate from paired measurements on transmitted Bell pairs.","Combining the inequality with the state-dependent qubit uncertainty relation yields the conditional bound $H(X_A|X_B)+H(Z_A|Z_B) \\ge 1+S(A|B)$ and the joint bound $H(X_AX_B)+H(Z_AZ_B) \\ge 2+S(AB)$.","The inequality is tight for a perfectly correlated classical bit and for a maximally entangled pair, pinning down the two physically distinct regimes in which the correlation budget closes exactly."],"supporting_citations":[{"why":"Supplies the operational interpretation of quantum mutual information as the asymptotic cost of erasing all correlations, which converts the score into an erasure-cost certificate.","marker":"[1]"},{"why":"Proposed the complementary-quantum-correlation relation and proved it for pure states and several structured mixed-state families, defining the conjecture this paper settles.","marker":"[7]"},{"why":"Fixed-purity numerical searches of two-qubit states found no violations, providing broad numerical evidence that the inequality holds.","marker":"[8]"},{"why":"Provides a sufficient condition covering additional two-qubit states, extending the partial evidence toward the full theorem.","marker":"[9]"},{"why":"Gives explicit counterexamples in every local dimension at least three, establishing the dimension boundary that makes the qubit case sharp and necessary.","marker":"[10]"},{"why":"Defines the Holevo information that bounds the classical information accessible from an ensemble, the central quantity in the half-information exclusion lemma.","marker":"[11]"},{"why":"Supplies relative-entropy data processing under local dephasing, the ingredient that joins the two one-sided tradeoffs into the full correlation budget.","marker":"[12]"},{"why":"Provides the hashing inequality that converts positive coherent information into achievable one-way entanglement-distillation rates.","marker":"[13]"},{"why":"Gives the quantum coding theorem that connects coherent information of the Choi state to a lower bound on quantum capacity.","marker":"[14]"}],"fun_headline_variants":["Qubit correlations: two tables replace tomography","Complementarity prevents overcounting in qubit pairs","Two local bases certify entanglement and capacity","Universal bound: qubit correlations from two settings","Tomography-free: two complementary measurement tables"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole result rests on the claim that the single-qubit function $g(x,y,z)=h(\\sqrt{x^2+y^2+z^2})-h(x)-\\tfrac{1}{2}h(z)$ is concave over the entire Bloch ball; if that concavity failed anywhere, the half-information tradeoff would not follow from Jensen's inequality, and the two-qubit inequality would lose its foundation.","fun_headline_variants_meta":{"raw":{"variants":["Qubit correlations: two tables replace tomography","Complementarity prevents overcounting in qubit pairs","Two local bases certify entanglement and capacity","Universal bound: qubit correlations from two settings","Tomography-free: two complementary measurement tables"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000286,"raw_usage":{"total_tokens":1679,"prompt_tokens":936,"completion_tokens":743,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":675}},"tokens_in":552,"tokens_out":743,"duration_ms":6716,"temperature":1.0,"reasoning_tokens":675,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:36:42.251574+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute, to high precision, the classical mutual informations $I(X_A:X_B)$ and $I(Z_A:Z_B)$ and the quantum mutual information $I(A:B)$ for any two-qubit state with Pauli measurements; a single state where $S_{XZ}$ exceeds $I(A:B)$ would falsify the central claim. Equivalently, search the Bloch ball for an ensemble of vectors $\\{v_c\\}$ and probabilities for which $g(\\bar{v})-\\sum_c p_c g(v_c)$ is negative, since that would violate the concavity lemma underpinning the proof.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the operational interpretation of quantum mutual information as the asymptotic cost of erasing all correlations, which converts the score into an erasure-cost certificate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Fixed-purity numerical searches of two-qubit states found no violations, providing broad numerical evidence that the inequality holds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides a sufficient condition covering additional two-qubit states, extending the partial evidence toward the full theorem."},{"cited_title":"Schneeloch, C","cited_arxiv_id":null,"evidence_quote":"Defines the Holevo information that bounds the classical information accessible from an ensemble, the central quantity in the half-information exclusion lemma."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies relative-entropy data processing under local dephasing, the ingredient that joins the two one-sided tradeoffs into the full correlation budget."},{"cited_title":"When Complementary Measurements Count the Same Classical Bit Twice: Counterexamples to CQC, ECQC, and Complementarity-Based Certification","cited_arxiv_id":"2608.03828","evidence_quote":"Gives the quantum coding theorem that connects coherent information of the Choi state to a lower bound on quantum capacity."}],"review_version":2}