{"id":"f4e6eba2-2dda-4759-9c13-4eff6bbc89bf","arxiv_id":"2507.05205","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Alternating minimization provably computes the doubly minimized Petz Rényi mutual information for all quantum states, with linear convergence for α∈(1,2] and O(1/n) convergence for α∈(1/2,1).","lead":"A quantum information paper proves that alternating minimization provably converges to the doubly minimized Petz Rényi mutual information for any finite-dimensional bipartite state, with explicit error rates for order parameters in (1/2,1) and (1,2]. This provides the first rigorous numerical recipe for a correlation measure that lacks a closed form and previously had no general algorithm.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the proofs are internally consistent; the only load-bearing external input is the joint concavity and uniqueness result of [17], which the paper cites rather than re-derives.","rationale":"The reader correctly identified reliance on [17] as the weakest external dependency, and I agree that this is the only place where the paper outsources a nontrivial analytic fact. However, I do not think this rises to a demonstrated flaw: citing a prior result is standard mathematical practice, and the needed concavity is a plausible consequence of known Lieb/Ando-type theorems. My own check of the derivations in Appendices C and D found no incorrect inequality sign, no missing support condition in the stated regimes, and no circular use of the main theorem. The contraction argument for alpha in (1,2] is self-contained apart from standard facts about Hilbert's projective metric and the fixed-point property of minimizers. The sublinear argument for alpha in (1/2,1) is more delicate, but all its steps, including the Fréchet derivative computation, the cancellation at (D.34)-(D.35), the Rényi-Pinsker bound, and the spectrum lower bound in Lemma 14, appear valid. I also checked the stopping criteria in Algorithms 1 and 2: they correctly use the proved bounds, with Algorithm 2 applying (3.20) to the increment x_{n-1} - x_n. The only caveat is that [17] is a preprint by the same author and is not re-derived here, so absolute certainty would require an independent verification of that result. This is a reason for moderate confidence, but not a reason to change the acceptance verdict.","tokens_in":29025,"tokens_out":19507,"duration_ms":208103,"concrete_test":"Independently verify the two properties imported from [17] for alpha in (1/2,1): (i) prove directly that (sigma_A, tau_B) maps to tr[rho_AB^alpha (sigma_A otimes tau_B)^(1-alpha)] is jointly concave, for example by applying the Lieb/Ando concavity theorem with A = sigma_A otimes I_B and B = I_A otimes tau_B; and (ii) prove strict convexity of the objective, or strict concavity of log Q_alpha, to justify uniqueness of the global minimizer. If both hold, Theorem 4 and the proposed algorithm stand; if either fails, the sublinear convergence claim collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined the main argument in good faith and found no internal error. For alpha in (1,2], Theorem 1 follows from Lemma 9 and the Birkhoff-Hopf theorem, with the contraction coefficient gamma = |1 - 1/alpha|; Corollaries 2 and 3 then follow by the fixed-point property of the global minimizer and the Hilbert-projective-metric estimate. For alpha in (1/2,1), Theorem 4 is also internally consistent: the convexity inequality at eq. (D.32) is applied in the correct direction, the derivative cancellations at (D.34)-(D.35) use exactly the first-order optimality of the alternating iterates, and the subsequent bounds via the Rényi Pinsker inequality and Lemma 14 are valid. The one genuinely load-bearing step not proved in the present text is the joint concavity of Q_alpha(rho_AB || sigma_A otimes tau_B) for alpha in (1/2,1), equivalently joint convexity of f, together with uniqueness of the global minimizer for that range; this is imported from the same author's preprint [17]. If that external result contained a gap, Theorem 4 and Algorithm 2 would need revision. I found no reason to think it does, and no circular dependence on the present paper.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies alternating minimization of the Petz divergence D_α(ρ_AB∥σ_A⊗τ_B) over states σ_A and τ_B, and proves convergence of the objective values to the doubly minimized Petz Rényi mutual information I_α^{↓↓}(A:B)_ρ. For α∈(1,2], it establishes a contraction of the alternating maps in Hilbert's projective metric with coefficient γ=|1−1/α| (Theorem 1), and from this derives linear convergence of the iterates (Corollary 2) and of the objective values at rate O(γ^{2n}) (Corollary 3), together with a termination-guaranteed algorithm. For α∈(1/2,1), it proves sublinear convergence of the objective values with an explicit O(1/n) bound (Theorem 4), again with an algorithm. The proofs are given in full in the appendices, and Appendix A recovers and extends the classical results of [22].","tokens_in":29282,"tokens_out":14231,"duration_ms":153856,"significance":"If the results are correct, they fill a genuine gap: no closed form is known for the doubly minimized PRMI, and prior non-asymptotic convergence analyses were limited to classical-classical states. The linear-rate result for α∈(1,2] is a clean quantum extension of the classical Hilbert-metric contraction argument, and the sublinear O(1/n) result for α∈(1/2,1) is the first quantum non-asymptotic guarantee in that parameter range. The paper is carefully written and provides explicit constants, finite-horizon termination statements, and complete appendix proofs; Remark 3 usefully identifies why the same approach fails for α<1/2. The main caveat is that the α<1 result depends on two load-bearing properties imported from the author's own unpublished preprint [17].","major_comments":[{"comment":"Theorem 4 is load-bearing on external results from the author's preprint [17] (arXiv:2406.01699): the inequality at (D.32) uses the joint convexity of f(σ_A,τ_B)=−Q_α(ρ_AB∥σ_A⊗τ_B), and the equality at (D.27)-(D.28) uses the uniqueness of the global minimizer stated in (2.11). Neither property is proved or stated as a precise theorem in this manuscript, and [17] is an unpublished preprint by the same author. Since the entire α∈(1/2,1) convergence claim rests on these facts, the manuscript should either include their statements with proofs or cite a peer-reviewed version; as it stands, Theorem 4 is conditional on [17]. The analogous fixed-point property used in (C.14)-(C.15) for the α∈(1,2] results is also cited from [17] and should be given the same treatment.","section":"Section 3B, Appendix D, Eq. (D.32) and Eq. (2.11)"}],"minor_comments":[{"comment":"The definition of δ in Theorem 1(b) is difficult to read because the line breaks separate the exponents from the operators; please typeset it as an explicit product of two operator norms with clear parentheses.","section":"Eq. (3.10)-(3.11)"},{"comment":"In Corollary 7(a) the condition \"If α>1/2\" appears inside a statement whose preamble fixes α∈[1/2,1)∪(1,∞); consider stating the corollary directly for α∈(1/2,1)∪(1,∞) to avoid ambiguity about the α=1/2 endpoint.","section":"Appendix A, Corollary 7"},{"comment":"Algorithm 1 refers to \"c0 as in (C.65)\", but (C.65) defines several quantities; please refer explicitly to Proposition 13 and its definition c0:=−2log min{min spec(σ̃_0), c_A}.","section":"Algorithm 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically sound in its internal derivations; I found no error in the contraction argument or in the sublinear-convergence proof. My recommendation is driven entirely by the load-bearing dependence on the author's unpublished preprint [17] for the α∈(1/2,1) half of the paper. If the editor is willing to accept an arXiv preprint as a citable source, the paper could be accepted after minor clarifications; otherwise the author should be asked to include the needed statements and proofs, or to update the reference to a peer-reviewed version."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Jim,\n\nThe short version: this is a solid, carefully written paper that gives the first provable convergence guarantees for alternating minimization of the doubly minimized Petz Rényi mutual information for arbitrary finite-dimensional quantum states. For α∈(1,2] they get linear convergence of the objective value with rate |1−1/α|^{2n}, and for α∈(1/2,1) they get sublinear O(1/n). Previous work only handled classical-classical states, so this is a real step forward.\n\nWhat I like: the proof structure is honest and complete. The contraction argument via Hilbert's projective metric and Birkhoff–Hopf for α∈(1,2] is clean, and the sublinear proof for α∈(1/2,1) is a genuinely different method, using joint convexity and Fréchet derivative bounds. The authors are upfront that they could not extend the classical linear-convergence proof to that range. The explicit constants are given, and the counterexample for α<1/2 is a nice sanity check.\n\nThe soft spot is the dependency, not the internal math. The sublinear result relies on joint concavity of Q_α and uniqueness of the global minimizer, both taken from the author's companion paper [17] and not re-derived. I read the argument and it is applied correctly; the stress-test note agrees. But for a theorem whose correctness hinges on an unpublished preprint by the same author, a referee should either verify [17] or require the author to include the needed statements as lemmas. That said, this is a normal citation pattern, not a circularity.\n\nMinor quibble: the paper is titled “computing” but contains no numerical experiments. For a theory paper that is fine, but the algorithms are presented without any demonstration of practical behavior. The per-iteration cost is not discussed. That would be worth adding.\n\nWho is this for: quantum information theorists working on Rényi quantities, especially anyone who needs to actually evaluate I_α↓↓ for a state. It deserves a serious referee. I would cite it and bring it to reading group if anyone is working on this. Recommend: accept with the usual request to clarify the dependency on [17] and maybe add a small numerical example.","headline":"Solid, carefully proved convergence results for computing the doubly minimized Petz Rényi mutual information; the main dependency is on the author's own prior work, but the math here checks out and the paper deserves a serious referee.","tokens_in":29808,"tokens_out":2809,"would_cite":true,"duration_ms":31772,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P45","94A17","90C25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Alternating minimization provably computes the doubly minimized Petz Rényi mutual information for all finite-dimensional quantum states.","keywords":["quantum information theory","Petz Rényi mutual information","alternating minimization","Hilbert's projective metric","convergence rates","Rényi divergence","bipartite quantum states"],"falsifier":"Take a concrete two-qubit state, such as a full-rank mixture of $\\lvert 00\\rangle$, $\\lvert 11\\rangle$, and a small noise term, and run the alternating iteration exactly for $\\alpha\\in(\\frac{1}{2},1)$. If for any $n$ the inequality $|x_n-I_\\alpha^{\\downarrow\\downarrow}| \\le \\max(\\frac{3}{2} c_0^2, 2x_0)/n$ fails, the sublinear theorem is false; likewise, for $\\alpha\\in(1,2]$, computing the ratio $d_H(N_{A\\to B}(\\sigma), N_{A\\to B}(\\tilde\\sigma))/d_H(\\sigma,\\tilde\\sigma)$ for two random positive marginals and finding a value larger than $\\gamma = |1-1/\\alpha|$ would falsify the contraction theorem.","tokens_in":28777,"feed_emoji":"🔁","tokens_out":13289,"duration_ms":124737,"temperature":0.7,"pith_summary":"The paper proves that the doubly minimized Petz Rényi mutual information, a one-parameter family of correlation measures defined as the smallest Petz divergence between a bipartite quantum state and a product state, can be computed by alternating minimization for every finite-dimensional bipartite state. For Rényi order $\\alpha \\in (1,2]$, the objective value after $n$ alternating rounds converges linearly, with error of order $|1 - 1/\\alpha|^{2n}$; for $\\alpha \\in (\\frac{1}{2},1)$, it converges sublinearly, with error $O(1/n)$. These are the first non-asymptotic convergence guarantees that apply to all quantum states, not only to classical-classical states. The paper turns the bounds into two explicit algorithms whose iteration counts can be fixed in advance from a desired error tolerance.","feed_headline":"Provable algorithm computes Petz Rényi mutual information","feed_subtitle":"Converges linearly for Rényi order 1 to 2 and sublinearly for 1/2 to 1, for any bipartite quantum state.","key_machinery":"Two separate mechanisms carry the argument. For $\\alpha\\in(1,2]$, the working tool is Hilbert's projective metric on the cone of positive semidefinite operators: each exact-minimizer update $N_{A\\to B}$ and $N_{B\\to A}$ is homogeneous and order-preserving or order-reversing in powers of the marginals, and a classical contraction theorem for positive linear maps bounds its Lipschitz constant by $\\gamma = |1-1/\\alpha|$; a full round therefore contracts by $\\gamma^2$. For $\\alpha\\in(\\frac{1}{2},1)$, the proof does not use the Hilbert metric. Instead it exploits the joint concavity of $Q_\\alpha(\\rho_{AB}\\|\\sigma_A\\otimes\\tau_B) = \\mathrm{tr}[\\rho_{AB}^{\\alpha}(\\sigma_A\\otimes\\tau_B)^{1-\\alpha}]$, the uniqueness of the global minimizer, and a Rényi–Pinsker inequality to relate the gap $x_{n-1}-x_n$ to the remaining error, producing a recursion that solves to $O(1/n)$.","core_discovery":"For any fixed $\\rho_{AB}$, alternating minimization updates the marginal $\\tau_B$ to the exact minimizer $\\hat\\tau_B = (\\mathrm{tr}_A[\\rho_{AB}^{\\alpha}\\sigma_A^{1-\\alpha}])^{1/\\alpha} / \\mathrm{tr}[(\\mathrm{tr}_A[\\rho_{AB}^{\\alpha}\\sigma_A^{1-\\alpha}])^{1/\\alpha}]$, then updates $\\sigma_A$ in the same way, and the sequence $x_n = D_\\alpha(\\rho_{AB}\\|\\sigma_A^{(n)}\\otimes\\tau_B^{(n)})$ decreases monotonically to the doubly minimized PRMI $I_\\alpha^{\\downarrow\\downarrow}(A:B)_\\rho$ for every $\\alpha$ in $(\\frac{1}{2},1)\\cup(1,2]$. For $\\alpha\\in(1,2]$, the contraction argument in Hilbert's projective metric yields $|x_n - I_\\alpha^{\\downarrow\\downarrow}| \\le \\frac{1}{\\alpha-1}[\\exp((\\alpha-1)(1+\\gamma)\\gamma^{2n}d_H(\\sigma_A^{(0)},\\hat\\sigma_A))-1]$ with $\\gamma = 1 - 1/\\alpha$, an explicit linear rate. For $\\alpha\\in(\\frac{1}{2},1)$, the proof uses joint concavity of the trace function $Q_\\alpha$ together with a Rényi–Pinsker inequality to obtain $|x_n - I_\\alpha^{\\downarrow\\downarrow}| \\le \\max(\\frac{3}{2} c_0^2, 2x_0)/n$, a sublinear $O(1/n)$ rate. The $\\alpha=1$ case is degenerate and reaches the minimum in one step; for $\\alpha\\in(0,\\frac{1}{2}]$, alternating minimization is shown not to converge to the global minimum in general.","pith_inferences":["The Hilbert-metric technique for $\\alpha\\in(1,2]$ is the natural template to attack the sandwiched Rényi mutual information, whose partial minimizers are not explicit; finding such formulas would let the same contraction proof run for that family.","The sublinear $O(1/n)$ rate for $\\alpha\\in(\\frac{1}{2},1)$ is likely conservative; numerical experiments on low-dimensional states could reveal much faster actual convergence, and a local strong-convexity analysis might upgrade the guarantee.","The $\\alpha=\\frac{1}{2}$ endpoint is connected to reflected-entropy-like measures mentioned in the introduction; the algorithm offers a numerical route to those quantities by taking a limit along $\\alpha\\in(\\frac{1}{2},1)$."],"forward_implications":["For $\\alpha\\in(1,2]$, the objective error after $n$ full alternating rounds is at most $\\frac{1}{\\alpha-1}[\\exp((\\alpha-1)(1+\\gamma)\\gamma^{2n}d_H(\\sigma_A^{(0)},\\hat\\sigma_A))-1]$, so the iteration count needed to reach a targeted precision is known before running the algorithm.","For $\\alpha\\in(\\frac{1}{2},1)$, the same iteration reaches precision $\\epsilon$ after a number of rounds no larger than $\\max(\\frac{3}{2} c_0^2, 2x_0)/\\epsilon$, again giving a finite stopping rule.","The alternating sequence of states converges to the unique global minimizer of the PRMI optimization problem for every $\\alpha\\in(\\frac{1}{2},1)\\cup(1,2]$.","The algorithms output a value $x$ certified to satisfy $|x - I_\\alpha^{\\downarrow\\downarrow}(A:B)_\\rho| \\le \\epsilon_0$ for any prescribed $\\epsilon_0>0$, for all quantum states in the stated range."],"supporting_citations":[{"why":"Supplies the joint-concavity/convexity of $Q_\\alpha$, the uniqueness of the global minimizer, and the CC-state reduction on which both convergence proofs rest.","marker":"[17]"},{"why":"Establishes linear convergence for the classical doubly minimized RMI in Hilbert's projective metric, the template the quantum $\\alpha\\in(1,2]$ proof extends.","marker":"[22]"},{"why":"Gives the quantum Sibson identity and the explicit formula for the optimal marginal that defines the alternating update step.","marker":"[13]"},{"why":"Together with [13], provides the partial-minimizer characterization used in the iteration rule.","marker":"[9]"},{"why":"Provides the Rényi–Pinsker inequality used to convert $Q_\\alpha$-gaps into objective-value gaps in the sublinear proof.","marker":"[25]"},{"why":"Contains the cone-contraction theorem and Hilbert-metric properties used to derive the $\\gamma$-contraction bound.","marker":"[26]"},{"why":"Supplies additivity of Hilbert's projective metric under tensor products, used to pass from state convergence to objective convergence.","marker":"[28]"},{"why":"Gives general alternating-minimization convergence results whose smoothness assumptions the paper cannot verify, motivating the self-contained sublinear proof.","marker":"[23]"}],"fun_headline_variants":["Alternating minimization provably computes Petz Rényi MI for any state","Linear and sublinear convergence for doubly minimized Petz Rényi MI","Any quantum state: alternating minimization solves Petz Rényi MI","Petz Rényi MI: alternating minimization converges with proven rates","Doubly minimized Petz Rényi MI computable via alternating minimization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The premise that has to hold is that the surface being minimized curves upward in both marginal directions at once for orders between 1/2 and 1, and that the minimum point is unique; the paper imports both facts from its companion work, and the sublinear proof depends on them.","fun_headline_variants_meta":{"raw":{"variants":["Alternating minimization provably computes Petz Rényi MI for any state","Linear and sublinear convergence for doubly minimized Petz Rényi MI","Any quantum state: alternating minimization solves Petz Rényi MI","Petz Rényi MI: alternating minimization converges with proven rates","Doubly minimized Petz Rényi MI computable via alternating minimization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000715,"raw_usage":{"total_tokens":3309,"prompt_tokens":1132,"completion_tokens":2177,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":748,"completion_tokens_details":{"reasoning_tokens":2082}},"tokens_in":748,"tokens_out":2177,"duration_ms":21304,"temperature":1.0,"reasoning_tokens":2082,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:30:31.530971+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a concrete two-qubit state, such as a full-rank mixture of $\\lvert 00\\rangle$, $\\lvert 11\\rangle$, and a small noise term, and run the alternating iteration exactly for $\\alpha\\in(\\frac{1}{2},1)$. If for any $n$ the inequality $|x_n-I_\\alpha^{\\downarrow\\downarrow}| \\le \\max(\\frac{3}{2} c_0^2, 2x_0)/n$ fails, the sublinear theorem is false; likewise, for $\\alpha\\in(1,2]$, computing the ratio $d_H(N_{A\\to B}(\\sigma), N_{A\\to B}(\\tilde\\sigma))/d_H(\\sigma,\\tilde\\sigma)$ for two random positive marginals and finding a value larger than $\\gamma = |1-1/\\alpha|$ would falsify the contraction theorem.","supporting_citations":[{"cited_title":"Operational Interpretation of the Sandwiched Rényi Divergence of Order 1/2 to 1 as Strong Converse Exponents","cited_arxiv_id":null,"evidence_quote":"Establishes linear convergence for the classical doubly minimized RMI in Hilbert's projective metric, the template the quantum $\\alpha\\in(1,2]$ proof extends."},{"cited_title":"Let α ∈ ( 1 2 , 1), ρAB ∈ S(AB), σ(0) A ∈ S∼ρA(A)","cited_arxiv_id":null,"evidence_quote":"Gives the quantum Sibson identity and the explicit formula for the optimal marginal that defines the alternating update step."},{"cited_title":"All of the following hold","cited_arxiv_id":null,"evidence_quote":"Together with [13], provides the partial-minimizer characterization used in the iteration rule."},{"cited_title":"Seshadreesan, and Mark M","cited_arxiv_id":null,"evidence_quote":"Provides the Rényi–Pinsker inequality used to convert $Q_\\alpha$-gaps into objective-value gaps in the sublinear proof."},{"cited_title":"Coding Theorems for Compound Problems via Quantum Rényi Divergences.IEEE Transactions on Information Theory, 61(6):2997–3012, 2015","cited_arxiv_id":null,"evidence_quote":"Contains the cone-contraction theorem and Hilbert-metric properties used to derive the $\\gamma$-contraction bound."}],"review_version":1}