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This paper argues that Petz's original proof of the monotonicity of relative entropy relies on a flawed contractive Jensen step, and that the proof is restored only by recognizing that V_ρ is an isometry.

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2026-08-04 16:59 UTC pith:HDXSL2U4

load-bearing objection A useful teaching paper on two classic DPI proofs; the math of the correction and counterexamples checks out, but the claim that Petz's original proof was flawed is not documented, and the Uhlmann section has two patchable gaps. the 2 major comments →

arxiv 2509.11221 v1 pith:HDXSL2U4 submitted 2025-09-14 quant-ph math-phmath.MP

On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches

Santiago Matheus , Francesco Bottacin , Edoardo Provenzi This is my paper
classification quant-ph math-phmath.MP
keywords relative entropydata processing inequalitymonotonicityoperator Jensen inequalitypartial tracequantum channelsinterpolation of sesquilinear formsPetz recovery map
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper re-examines two classic proofs that relative entropy is monotone under quantum channels, a property known as the data processing inequality. It claims that Petz's proof, as reconstructed here, has a specific gap: showing that the bridging operator V_ρ is a contraction does not justify the contractive Jensen operator inequality, because the integrands g_ξ(x)=(x+ξ)^{-1}-(1+ξ)^{-1} are positive at x=0. The fix is the stronger identity V†_ρ V_ρ = id, which makes V_ρ an isometry and makes the needed Jensen inequality valid. The paper then proves the same monotonicity for non-invertible density operators via a regularization limit, and develops Uhlmann's alternative proof using interpolation of positive sesquilinear forms, which needs no invertibility assumption. If right, the paper pins down exactly which operator property is doing the work in the data processing inequality and reconciles two proof strategies.

Core claim

The paper's central claim is that monotonicity of relative entropy under partial trace follows from the isometry identity V†_ρ V_ρ = id, not from the weaker contraction property of V_ρ. Petz's original approach is reconstructed as applying the contractive Jensen inequality to V_ρ; the paper shows this step is invalid because, in the integral representation of -log, the functions g_ξ(x)=(x+ξ)^{-1}-(1+ξ)^{-1} have g_ξ(0)>0, making the needed Jensen inequality false even for scalars. The isometry identity restores the proof by switching to the isometric form of Jensen's inequality, and the argument extends to non-invertible states by a limiting procedure. Uhlmann's interpolation-of-forms proof

What carries the argument

The central object is the weighted adjoint of the partial trace, V_ρ = R_{ρ^{1/2}} ∘ Tr†_b ∘ R_{Tr_b(ρ)^{-1/2}}, which maps the reduced-state Hilbert space into the joint-system Hilbert space and sends Tr_b(ρ)^{1/2} to ρ^{1/2}. The key identity is V†_ρ V_ρ = id, the isometry condition that converts an invalid contractive Jensen step into a valid isometric Jensen inequality. In Uhlmann's route, the machinery is interpolation of positive sesquilinear forms: compatible representations give forms γ^t_{α→β} represented by A^{1-t}B^t, and the proof uses the order-monotonicity theorem plus the pullback property ψ^*(γ^t_{α→β}) ≤ γ^t_{ψ^*α→ψ^*β} for ψ = Tr†_b.

Load-bearing premise

The paper's diagnosis that Petz's proof is flawed presupposes that Petz actually argued from the contraction property alone; if the original already invoked the isometry V†_ρ V_ρ = id, then the alleged flaw is in the reconstruction, not in Petz's proof.

What would settle it

Look at the exact passage in Petz's 2003 paper where the Jensen inequality is applied: if it already states or proves V†_ρ V_ρ = id, the flaw claim cannot stand historically; separately, the scalar counterexample with α=0.5, ξ=0.5, x=1 shows (αxα+ξ)^{-1} > α(x+ξ)^{-1}α, so the contraction-based inequality is genuinely false.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • The corrected Petz proof shows that the data processing inequality for relative entropy follows from an isometric embedding of the reduced-state Hilbert space, not merely from contractivity.
  • Non-invertible density operators, including pure states, are covered: the support inclusion supp(ρ)⊆supp(σ) passes to partial traces, and the regularized definition lets one take the ε→0 limit of the invertible-case inequality.
  • Uhlmann's interpolation proof gives the same monotonicity for all finite-dimensional states without an invertibility assumption, and its mechanism is the order-preserving property of interpolations under pullback.
  • Petz's claimed extension of the proof to adjoints of unital Schwarz maps does not survive the correction, because V_ρ is an isometry only for the partial trace; that broader monotonicity requires a different argument.
  • Since V_σ is identified with a component of the Petz recovery map, the same operator that restores the proof is also the one governing recoverability and saturation of the data processing inequality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If Petz's original text is found to contain the isometry argument already, the historical claim of a flaw collapses; the mathematical lesson that contraction alone is insufficient would still stand independently as a caution for reconstructed proofs.
  • The same obstruction—functions g_ξ positive at zero—can serve as a quick diagnostic for other operator-convex functionals: attempts to prove data-processing-type inequalities via contractive Jensen will fail on such functionals, and the fix should be to look for an isometry or a kernel with f(0)≤0.
  • Because V_σ is a component of the Petz recovery map, one can test whether quantitative versions of the data processing inequality can be stated directly in terms of how close V_σ is to a unitary, tying the proof mechanism to recoverability in a transparent way.
  • Uhlmann's interpolation parameter t runs through [0,1]; a natural extension not in the paper is to check whether the same order-monotonicity yields t-parametrized data-processing inequalities for the whole interpolation curve, not just its derivative at t=0.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper revisits two classical proofs of the monotonicity of quantum relative entropy under partial trace: Petz's operator-algebraic proof and Uhlmann's proof via interpolation of positive sesquilinear forms. Section 2 collects preliminaries on operator convexity, partial trace, and the support-based versus regularized definitions of relative entropy. Section 3 reconstructs Petz's argument, claims that his original use of the contractive Jensen operator inequality is flawed because the contraction V_rho does not satisfy the required inequality for g_xi(x)=(x+xi)^-1-(1+xi)^-1, and gives scalar counterexamples (Figs. 1-2). The correction, attributed to Petz and Nielsen [16], is that V_rho is actually an isometry, V_rho^dagger V_rho=id, which legitimizes the isometric Jensen inequality. The section extends the proof to non-invertible states via epsilon-regularization and a support-inclusion lemma. Section 4 develops Uhlmann's theory of interpolations of positive forms, defines relative entropy as a derivative of an interpolation, and proves monotonicity under partial trace using the pullback by the adjoint of the partial trace.

Significance. The mathematical core of the paper is sound and useful: the explicit scalar counterexamples showing that a contraction does not suffice for the Jensen-type inequality, the computation that V_rho is an isometry, and the support lemma for partial traces are correct and are presented cleanly. The paper also provides a reasonably accessible finite-dimensional treatment of Uhlmann's interpolation approach, which is often regarded as opaque. If the historical attribution to Petz is sustained, the paper identifies a genuinely subtle pitfall in a standard proof and explains the mechanism by which the proof is repaired. However, the historical claim is not documented, and the proof of the non-invertible case in Uhlmann's framework has a gap. These issues affect the paper's central novelty and its advertised generality.

major comments (2)
  1. [§3, Eqs. (66)–(68), (73)–(77), and Section 3.2 final remark] The paper's central novelty is the assertion that Petz's original proof in [18] used the contractive Jensen inequality with a mere contraction V_rho and that this step is flawed. The manuscript never quotes [18] nor provides an equation number or page reference for the offending step. If Petz's proof already used the isometry V_rho^dagger V_rho=id, or applied the isometric Jensen inequality directly, then the 'subtle flaw' is an artifact of the authors' reconstruction. This matters because the abstract's claim and the final remark about adjoints of unital Schwarz maps both depend on the attribution. Please supply the exact step from [18], or explicitly reframe the claim as a flaw in the contractive-Jensen strategy rather than in Petz's original proof.
  2. [§4.2, Theorem 4.6, Eqs. (132)–(136)] For non-invertible rho and sigma, the proof that S_{rho||sigma}(id,id)=S(rho||sigma) is incomplete. The definition (132) uses a liminf of a difference quotient of the original interpolation gamma^t_{rho_L -> sigma_R}. The proof instead considers the epsilon-regularized forms S_{rho_epsilon||sigma_epsilon}(id,id) and shows that their limit equals the regularized relative entropy (40). To conclude the theorem, one must justify lim_{epsilon->0} S_{rho_epsilon||sigma_epsilon}(id,id)=S_{rho||sigma}(id,id), e.g. by a continuity or interchange argument, or by a direct computation with the non-invertible functional calculus. As written, the equality (133) is not established for the singular case, which is precisely the case Uhlmann's method is claimed to cover automatically.
minor comments (4)
  1. [§3, Eqs. (75)–(76)] The notation in these inequalities appears inconsistent: both display Delta^a_{rho,sigma}, but the argument requires comparing (V_rho^dagger Delta_{rho,sigma} V_rho + xi)^{-1} with V_rho^dagger (Delta_{rho,sigma}+xi)^{-1} V_rho. Please correct the subscript/superscript so that the displayed inequalities match the quantities used in the proof.
  2. [§3.2, Eq. (90)] Since rho_epsilon=rho+epsilon id_{ab}, one has Tr_b(rho_epsilon)=Tr_b(rho)+epsilon dim(H_b) id_a, not Tr_b(rho)+epsilon id_a. The limit in (91) is unaffected, but the identity should be stated explicitly to avoid confusion with the regularization on H_a.
  3. [§4.2, Remark] The remark states that f_t(x,y)=x^{1-t}y^t is 'jointly operator convex' in (x,y). This is not correct: for scalar x,y and t=1/2, sqrt(xy) is concave, not convex. The intended convexity is probably that t -> A^{1-t}B^t is convex for commuting positive A,B. Please revise or delete the claim.
  4. [References] Reference [16] (Nielsen and Petz) lacks a year. Please add the publication year to complete the citation.

Circularity Check

0 steps flagged

No circular reasoning detected: the monotonicity derivations are self-contained and import only standard external theorems.

full rationale

The paper does not fit any of the circularity patterns. The main derivations are self-contained: Section 2.1 proves the equivalence between the support-based and regularized definitions of relative entropy rather than assuming it; Section 3 demonstrates the failure of the contractive Jensen inequality by explicit scalar counterexamples (Figs. 1–2) and then restores Petz's strategy using the isometric identity V†_ρ V_ρ = id, which licenses the isometric Jensen inequality (Theorem 2.1(iii)); Section 4 provides an independent route via interpolation of positive sesquilinear forms, citing standard external results (Hansen–Pedersen, Pusz–Woronowicz, Stinespring). There is no fitted parameter later renamed as a prediction, no definition of the target quantity smuggled into the hypotheses, and no load-bearing self-citation by the present authors. The only notable weakness is historical rather than circular: the claim that Petz's original proof in [18] relied on the contractive Jensen inequality with a mere contraction is never supported by a quotation, equation number, or page reference from [18]. That attribution gap affects the novelty of the 'flaw' claim, but it does not make the paper's own mathematical derivation circular, since the invalid-contraction argument and its isometric correction are demonstrated directly and independently of what Petz actually wrote.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claims rest on standard operator-theoretic background (Hansen-Pedersen, Stinespring, Pusz-Woronowicz) plus two assertions the paper neither proves nor cites (Theorem 4.3 maximality, Theorem 4.6 continuity) and one historical reading of [18]. No free parameters are fitted: the numbers in Figures 1-2 are illustrative counterexample values. No invented entities are postulated; V_rho is a constructed operator, not an entity with independent physical content.

axioms (6)
  • standard math Hansen-Pedersen Jensen operator inequality and its contractive and isometric forms (Theorem 2.1, Corollary 2.1)
    Gates both the diagnosis of the Petz flaw (contractive form requires f(0) ≤ 0, violated by g_xi) and the correction (isometric form). Invoked in Section 3 and Section 3.1.
  • standard math Stinespring dilation theorem and the CPTP structure of partial trace
    Used in the introduction (eq. (45)) to reduce the data processing inequality for general channels to monotonicity under partial trace.
  • standard math Pusz-Woronowicz functional calculus for positive sesquilinear forms (Theorem 4.2)
    Underpins the whole interpolation construction in Section 4.1; the proof is delegated to reference [21], not reproduced.
  • standard math Maximality of the geometric mean among dominated forms (Theorem 4.3)
    Load-bearing for Theorems 4.4 and 4.5 and hence for the Uhlmann monotonicity proof. The paper's own proof contains an unjustified positivity claim (see red flags), so the reader must rely on the literature for this background fact.
  • ad hoc to paper Continuity of the relative-entropy form under epsilon-regularization, S_{rho||sigma}(id,id) = lim_epsilon S_{rho_epsilon||sigma_epsilon}(id,id)
    Asserted without proof at the end of Theorem 4.6 to extend the interpolation formula to non-invertible states; not established from the lim inf definition (132).
  • domain assumption Fidelity of the reconstruction of Petz's argument in [18] (that Petz used the contractive Jensen inequality on a contraction)
    The paper's headline flaw claim presupposes this historical reading; no quote or page from [18] is provided, so the attribution cannot be checked from the text.

pith-pipeline@v1.3.0-alltime-deepseek · 22515 in / 60174 out tokens · 570328 ms · 2026-08-04T16:59:13.578480+00:00 · methodology

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Cite this review

Pith. "Pith review of On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches." pith.science (2026). https://pith.science/paper/HDXSL2U4

@misc{pith2026250911221,
  author       = {Pith},
  title        = {Pith review of: On the Monotonicity of relative entropy: A Comparative Study of Petz's and Uhlmann's Approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HDXSL2U4}},
  note         = {Machine review of arXiv:2509.11221}
}
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read the original abstract

We revisit the monotonicity of relative entropy under the action of quantum channels, a foundational result in quantum information theory. Among the several available proofs, we focus on those by Petz and Uhlmann, which we reformulate within a unified, finite-dimensional operator-theoretic framework. In the first part, we examine Petz's strategy, identify a subtle flaw in his original use of Jensen's contractive operator inequality, and point out how it was corrected to restore the validity of his line of reasoning. In the second part, we develop Uhlmann's approach, which is based on interpolations of positive sesquilinear forms and applies automatically also to non-invertible density operators. By comparing these two approaches, we highlight their complementary strengths: Petz's method is more direct and clear, Uhlmann's is more abstract and general. Our treatment aims to clarify the mathematical structure underlying the monotonicity of relative entropy and to make these proofs more accessible to a broader audience interested in both the foundations and the applications of quantum information theory.

Figures

Figures reproduced from arXiv: 2509.11221 by Edoardo Provenzi, Francesco Bottacin, Santiago Matheus.

Figure 1
Figure 1. Figure 1: Comparison of (αxα + ξ) −1 and α(x + ξ) −1α, illustrating the failure of the Jensen-type inequality in the scalar case, with α = 0.5 and ξ = 0.5. Note that the inequality − log(V †XV ) ≤ −V † log(X)V is also false for a generic contraction V , as we show in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Counterexample showing that the inequality [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

discussion (0)

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Reference graph

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