{"id":"59347ff5-5355-496d-9883-0347357cb429","arxiv_id":"2608.07246","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A rigorous two-sided bound on relative free energy is generalized to unbounded perturbations in von Neumann algebras, with variational formulas, and proposed as a thermodynamic entanglement criterion.","lead":"This paper extends the two-sided Bogoliubov inequality, which bounds free energy differences by expectation values of an interaction term, to infinite-dimensional quantum systems described by von Neumann algebras. It also proposes using the resulting free energy bounds as a thermodynamic way to estimate quantum correlations and entanglement, a claim that is mathematically motivated but not yet proven.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The theorem's proof is sound under (A1)-(A4), but the central claim that the relative free energy quantifies entanglement is asserted, not derived, and is unlikely to hold as stated.","rationale":"The reader's verdict (CONDITIONAL) correctly identifies two issues: the technical assumption (A4) in Lemma 5.1 and the unsupported physical interpretation linking F to entanglement. I regard the physical identification as the more load-bearing concern because the theorem itself is a valid conditional statement whose proof I could not find a fatal flaw in, whereas the abstract's 'quantification of entanglement' claim is not a theorem but an interpretive leap. The paper provides rigorous arguments for the two-sided inequality and variational bounds; credit is due for extending Dereziński-Jakšić-Pillet unbounded perturbation theory and Petz's variational principles. However, the criterion's operational meaning is left vague: F is never compared with standard entanglement measures, and the only explicit field-theoretic example is isospectral, yielding F=0. The conclusion's own caveat about physical interpretation in relativistic settings is in tension with the unqualified abstract statement. Thus the correct verdict remains CONDITIONAL: the mathematics is acceptable, but the central physical claim needs to be either proven or substantially softened. I do not recommend REJECT because the mathematical results stand on their own, and I do not recommend ACCEPT because the advertised entanglement quantification is not delivered.","tokens_in":34463,"tokens_out":26028,"duration_ms":241481,"concrete_test":"Implement the finite-dimensional two-qubit analog: choose H0 = -beta^{-1} log(rho_A (x) rho_B) with rho_A = rho_B = I/2, and select U so that the Gibbs state at inverse temperature beta is (a) the classically correlated state rho_c = 1/2(|00><00| + |11><11|) and (b) a mixed entangled state with the same reduced states (e.g., a Werner state). Compute F(rho,rho_0) exactly via Eq. (28) for both cases. If F(rho_c,rho_0) > 0 for the classically correlated state, or if F fails to correlate with an established entanglement measure across the family, then the claimed 'quantification of entanglement' is not supported by the theorem.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 5.4 and its supporting lemmas are internally consistent: granting (A1)-(A4), Lemma 5.1 gives S_M(omega_U,omega) = -beta omega_U(U) - log||Omega_{-beta U}||^2, and non-negativity of the Araki-Uhlmann relative entropy then yields the two-sided bound. The load-bearing weakness is the abstract's claim that these developments yield 'a physically well-founded thermodynamic criterion for the quantification of entanglement.' No standard property of an entanglement measure is established: monotonicity under LOCC, vanishing on all separable states, and faithfulness are not verified. F(omega_U,omega) is a free-energy difference between a coupled and an uncoupled KMS state; it weights all correlations generated by U, classical and quantum alike. The sole infinite-dimensional example (Section 5.3) uses an inner Weyl perturbation whose free energy vanishes by unitary equivalence, so it cannot demonstrate that F tracks entanglement. The conclusion itself concedes that the physical interpretation 'needs to be handled with some care' in relativistic QFT, which undercuts the unqualified statement in the abstract.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper generalizes the quantum-mechanical two-sided Bogoliubov inequality to arbitrary von Neumann algebras. The main theorem (Theorem 5.4) states that for a faithful (τ,β)-KMS state ω with vector representative Ω and a perturbation U in the class S₂(L,Ω), the relative free energy F(ω_U,ω)=ω_U(U)+β^{-1}S_M(ω_U,ω) satisfies ω_U(U)≤F(ω_U,ω)≤ω(U). The proof combines the Dereziński–Jakšić–Pillet unbounded perturbation theory with a new identity (Lemma 5.1) expressing S_M(ω_U,ω) as -βω_U(U)-log‖Ω_{-βU}‖². The paper also proves variational characterizations of F (Theorem 5.15), extends Petz-type variational principles to unbounded perturbations, discusses a bosonic scalar-field example with an inner Weyl perturbation (Section 5.3), and shows a reduction to the finite-dimensional case under additional hypotheses (Proposition 5.7). The mathematical core is presented in detail, in particular the extensions of Araki's bounded-perturbation formulas and the strong-resolvent convergence arguments.","tokens_in":34652,"tokens_out":7890,"duration_ms":82967,"significance":"If the main theorem is correct, it is a useful rigorous result: it provides computable upper and lower bounds for a relative free energy in infinite-dimensional systems and extends known variational principles to unbounded perturbations. The paper's strengths are that the proofs are detailed, the reduction of the main inequality to non-negativity of relative entropy is transparent, and several auxiliary results (e.g., Lemma 5.1 and Lemma 4.17) are original extensions of the Dereziński–Jakšić–Pillet framework. However, the advertised physical conclusion that this yields a well-founded thermodynamic criterion for entanglement quantification is not derived: no standard entanglement-measure property is established for F(ω_U,ω), and the only application considered is a unitary (inner) perturbation with identically vanishing free energy. The technical content is therefore plausible and publishable as a mathematical contribution, but the physical claim, as stated in the title and abstract, is not supported by the presented results.","major_comments":[{"comment":"The paper claims that the two-sided Bogoliubov inequality yields 'a physically well-founded thermodynamic criterion for the quantification of entanglement'. No property of an entanglement measure is proved: F(ω_U,ω) is not shown to vanish on separable states, to be monotone under local operations and classical communication, or to be faithful. As a free-energy difference induced by an arbitrary interface operator U, it weights all correlations generated by U, classical and quantum alike. The conclusion in §6 itself notes that the physical interpretation 'needs to be handled with some care' in relativistic QFT. The authors should either prove at least one standard entanglement-measure property for F or rephrase the abstract, title, and introduction as proposing a thermodynamic estimate of correlations rather than a criterion for entanglement.","section":"Abstract and §1.1, §6"},{"comment":"The bosonic scalar-field example cannot demonstrate that F tracks entanglement. The perturbation is inner: by Lemma 5.8, L+U = W L W* with W∈M, so the perturbed state is unitarily equivalent to ω, and Lemma 5.9 gives F(ω_U,ω)=0 identically. The two-sided inequality then collapses to the condition G_c(f,Lf)≤0, i.e., to positivity of the relative entropy. In addition, the example does not verify that U belongs to S₂(L,Ω): assumptions (A2) and (A4) are never checked. Please either verify the required assumptions and analyze a genuinely non-inner perturbation, or present the example only as a consistency check of the bounds.","section":"Section 5.3"},{"comment":"The claimed reduction to the quantum-mechanical two-sided Bogoliubov inequality is only conditional. The proposition assumes, beyond the hypotheses of Theorem 2.1, the Hilbert-Schmidt condition ‖e^{-βV/2}e^{-βH0/2}‖_HS<∞ (which, as noted in footnote 3, is automatic only for lower semi-bounded V) and treats (A4) as a standing hypothesis. Thus Eq. (27) is shown to imply Eq. (6) in a restricted subclass of finite-dimensional systems, not in the full generality of Theorem 2.1. The wording 'reduces to' should be qualified to reflect these extra assumptions, or the extra assumptions should be derived from the hypotheses of Theorem 2.1.","section":"Proposition 5.7"},{"comment":"The lower bound ω_U(U)≤F(ω_U,ω) depends essentially on assumption (A4), βω_U(U_+)<∞. This assumption is not implied by (A1)–(A3) and is not mentioned in the informal statement of the theorem in §1.2. Moreover, in the proof of Lemma 5.1 the assertion that W:=V+β^{-1}log‖Ω_{-βV}‖² Id belongs to S₂(L,Ω) is stated without verification. The statement is plausible, but it should be checked explicitly because the proof of the lemma relies on Corollary 4.18 for W. Please state the domain of validity of Theorem 5.4 precisely whenever the theorem is advertised informally, and justify the S₂ membership in Lemma 5.1.","section":"Lemma 5.1 and Theorem 5.4"}],"minor_comments":[{"comment":"The modular conjugation J is anti-unitary, not self-adjoint in the usual operator-theoretic sense; the phrase 'J is a bounded, self-adjoint, bijective operator' should be replaced by 'anti-unitary involution' or 'satisfying J²=Id and J*=J as an anti-linear operator'.","section":"Lemma 4.9"},{"comment":"The discussion after Assumption (A3) uses the inequality x> -e^{-x} to show βω(V_-)>-∞. The text states 'βω(V_-)>-∞', which is correct, but the phrase immediately before 'βω(V)_>-∞' should be 'βω(V)>-∞' (the subscript V_- appears to be a typographical artifact). Clarify that V=V_++V_- and that (A3) only controls the negative part of the expectation.","section":"Section 5.1"},{"comment":"The notation -L_{V_n}+V_n→-L_V+V in strong resolvent sense is fine, but it would help the reader if the proof explicitly stated that the domain equality dom(L-V)=J(dom(L)∩dom(V)) follows from the identities dom(L)=J dom(L) and dom(JVJ)=J dom(V), since this is used in the application of Proposition B.1.","section":"Section 4.4, Lemma 4.12(c)"}],"recommendation":"major_revision","confidential_remarks":"The mathematical core of the manuscript is solid and likely correct, and the extension of the Dereziński–Jakšić–Pillet framework is a meaningful contribution. The main obstacle to acceptance is the mismatch between the strong physical claim in the title/abstract ('thermodynamic criterion for the quantification of entanglement') and the actual content, which establishes inequalities for relative free energy but no entanglement-measure properties. The Section 5.3 example, being inner, is also not evidence for the entanglement interpretation. I would support publication after the authors either prove standard entanglement-measure properties or substantially temper the physical claims, and after the technical gaps in Proposition 5.7 and the example are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: the main theorem is correct as stated, and the paper is worth engaging. Theorem 5.4 gives a two-sided bound on a relative free energy for unbounded perturbations in general von Neumann algebras; the lower bound is new, and the variational principles (Propositions 5.11, 5.14, Theorem 5.15) extend Petz's bounded-perturbation results. The proofs follow the Dereziński-Jakšić-Pillet framework carefully; Lemma 5.1 is a neat identity, and the use of non-negativity of Araki-Uhlmann relative entropy is clean. Credit where due: this is formal, reproducible mathematics with detailed arguments.\n\nThe soft spots are real but not fatal. Assumption (A4) is load-bearing for the lower bound; if beta omega_V(V_+) is infinite, the identification in Lemma 5.1 can fail and the lower bound is not guaranteed. That is a genuine technical limitation, and the paper states it, but it is still a limitation. The reduction to the B(H) case (Prop 5.7) requires extra trace-class and relative-boundedness assumptions, so the generalization is not fully equivalent to the starting point. Minor.\n\nThe main overclaim is the abstract's statement that this yields \"a physically well-founded thermodynamic criterion for the quantification of entanglement.\" What is proved is an inequality for a free-energy difference. None of the standard properties of an entanglement measure are verified: no LOCC monotonicity, no vanishing on all separable states, no faithfulness. The sole infinite-dimensional example (Section 5.3) is an inner Weyl perturbation; the free energy vanishes by unitary equivalence, so it cannot demonstrate entanglement tracking. The conclusion itself warns that \"the physical interpretation... needs to be handled with some care\" in relativistic QFT, which undercuts the unqualified abstract claim. That tension should be fixed.\n\nWho is this for? Mathematicians and mathematical physicists working on operator algebras, KMS states, and perturbation theory. It is not a physics paper that establishes an entanglement measure; it is a rigorous math paper with an ambitious physical gloss. I would send it to a serious referee, and I would expect the referee to request revisions that temper the physical claims and clarify the status of (A4). The mathematics itself deserves publication.","headline":"The math is solid and the unbounded-perturbation lower bound plus variational principles are genuinely new; the entanglement interpretation in the abstract is an overclaim that should be tempered.","tokens_in":35216,"tokens_out":2272,"would_cite":true,"duration_ms":22106,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L10","46L55","82B10","81T05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a two-sided Bogoliubov inequality for the relative free energy of faithful KMS states on von Neumann algebras, bounding $F(\\omega_U,\\omega)$ between the perturbed and unperturbed expectations of the interaction operator…","keywords":["relative entropy","KMS states","von Neumann algebras","unbounded perturbation theory","two-sided Bogoliubov inequality","relative free energy","quantum entanglement","Araki-Uhlmann relative entropy"],"falsifier":"Take a translation-invariant harmonic-chain or free Bose-field model with a faithful KMS state, and choose as $U$ a non-inner, unbounded, long-range coupling between two half-systems for which $\\beta\\omega_U(U_+)=+\\infty$ while assumptions (A1)–(A3) still hold; compute $S_M(\\omega_U,\\omega)$ directly from the relative modular operator and compare with the two candidate formulas. If $S_M(\\omega_U,\\omega) \\neq -\\beta\\omega_U(U)-\\log\\|\\Omega_{-\\beta U}\\|^2$, or if $\\omega_U(U) > F(\\omega_U,\\omega)$, then Lemma 5.1 and the lower bound fail outside (A4). A finite-truncation numerical extrapolation of the same setup can already indicate whether the inequality is violated in the limit.","tokens_in":34253,"feed_emoji":"♾️","tokens_out":13224,"duration_ms":122537,"temperature":0.7,"pith_summary":"The paper aims to move the two-sided Bogoliubov inequality—a finite-system bound on the free-energy cost of splitting an interacting system into independent parts—into the setting of von Neumann algebras, the natural observable algebra for infinitely many degrees of freedom. Its main theorem states that for a faithful KMS state $\\omega$ (the algebraic equilibrium state at inverse temperature $\\beta$) and a self-adjoint perturbation $U$ affiliated with the algebra, the relative free energy $F(\\omega_U,\\omega)$ between the perturbed and unperturbed states satisfies $\\omega_U(U) \\leq F(\\omega_U,\\omega) \\leq \\omega(U)$. Since $F$ is also shown to equal $-\\beta^{-1} \\log \\|\\Omega_{-\\beta U}\\|^2$, the two sides are computable from expectation values of the interaction alone, bypassing partition functions and entropy. The authors present this as a physically well-founded thermodynamic criterion for quantifying entanglement in infinite-dimensional systems, with a bosonic scalar-field example and variational refinements that reduce to the original finite-dimensional inequality when the algebra is all bounded operators on a Hilbert space.","feed_headline":"Bogoliubov free-energy bound reaches infinite systems","feed_subtitle":"New theorem bounds the cost of separating subsystems by two energy averages, giving a thermodynamic route to entanglement.","key_machinery":"The engine of the proof is the Araki–Uhlmann relative entropy $S_M(\\psi,\\varphi)$, built from the relative modular operator $\\Delta_{\\Psi,\\Phi}$ in the standard-form representation of a von Neumann algebra, together with the unbounded perturbation theory of KMS states. For a separating vector $\\Omega$, the modular operator is $\\Delta_\\Omega = e^{-\\beta L}$ with $L$ the standard Liouvillian, and the perturbed vector $\\Omega_{-\\beta U}=e^{-\\beta(L+U)/2}\\Omega$ is obtained as a weak limit of bounded approximations $e^{-\\beta(L+U_n)/2}\\Omega$. The technical core is an extension of the bounded-perturbation relative-modular identities to unbounded $U$: $\\log\\Delta_{\\Omega,\\Omega_{-\\beta U}}=\\log\\Delta_\\Omega+\\beta J U J$ (with $J$ the modular conjugation), and its companion $\\log\\Delta_{\\Omega_{-\\beta U},\\Omega}=\\log\\Delta_{\\Omega_{-\\beta U}}+\\beta U$. Under assumption (A4), $\\beta\\omega_U(U_+)<\\infty$, Lemma 5.1 converts the second identity into $S_M(\\omega_U,\\omega)=-\\beta\\omega_U(U)-\\log\\|\\Omega_{-\\beta U}\\|^2$, so $F(\\omega_U,\\omega)=-\\beta^{-1}\\log\\|\\Omega_{-\\beta U}\\|^2$; nonnegativity of relative entropy in the two orders yields both sides of the sandwich.","core_discovery":"The paper's central discovery is Theorem 5.4. In a standard-form von Neumann algebra $M$ with a $W^*$-dynamics $\\tau$ and a faithful $(\\tau,\\beta)$-KMS state $\\omega$ at inverse temperature $\\beta>0$, take any self-adjoint operator $U$ affiliated with $M$ that satisfies assumptions (A1)–(A4) involving the standard Liouvillian $L$ and the separating vector $\\Omega$. Define the relative free energy $F(\\omega_U,\\omega) = \\omega_U(U) + \\beta^{-1} S_M(\\omega_U,\\omega)$, where $S_M$ is the Araki–Uhlmann relative entropy and $\\omega_U$ is the perturbed state constructed from $\\Omega_{-\\beta U}=e^{-\\beta(L+U)/2}\\Omega$. Then $\\omega_U(U) \\leq F(\\omega_U,\\omega) \\leq \\omega(U)$. The proof uses Lemma 5.1, which identifies $S_M(\\omega_U,\\omega) = -\\beta\\omega_U(U) - \\log\\|\\Omega_{-\\beta U}\\|^2$, so $F = -\\beta^{-1}\\log\\|\\Omega_{-\\beta U}\\|^2$, and the bounds follow from nonnegativity of relative entropy in the two orders. Theorem 5.15 adds variational equalities: the infimum over normal states $\\psi$ of $\\psi(U)+\\beta^{-1}S_M(\\psi,\\omega)$ and the supremum over affiliated operators $V$ of $\\omega_U(U-V)-\\beta^{-1}\\log\\|\\Omega_{-\\beta V}\\|^2$ are both exactly $F(\\omega_U,\\omega)$. When $M=\\mathcal{B}(H)$, Proposition 5.7 shows the theorem reduces to the original quantum-mechanical two-sided Bogoliubov inequality; a free scalar-field example with an inner Weyl perturbation gives a case where $F=0$ and the inequality becomes the positivity of relative entropy.","pith_inferences":["Because the gap $\\omega(U)-\\omega_U(U)$ vanishes exactly for inner Weyl perturbations in the scalar-field example, it is tempting to read this gap as an operational measure of non-inner correlation; the paper suggests thermodynamic relevance but does not propose a measurement protocol, so that step is ours.","The Donsker–Varadhan form of the variational principle is the mathematical core of quantum hypothesis testing, so the relative free energy plausibly admits an interpretation as the optimal-error asymmetry of distinguishing $\\omega$ from $\\omega_U$; the paper does not draw this information-theoretic link.","A testable extension is to compute both sides of the inequality numerically in truncated harmonic-chain or Fock-space models with non-inner, unbounded couplings; this could reveal how tight the bounds are and how quickly the free-energy gap converges as a function of coupling strength.","The same perturbation framework likely yields two-sided bounds for other relative entropies, such as Rényi-type divergences, rather than only the Araki–Uhlmann entropy; that would give a one-parameter family of thermodynamic correlation measures. This is our extrapolation, not a result in the paper."],"forward_implications":["For any faithful KMS state on a von Neumann algebra, the free energy of separation can be computed from two expectation values, $\\omega(U)$ and $\\omega_U(U)$, with no partition-function or entropy evaluation.","The lower bound in Theorem 5.4 is a genuinely new estimate, so the theorem gives a two-sided thermodynamic handle on correlations in infinite-dimensional systems, not merely a re-derived Peierls–Bogoliubov inequality.","The variational equalities of Theorem 5.15 allow the bounds to be optimized over states and perturbations, which the paper connects to materials design with prescribed quantum characteristics.","In the free scalar-field application, an inner Weyl perturbation yields a unitarily equivalent perturbed state, $F=0$, and the inequality collapses to positivity of relative entropy; in geometric settings the same relative-entropy gap links to horizon entropy–area relations.","When $M=\\mathcal{B}(H)$, the theorem reduces to the original quantum-mechanical two-sided Bogoliubov inequality, so the finite-dimensional result is contained as a special case."],"supporting_citations":[{"why":"Supplies the unbounded perturbation theory of W*-dynamics, Liouvilleans and KMS states: the assumptions (A1)–(A3), existence of the perturbed vector, and the relative-entropy and Peierls–Bogoliubov identities used in the proof.","marker":"[22]"},{"why":"Provides the relative modular operator and the bounded-perturbation identity connecting logarithms of relative modular operators, which the paper extends to unbounded perturbations.","marker":"[7]"},{"why":"Establishes the bounded-perturbation Gibbs and Donsker–Varadhan variational principles for relative entropy that the paper extends to the unbounded setting.","marker":"[49]"},{"why":"Introduces the quantum-mechanical two-sided Bogoliubov inequality and free-energy-of-separation framework that the paper generalizes and recovers when the algebra is the full bounded-operator algebra.","marker":"[55]"},{"why":"Supplies the Golden–Thompson and Peierls–Bogoliubov inequalities for general von Neumann algebras, used in the perturbed-vector construction and in bounding the norm of the perturbed vector.","marker":"[3]"}],"fun_headline_variants":["Bogoliubov inequality extended to von Neumann algebras","Free-energy bounds give thermodynamic entanglement test","New two-sided Bogoliubov bound for infinite systems","Variational free energy proves entanglement criterion","Bogoliubov inequality now covers infinite-dimensional entanglement"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The theorem holds only if the positive part of the interaction energy has finite expectation in the perturbed state (assumption (A4)); separately, calling the resulting free energy a measure of entanglement is a physical identification argued by analogy, not proven.","fun_headline_variants_meta":{"raw":{"variants":["Bogoliubov inequality extended to von Neumann algebras","Free-energy bounds give thermodynamic entanglement test","New two-sided Bogoliubov bound for infinite systems","Variational free energy proves entanglement criterion","Bogoliubov inequality now covers infinite-dimensional entanglement"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000336,"raw_usage":{"total_tokens":1928,"prompt_tokens":1080,"completion_tokens":848,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":696,"completion_tokens_details":{"reasoning_tokens":775}},"tokens_in":696,"tokens_out":848,"duration_ms":9180,"temperature":1.0,"reasoning_tokens":775,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T11:53:26.084235+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a translation-invariant harmonic-chain or free Bose-field model with a faithful KMS state, and choose as $U$ a non-inner, unbounded, long-range coupling between two half-systems for which $\\beta\\omega_U(U_+)=+\\infty$ while assumptions (A1)–(A3) still hold; compute $S_M(\\omega_U,\\omega)$ directly from the relative modular operator and compare with the two candidate formulas. If $S_M(\\omega_U,\\omega) \\neq -\\beta\\omega_U(U)-\\log\\|\\Omega_{-\\beta U}\\|^2$, or if $\\omega_U(U) > F(\\omega_U,\\omega)$, then Lemma 5.1 and the lower bound fail outside (A4). A finite-truncation numerical extrapolation of the same setup can already indicate whether the inequality is violated in the limit.","supporting_citations":[{"cited_title":"Dereziński, V","cited_arxiv_id":null,"evidence_quote":"Supplies the unbounded perturbation theory of W*-dynamics, Liouvilleans and KMS states: the assumptions (A1)–(A3), existence of the perturbed vector, and the relative-entropy and Peierls–Bogoliubov identities used in the proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the relative modular operator and the bounded-perturbation identity connecting logarithms of relative modular operators, which the paper extends to unbounded perturbations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the bounded-perturbation Gibbs and Donsker–Varadhan variational principles for relative entropy that the paper extends to the unbounded setting."},{"cited_title":"Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras","cited_arxiv_id":"2501.04564","evidence_quote":"Introduces the quantum-mechanical two-sided Bogoliubov inequality and free-energy-of-separation framework that the paper generalizes and recovers when the algebra is the full bounded-operator algebra."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Golden–Thompson and Peierls–Bogoliubov inequalities for general von Neumann algebras, used in the perturbed-vector construction and in bounding the norm of the perturbed vector."}],"review_version":1}