{"id":"eaccdc85-3627-4d36-ab84-514fac009a74","arxiv_id":"2602.00595","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"An outer-approximation algorithm certified by support functions computes tight entropic uncertainty bounds for arbitrary finite-dimensional measurements, showing standard analytical bounds are often loose.","lead":"This paper introduces a numerical method that computes the tightest possible entropic uncertainty bounds for arbitrary quantum measurements by surrounding the quantum probability set with shrinking polytopes. It shows common analytical bounds are loose and uses the improved bounds to strengthen quantum steering detection.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Adaptive cutting-plane convergence is asserted, not established; without it the 'preassigned numerical precision' guarantee for Algorithm 1 is unsupported.","rationale":"The central claim is computational certification. The load-bearing condition is that the loop terminates with a small gap. The paper's convergence paragraph is a non-sequitur: it discusses exhaustion of directions, not the adaptive gradient directions actually used. Without a proof of convergence (or a complexity bound), the 'preassigned numerical precision' claim is not justified. This does not force rejection: the method may be correct, and Fig. 1 provides evidence for one instance; the issue is a missing proof that could be supplied. Hence a conditional verdict remains appropriate. On the Rényi point: the reader's first premise is not a load-bearing attack. Rényi α>1 is not concave, but its upper level sets are sublevel sets of the convex function p ↦ Σ p_i^α, so H_R^α is quasi-concave; the minimum over a polytope is still attained at a vertex. Thus the lower-bound argument can be repaired by replacing concavity with quasi-concavity. The real problem is convergence of the outer approximation, which the reader also flagged as the second load-bearing premise. The abstract's athermality and variance-recovery claims are unsupported in the body, but they are secondary to the main EUR claim.","tokens_in":10045,"tokens_out":23598,"duration_ms":265166,"concrete_test":"Using the repository code [38], run Algorithm 1 with Shannon entropy on a fixed qutrit three-measurement instance from Fig. 3 (e.g., a=0.3, φ=0) and independently compute h(E) to high accuracy via a dense search over the Bloch-sphere parameterization of qutrit states (or a global optimizer). Record h−, h+, and Gap for 10^2, 10^3, and 10^4 iterations. If Gap fails to fall below, say, 10^-6, or if h− saturates above the independent h(E) within tolerance, the claimed preassigned precision is not delivered. For a sharper failure search, also test a POVM whose quantum probability space Z has a flat face aligned so that repeated gradient cuts hit the same face; check whether convergence stalls.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central promise is that Algorithm 1 produces tight EUR lower bounds 'with a preassigned numerical precision' for arbitrary finite-dimensional POVMs. For this it is not enough that each cut is a valid outer approximation; the loop must terminate with Gap = h+ − h− ≤ ε. The only argument offered in the 'Convergence and discussion' paragraph is that as the set of included directions u grows, the outer polytope converges to P in the Hausdorff metric. This does not apply to the actual algorithm: u is not an exhaustive direction sequence but the gradient −∇H at the current outer-polytope minimizer. No proof is given that these adaptive normals expose every relevant face of Z, that h− → h(E), or that h+ → h(E). The upper bound h+ is obtained by projecting the outer minimizer through the ground state of Ω(−∇H); this is a heuristic, and even if P_k → Z, h+ could remain strictly above h(E), so Gap need not vanish. Thus the 'preassigned precision' claim has a genuine unproved step for all entropies considered. The Rényi concavity worry is less severe: Rényi α>1 is not concave, but it is quasi-concave on the simplex, so the vertex lower bound h− = min_{V_k} H remains valid; the proof of Eq. (25) is overstrong, but the lower-bound conclusion survives.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces a support-function-based outer-approximation method for computing h(E) = inf_ρ H(p_E(ρ)), the minimal entropy of an effective POVM obtained by concatenating several measurements, and applies it to entropic uncertainty relations (EURs) for Shannon, Tsallis, and Rényi entropies. The main steps are: an affine SVD reduction of the quantum probability space P(E) to a lower-dimensional convex set Z; the spectral identity σ_P(u) = λ_max(Σ_i u_i E_i) for the support function; and an iterative algorithm that maintains an outer polytope P_k ⊇ Z, obtains a lower bound h− by minimizing H over the vertices of P_k, an upper bound h+ from the ground state of the gradient observable Ω(∇H), and refines P_k with a support-function cutting plane. The authors claim that Algorithm 1 yields tight EUR bounds 'with a preassigned numerical precision', benchmark it against Maassen–Uffink, Coles–Piani, and majorization bounds for qutrit measurements, and use the tighter bounds to improve steering detection thresholds. The abstract additionally promises recovery of the variance-based uncertainty relations of [32] and certification of maximal athermality resources; neither of these appears in the body.","tokens_in":1962,"tokens_out":2139,"duration_ms":485702,"significance":"The reformulation h(E) = min_{z∈Z} H(s+Qz) and the oracle σ_P(u) = λ_max(Ω(u)) are clean and correct, and the sandwich structure—valid lower and upper bounds at every iteration—is a genuine strength: the method is parameter-free and the code is promised in the linked repository [38]. The comparisons in Figs. 2–4 provide useful numerical evidence that standard analytical and majorization bounds are not tight for the examined asymmetric qutrit settings. If the convergence and differentiability gaps identified below are resolved, this would be a broadly applicable computational tool for quantum resource theories. As written, however, the central certification claim is not established, the treatment of Rényi α>1 rests on a false concavity premise, and the abstract announces content missing from the body. The significance of the paper is therefore conditional on substantial revision.","major_comments":[{"comment":"The central claim that Algorithm 1 yields tight EUR bounds 'with a preassigned numerical precision' is not supported. The convergence paragraph argues that as the set of included directions u grows, the outer polytope converges to P in the Hausdorff metric. That is true for exhaustive direction sets, but the algorithm adds only the adaptive normal g = −∇H(p_poly) (lines 8 and 16–17; Eq. (27)); no proof is given that these adaptively generated normals expose all relevant faces of Z, nor that the upper-bound point p_real from the ground state of Ω(g) (line 10) satisfies H(p_real) → h(E). Even if P_k → Z in Hausdorff distance, h+ need not approach h(E), so the stopping criterion Gap ≤ ε is not guaranteed to be reached. The cut can also fail to remove the current vertex when that vertex attains the supporting-plane value min_P⟨g,·⟩ while remaining infeasible; no rule for this case is given,","section":"Outer approximation algorithm, 'Convergence and discussion', Algorithm 1"},{"comment":"Eq. (25) justifies the vertex lower bound by asserting 'the entropy function is concave', and the Introduction states that the uncertainty functional is concave in ρ. This is false for Rényi entropy with α>1, which is explicitly included in Eqs. (4) and (9). Rényi entropy is quasi-concave on the probability simplex for all α>0 (for α>1, Σ_i p_i^α is convex, so the superlevel sets are convex). The vertex lower bound remains valid under quasi-concavity, and the gradient-based cut inherits the needed first-order condition, so the results are repairable. However, the stated premise is incorrect as written, the derivation of Eq. (25) must be corrected, and the claim that Eq. (4) satisfies all of Deutsch's requirements should be qualified.","section":"Eqs. (4), (9), (25); Introduction"},{"comment":"The abstract announces two results that do not appear in the body: recovery of the exact variance-based uncertainty relations of [32] and determination of the maximal athermality resource from restricted measurements. The manuscript contains neither a variance-based UR computation nor any athermality analysis; the applications actually reported are Shannon/Tsallis EURs and steering thresholds (Figs. 2–4). The abstract and conclusion must be reconciled with the actual content, either by adding the missing sections or by removing these claims.","section":"Abstract"},{"comment":"The curves labeled 'qOptimal' are presented as the true quantum-optimal bounds, but the paper does not report the tolerance ε, the number of iterations, the vertex counts, or the converged gaps for the parameter scans behind these figures. Given that the certified-precision statement is unsupported (see first major comment), the numerical 'optimality' labels should be backed by the actual termination data. I also recommend an independent sanity check in a case where the exact bound is known (e.g., mutually unbiased bases, or the Schwonnek variance benchmark mentioned in the abstract).","section":"Figs. 2–4; 'Application and discussion'"}],"minor_comments":[{"comment":"The main text says 'A proof of Eq. (8) is provided in Section A' and cross-references are informal; use a consistent appendix citation and place it at Eq. (10) as well as Eq. (8).","section":"Eqs. (8), (10); Appendix A"},{"comment":"Reference [15] misspells 'Sánchez-Ruiz' as 'Sánches-Ruiz'; reference [21] lacks standard volume/page information and has an irregular DOI listing.","section":"References [15], [21]"},{"comment":"The statement that Eq. (4) 'satisfies all the requirements for an uncertainty measure proposed by Deutsch' needs qualification: for Shannon and Tsallis entropies concavity holds, but for Rényi α>1 only quasi-concavity is available (see major comment 2).","section":"Introduction, Eq. (4)"},{"comment":"The caption should state the tolerance ε used and clarify whether the red square marks the vertex minimizing H over the current outer polytope or the final returned point.","section":"Fig. 1"},{"comment":"The inequality with p^{(k)}_{ij} introduces assemblage notation without definition; readers need either a brief restatement or a precise pointer to the conventions of [13].","section":"Steering inequality (p. 5)"},{"comment":"The paper should specify the numerical implementation: vertex-enumeration library and method, floating-point tolerance for vertex computation, and the finite-gradient or subgradient rule used when the minimizing vertex has zero probability components (relevant also to major comment 1).","section":"Algorithm 1; implementation details"}],"recommendation":"major_revision","confidential_remarks":"The manuscript appears to be an incomplete version of the announced program: the abstract and title promise variance-based URs and athermality certification, but the body contains only entropic results. The main technical obstacle is the absence of a convergence/termination analysis for the adaptive cutting-plane sequence, together with the undefined gradient at boundary vertices; a one-dimensional test shows the cut is properly separating while the vertex is infeasible, and concavity forces h+ → h(E) in the limit, but no finite-iteration guarantee is given. I recommend requesting the source code for an implementation audit and, if possible, involving a referee with expertise in outer-approximation methods for concave minimization."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take after reading it through. The genuinely new thing is the reduction: any N-measurement entropic uncertainty functional built from Shannon/Tsallis entropies can be viewed as the entropy of a single effective POVM, so the problem becomes minimizing H(p_E(ρ)). That's a real simplification, and the dual/support-function reformulation with λ_max as the support oracle is the right kind of machinery, directly in the spirit of Schwonnek et al. The sandwich construction — vertices of the outer polytope for a lower bound and a ground-state projection for an upper bound — is sensible for concave entropies, and the numerical examples show it working in practice. I also like that they shipped code; that's reproducible evidence.\n\nThe soft spots are real, though. The 'preassigned numerical precision' claim is not backed by any convergence proof. The paragraph on convergence says that as the set of directions u grows, outer polytopes converge to P in Hausdorff metric. That's true for exhaustive directions, but the algorithm chooses directions adaptively as gradients at the current minimizer, and no argument shows those normals reveal all faces needed to drive h− and h+ together. The upper-bound step is a heuristic projection. So the central guarantee is asserted, not established. If the authors present it as a heuristic numerical tool, that's fine, but then they should not call it a certified bound.\n\nSecond, Eq. (25) relies on concavity of H. That's fine for Shannon and Tsallis, but Rényi α>1 is not concave. The lower bound may still survive because Rényi is quasi-concave, so the minimum over a polytope is still at a vertex, but the authors need to say that and prove it. Right now the claim as written is unsupported for part of their advertised domain.\n\nThird, the abstract advertises two things the body never delivers: recovering the variance-based relations of Schwonnek et al. and a maximal athermality resource application. I searched the text; neither appears. That's a mismatch that has to be fixed before this goes out.\n\nThere are also smaller issues: the entropy gradient can blow up when a polytope vertex hits zero probability, and vertex-enumeration complexity is not fully discussed. But those are minor.\n\nBottom line: this paper has a solid core and deserves a serious referee. I'd send it out, but ask the authors to either prove or explicitly drop the precision guarantee, correct the Rényi argument, and align the abstract with the actual content. For my own work, the h(E) reformulation is worth citing; I'd bring it to a reading group if someone wants to discuss cutting-plane methods in quantum info.","headline":"A clean new reduction plus a working numerical tool for entropic URs, undercut by an unproven convergence claim and an abstract that promises sections that aren't there.","tokens_in":10834,"tokens_out":6178,"would_cite":true,"duration_ms":63128,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","81P15","90C26"],"pacs":["03.65.Ta","03.67.-a"],"model":"deepseek-v4-flash","headline":"This paper claims that optimal entropic uncertainty relations for arbitrary finite-dimensional quantum measurements can be computed to any desired precision using a support-function outer-approximation of the quantum probability space.","keywords":["entropic uncertainty relations","quantum probability space","outer approximation","support function","POVM","quantum steering","concave minimization","resource certification"],"falsifier":"For a single qubit and a fixed POVM, compute the algorithm's lower bound with the alpha-generalized entropy (alpha = 2) and compare it to the exact minimum obtained by scanning all pure states over the Bloch sphere; if the algorithm's bound ever exceeds the exact minimum, the concavity premise fails. For the convergence claim, a single run on a small POVM where the gap stagnates above the chosen tolerance would falsify the universal 'preassigned precision' statement.","tokens_in":9951,"feed_emoji":"🎯","tokens_out":8151,"duration_ms":89009,"temperature":0.7,"pith_summary":"Entropic uncertainty relations bound the total uncertainty of several measurements by a state-independent number, but finding the optimal (tight) bound is a non-convex minimization generally believed hard. This paper claims that for any finite set of POVMs the optimal bound can be computed to any desired precision: it combines all measurements into a single effective POVM, maps the accessible probability distributions to a low-dimensional convex set, and approximates that set from outside by polytopes. The crucial move is that the boundary of the quantum probability set is available through largest-eigenvalue computations, which serve both to produce a cut that separates an invalid candidate from the true set and to produce a valid physical upper bound. The paper shows on examples that standard analytical and majorization-based bounds are loose for generic measurements, and that the computed tight bounds materially improve steering detection thresholds.","feed_headline":"Compute tight uncertainty bounds to any preset precision","feed_subtitle":"Outer-approximating the quantum probability set turns a hard search into spectral cuts and sharper steering tests.","key_machinery":"The load-bearing identity is sigma_P(u) = lambda_max(sum_i u_i E_i), which expresses the support function of the accessible probability set as an eigenvalue problem. The algorithm reduces the optimization to a low-dimensional Z-space via the singular-value decomposition of the affine Bloch map, then alternates between minimizing the entropy at vertices of an outer polytope (lower bound) and evaluating the entropy at the ground state of the gradient Hamiltonian (upper bound), adding a cutting plane from the support-function oracle until the gap closes.","core_discovery":"The central claim is that the map from states to outcome probabilities for any POVM spans a convex set whose support function is the largest eigenvalue of an effective observable, and that this oracle turns concave minimization over quantum states into an outer-approximation scheme. Iterating vertex checks and spectral cuts produces certified lower and upper bounds whose gap can be driven below any prescribed tolerance. The paper demonstrates the method on random POVMs in dimension 100 and on qutrit two- and three-measurement settings, where it outperforms known analytical bounds.","pith_inferences":["Because the computational cost is governed by the rank of the affine probability map, not the Hilbert-space dimension, the method may scale to high-dimensional systems with few outcomes; the paper only demonstrates this on one random example, so this remains a promising inference.","The support-function oracle could plausibly be adapted to certify boundaries of other quantum convex sets, such as separable states, by replacing the largest-eigenvalue step with a semidefinite relaxation; this extension is not in the paper.","The concavity restriction marks a real boundary of applicability: for the alpha-generalized entropy with alpha > 1 the vertex lower bound is not guaranteed, so a different lower-bounding mechanism would be needed to extend the method to those functionals.","If the computed bounds are as tight as the examples indicate, they could be used to benchmark or replace analytical EURs in quantum-cryptography security proofs, where every bit of tightness matters."],"forward_implications":["For any finite-dimensional POVM and any concave entropy in the listed family, the algorithm returns certified lower and upper bounds on the optimal uncertainty; the gap can be driven below any preassigned tolerance.","The known analytical and majorization-based entropic bounds are substantially non-tight for generic measurement settings, so previously reported EURs understate the actual uncertainty limit.","Tighter EURs directly lower the visibility threshold for steering detection: the qutrit example shows steerability certified against more white noise than the majorization-based threshold allows.","The method handles multiple measurements (N > 2) and asymmetric, non-ideal measurement settings without requiring new analytical formulas.","The same optimization framework is claimed to apply to other convex resources, such as certifying maximal athermality from restricted measurements."],"fun_headline_variants":["Tight quantum uncertainty bounds to any preset precision","Spectral cuts tighten quantum uncertainty bounds","Certified quantum limits from geometric optimization","Tight uncertainty relations via spectral cuts","Support functions unlock tight quantum measurement bounds"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The certified lower bound assumes the entropy is concave over the outer polytope; that holds for Shannon and Tsallis entropies but not for the alpha-generalized entropy with alpha > 1, and the paper's additional claim that the cutting planes converge to the exact boundary in Hausdorff metric is asserted without proof.","fun_headline_variants_meta":{"raw":{"variants":["Tight quantum uncertainty bounds to any preset precision","Spectral cuts tighten quantum uncertainty bounds","Certified quantum limits from geometric optimization","Tight uncertainty relations via spectral cuts","Support functions unlock tight quantum measurement bounds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000865,"raw_usage":{"total_tokens":3565,"prompt_tokens":703,"completion_tokens":2862,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":2799}},"tokens_in":447,"tokens_out":2862,"duration_ms":20940,"temperature":1.0,"reasoning_tokens":2799,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T05:58:38.505173+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a single qubit and a fixed POVM, compute the algorithm's lower bound with the alpha-generalized entropy (alpha = 2) and compare it to the exact minimum obtained by scanning all pure states over the Bloch sphere; if the algorithm's bound ever exceeds the exact minimum, the concavity premise fails. For the convergence claim, a single run on a small POVM where the gap stagnates above the chosen tolerance would falsify the universal 'preassigned precision' statement.","supporting_citations":[],"review_version":1}