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Uhlmann's theorem for relative entropies

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Pith's one-line read Uhlmann's equality now holds for every α-Rényi relative entropy

desk verdict A genuinely new result with an honest proof sketch, but the central proof has a repairable gap around the map E in Theorem 3.1 that the referee will need to see fixed. read the letter →

arxiv 2502.01749 v2 pith:TN74D5KV submitted 2025-02-03 quant-ph

Giulia Mazzola , David Sutter , Renato Renner This is my paper · ORCID
classification quant-ph MSC 81P4594A17
keywords Uhlmann'stheoremsandwichedRényirelativeentropyquantumfidelitymax-relativeregularizationmeasureddeFinettireductionspectralpinching
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Uhlmann's theorem says that for any two quantum states ρ_AB and σ_A, the fidelity between ρ_AB and a suitable extension of σ_A equals the fidelity of the reduced states. This paper extends that equality to the whole family of sandwiched α-Rényi relative entropies, which includes fidelity, relative entropy, and max-relative entropy as special cases. A plain single-copy equality is impossible for intermediate α, so the authors prove two substitutes: a regularized equality that holds asymptotically over many copies, and a measured version in which both sides are preceded by an optimized measurement. The results matter because they let a divergence defined on a small system be evaluated through extensions on a larger system, which is what makes Uhlmann's theorem useful in quantum information.

What carries the argument

The proof imports Uhlmann's original identity for fidelity and the known α > 1 case from [21], and then handles α ∈ [1/2, 1) by a two-step lifting of the classical construction. First, asymptotic spectral pinching with respect to $σ_A^{{⊗n}}$ and its pinched reduction ρ'_{A^n} makes the relevant operators commute, producing a feasible extension σ̄_{A^n B^n} of $σ_A^{{⊗n}}$ whose Rényi divergence from $ρ_AB^{{⊗n}}$ is close to D_α(ρ_A‖σ_A). Second, a refined postselection de Finetti reduction with fixed marginal σ_A (Equation (37), imported from [11]) bounds the support function of the extension set: (1/n) log h_{$C^{{σ_A}}$_{AB,n}}($τ^{{⊗n}}$) = log h_{$C^{{σ_A}}$_{AB}}(τ) + o(n)/n, which is the asymptotic multiplicativity that yields the measured inequality. Variational formulas for the measured Rényi divergence and Sion's minimax theorem then convert the support-function bound into single-letter statements.

What would settle it

Compute, for a specific pair of finite-dimensional states such as ρ_AB = |φ⟩⟨φ|_AB with |φ⟩ = √(1/4)|00⟩ + √(3/4)|11⟩ and σ_A = 1/2 I_A, the sequence (1/n) D_{3/2}($ρ_AB^{{⊗n}}$ ‖ $C^{{σ_A^{⊗n}}$}_{A^n B^n}) for n = 1, 2, 3, ... and check whether it converges to D_{3/2}(ρ_A‖σ_A). A limit below D_{3/2}(ρ_A‖σ_A) would disprove the regularized Uhlmann theorem for α = 3/2.

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Extended reading notes

Core claim

The central claim is that for any finite-dimensional ρ_AB, σ_A and any α ∈ [1/2, ∞], D_α(ρ_A‖σ_A) equals the regularized minimum over extensions of σ_A of D_α(ρ_AB‖·), namely D^∞_α(ρ_AB‖$C^{{σ_A}}$_AB) = lim_{n→∞} (1/n) D_α($ρ_AB^{{⊗n}}$‖$C^{{σ_A^{⊗n}}$}_{A^n B^n}). This is Theorem 3.1 for α ≠ 1 and Theorem 4.1 for α = 1, where the divergences are the sandwiched Rényi relative entropies (with the relative entropy and max-relative entropy at the endpoints). Because the unregularized equality fails for α in (1/2, ∞), the paper also proves the measured Uhlmann theorem: D_{α,M}(ρ_A‖σ_A) ≤ D_{α,M}(ρ_AB‖$C^{{σ_A}}$_AB) ≤ D_α(ρ_A‖σ_A), where the measured divergence maximizes over POVMs. Together these results establish that Uhlmann-type identities are not specific to fidelity but hold, in the appropriate sense, for the entire Rényi family.

Load-bearing premise

The argument stands on a de Finetti-type bound stating that any permutation-symmetric extension of $σ_A^{{⊗n}}$ is dominated, up to a polynomial factor, by a mixture of tensor powers of single-copy extensions of σ_A; if that bound fails, the support-function lemma and the measured Uhlmann inequality lose their single-letter form.

Editorial extensions

If this is right

  • For every α ∈ [1/2, ∞], the divergence D_α(ρ_A‖σ_A) can be expressed as a regularized minimization over extensions, giving a variational handle on Rényi divergences that does not require diagonalizing ρ_A.
  • For the relative entropy (α = 1), the regularized optimizer has an explicit form, making the equality constructive rather than an abstract limit.
  • The measured Uhlmann inequalities provide single-letter upper and lower bounds for the regularized quantity in terms of convex optimization problems, since D_{α,M}(ρ_AB‖C^{σ_A}_AB) and D_α(ρ_AB‖C^{σ_A}_AB) are convex.
  • For α ∈ (1/2, ∞), the single-copy quantity D_α(ρ_AB‖C^{σ_A}_AB) is not additive under tensor products, while at the endpoints α = 1/2 and α = ∞ it is exactly additive.
  • When [ρ_A, σ_A] = 0, the measured and regularized quantities coincide, recovering the classical case where regularization is unnecessary.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit optimizer in Remark 4.1 is formally a Petz recovery map; if that link is exploitable, the regularized Uhlmann equality may yield new recovery-error bounds for approximate quantum Markov chains.
  • The de Finetti bound's polynomial overhead depends on d_A^2 d_B^4; tightening this dependence would sharpen the single-letter measured inequality, and the paper's methods suggest testing whether the exponent can be improved.
  • One could test whether the measured Uhlmann inequality is tight under optimized POVMs for states with non-commuting ρ_A and σ_A, which would give a single-letter converse for the regularization.
  • The impossibility result in Appendix A suggests that any sufficient divergence obeying a data-processing inequality will need some regularization or measurement before an Uhlmann-type identity can hold; this may guide the search for other divergence families.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper generalizes Uhlmann's theorem for fidelity to sandwiched Rényi relative entropies D_α for α∈[1/2,∞]. The two main results are a regularized Uhlmann theorem, D_α(ρ_A‖σ_A)=D^∞_α(ρ_AB‖C^{σ_A}_{AB}) (Theorem 3.1, extended to α=1 in Theorem 4.1), and a single-letter measured Uhlmann theorem, D_{α,M}(ρ_A‖σ_A)≤D_{α,M}(ρ_AB‖C^{σ_A}_{AB})≤D_α(ρ_A‖σ_A) (Theorem 3.2, extended to α=1 in Theorem 4.2). The proofs use asymptotic spectral pinching, a postselection de Finetti reduction imported from [11], variational formulas for measured Rényi divergences, and Sion's minimax theorem. The paper also gives explicit constructions of extensions for the relative entropy case and discusses why a non-regularized Uhlmann equality cannot hold for intermediate α.

Significance. If correct, these results settle a natural question: Uhlmann-type equalities hold for all α-Rényi relative entropies in the regularized sense, and the measured version provides single-letter bounds. The paper cleanly separates the new regime α∈[1/2,1) and the measured results from previously known α>1 results, and it gives explicit optimizers for α=1. The proof is largely self-contained and clearly cites the one non-elementary input, the postselection de Finetti reduction [11, Cor. 3.2], which underpins the measured Uhlmann theorem via Lemma 3.2. The main technical defect I found is localized to the trace-preservation claim for the map E in Theorem 3.1; this is fixable without changing the structure of the proof.

major comments (1)
  1. [Theorem 3.1, Eqs. (41)–(49)] The map E defined in Eq. (41) as E(X)=ρ̄^{1/2}ρ̄_A^{-1/2}Xρ̄_A^{-1/2}ρ̄^{1/2} is claimed to be trace-preserving, but this is false in general. With V=ρ̄^{1/2}(ρ̄_A^{-1/2}⊗I_B), a direct computation gives tr_B[V†V]=ρ̄_A^{-1/2}tr_B[ρ̄]ρ̄_A^{-1/2}=Π_{supp(ρ̄_A)}, the projection onto the support of ρ̄_A, not the identity unless ρ̄_A has full rank. Consequently the DPI step in Eq. (48) is not justified as written. The gap is repairable: the proof of Eqs. (44)–(47) forces supp(σ_A^{⊗n})⊆supp(ρ̄_A), so one can extend E to a genuine TPCP map E'(X)=VXV†+KXK† with K=(I_A−Π_{supp(ρ̄_A)})⊗|φ⟩_B for a fixed normalized |φ⟩_B. This map agrees with E on ρ̄_A and σ_A^{⊗n}, and the DPI step becomes valid. The manuscript should be revised to either define this extension or otherwise justify the DPI step.
minor comments (4)
  1. [Lemma 3.3, Eqs. (95)–(100)] The proof applies Sion's minimax theorem with the set of positive definite matrices as one of the convex sets. This set is not compact, but the concern that this invalidates the argument does not land: Sion's theorem only requires compactness of one of the convex sets, and C^{σ_A}_{AB} is compact. I would nonetheless ask the authors to spell out the required semicontinuity and quasi-concavity/quasi-convexity conditions explicitly, especially for the α>1 case where the objective is concave in τ rather than convex.
  2. [Section 3.1, footnote 12] The notation ρ̄_A^{-1/2} is used for the inverse on the support of ρ̄_A; this should be defined explicitly as the pseudoinverse or with a phrase such as 'restricted to the support of ρ̄_A' to avoid ambiguity.
  3. [Lemma 3.3, Eq. (101)] The step labeled (⋄) is a scaling argument rather than a pointwise AM-GM comparison at the same τ; the current wording is terse. A short explanation that for every τ one can choose λ so that the sum at λτ equals the geometric term at τ would make the direction of the inequality clear.
  4. [Lemma 3.2, proof] The phrase 'a relaxed asymptotic version of the polar assumption introduced in [9, Lemma 8]' is informal and not needed for the proof. Since Lemma 3.2 is a workhorse for the measured Uhlmann theorem, consider removing this aside or explaining the connection in one sentence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new alpha<1 Uhlmann-type proof is constructed from pinching and operator monotonicity, and the cited prior results are independent published tools.

full rationale

The derivation is self-contained against external benchmarks. The genuinely new range alpha in [1/2,1) is proved constructively in Theorem 3.1 via the pinching construction (Equations (40)-(65)): the DPI gives one inequality and the pinching inequality together with operator monotonicity gives the other, with no parameter fitted to the target quantity and no use of the alpha<1 Uhlmann identity as an input. The alpha>1 and alpha=infinity cases are imported from published results ([21, Lemma 3.3] and [22, Corollary A.2]); although [21] shares authors with the present paper, it is a separately published theorem and its assumptions do not include the present claim, so this is legitimate citation rather than circularity. The alpha=1 regularized theorem follows by the continuity lemma B.6, and the measured theorems use the postselection de Finetti reduction (Equation (37)) from [11] only as a tool to establish asymptotic multiplicativity of support functions (Lemma 3.2); that reduction is a published structural result, not a disguised form of the Uhlmann identity. The explicit optimizers in Remarks 4.1 and 4.4 are derived from the independent multivariate Golden-Thompson inequality, not assumed. The only substantive concern in the proof is whether the map E in Equation (41) is trace-preserving as written: its Kraus operator satisfies Vdagger V = Pi_supp(bar_rho_A), not I, when bar_rho_A is rank-deficient, which is a rigor gap in the DPI step of Equations (48)-(49) and is potentially repairable; however, this is a correctness issue rather than a circularity, so it does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The central claim rests on standard theorems of quantum information and convex analysis. No free parameters are introduced and no new entities are postulated. The only subtle point is the implicit compactness assumption in the minimax applications, which is flagged in red_flags.

assumptions (10)
  • domain assumption Finite-dimensional Hilbert spaces; all density matrices are finite-dimensional and the relevant divergences are lower semi-continuous.
    Stated in the introduction and used throughout, e.g., to replace infimum by minimum in Eq. (1).
  • standard math Sandwiched Renyi relative entropy satisfies the data-processing inequality.
    Invoked in Eq. (3), Eq. (39), and the proof of Theorem 3.2; cited to [39, Theorem 4.18] and [13, Theorem 1].
  • standard math Measured Renyi and measured relative entropies satisfy the variational formulas (21), (22), (29), and (30).
    Used in Lemma 3.3 and Appendix C; cited to [2, Lemma 3] and [2, Lemma 1].
  • standard math Postselection de Finetti reduction with fixed marginal, Eq. (37).
    Imported from [11, Corollary 3.2]; underpins Lemma 3.2 and hence Theorem 3.2.
  • standard math Sion's minimax theorem.
    Used to swap infimum and supremum in Lemma 3.3 and Lemma 4.6; cited to [33]. The non-compact domain of tau is an unaddressed technical point flagged in red_flags.
  • standard math Properties of pinching maps, including the pinching inequality and commutation relations (I)-(VII).
    Used throughout Theorem 3.1 and Fact B.3; cited to [34, Section 3.1] and [21, Lemma 2.7].
  • standard math For permutation-invariant states on a tensor power Hilbert space, the number of distinct eigenvalues is poly(n).
    Used in Fact B.3 to bound the pinching penalty; cited to [21, Lemma 2.8].
  • standard math D_alpha is continuous and monotone in alpha, with limits D_1 = D and D_infinity = D_max.
    Used in Lemma B.6, Lemma B.7, and Theorem 4.1; cited to [25, Theorems 5 and 7] and [22, Corollary A.2].
  • standard math Additivity of D_alpha and D under tensor products.
    Used in Theorem 3.1 and Remark 4.1; standard for the sandwiched Renyi divergences.
  • standard math Multivariate Golden-Thompson inequality, Eq. (203).
    Used in Appendix C for the alternative proof of Theorem 4.2; cited to [35, 16, 34].

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Pith. "Pith review of Uhlmann's theorem for relative entropies." pith.science (2026). https://pith.science/paper/TN74D5KV

@misc{pith2026250201749,
  author       = {Pith},
  title        = {Pith review of: Uhlmann's theorem for relative entropies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TN74D5KV}},
  note         = {Machine review of arXiv:2502.01749}
}
abstract

Uhlmann's theorem states that, for any two quantum states $\rho_{AB}$ and $\sigma_A$, there exists an extension $\sigma_{AB}$ of $\sigma_A$ such that the fidelity between $\rho_{AB}$ and $\sigma_{AB}$ equals the fidelity between their reduced states $\rho_A$ and $\sigma_A$. In this work, we generalize Uhlmann's theorem to $\alpha$-R\'enyi relative entropies for $\alpha \in [\frac{1}{2},\infty]$, a family of divergences that encompasses fidelity, relative entropy, and max-relative entropy corresponding to $\alpha=\frac{1}{2}$, $\alpha=1$, and $\alpha=\infty$, respectively.

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