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Maximal R\'enyi Relative Entropy for $\alpha>2$

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Pith's one-line read For $\alpha>2$, the largest quantum R\'enyi relative entropy is $D_{\alpha,\alpha-1}$.

desk verdict Completes the maximal Rényi relative entropy characterization for α>2 with a clean two-sided proof; the main caveat is inherited external inputs, not internal gaps. read the letter →

arxiv 2608.01124 v1 pith:U5OR4XQX submitted 2026-08-02 quant-ph

Roberto Rubboli This is my paper · ORCID
classification quant-ph MSC 81P4594A17
keywords R\'enyirelativeentropymaximalquantumextensionalpha-zpreparedgeometricdivergenceGibbs-preservingoperationscatalyticcoherencegenerationmajorization
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

The paper completes the search for the maximal quantum extension of the classical R\'enyi relative entropies. It proves that every quantum relative entropy agreeing with $D_\alpha$ on classical states is bounded above by the geometric R\'enyi entropy $\hat D_\alpha$ for $\alpha\in[0,2]$ and by the $\alpha$-$z$ R\'enyi relative entropy $D_{\alpha,\alpha-1}$ for $\alpha\in(2,\infty]$, closing the one remaining gap. The proof works by showing that the regularized prepared R\'enyi relative entropy equals $D_{\alpha,\alpha-1}$, with an explicit asymptotically optimal preparation map. This characterization also yields necessary and sufficient conditions for catalytically assisted Gibbs-preserving operations to turn an energy-incoherent state into an energy-coherent one.

What carries the argument

The load-bearing object is the prepared R\'enyi relative entropy $D^P_\alpha(\rho\|\sigma)$, defined as the infimum of $D_\alpha(p\|q)$ over classical distributions $(p,q)$ and preparation channels sending $p$ to $\rho$ and $q$ to $\sigma$, together with its regularization over tensor powers. The paper proves that the regularized limit equals $D_{\alpha,\alpha-1}$ for $\alpha>2$ through two mechanisms: a lower bound obtained by applying the data-processing inequality of $D_{\alpha,\alpha-1}$, and an upper bound from an explicit preparation map constructed by pinching $\sigma$ onto the spectral projectors of $\rho$. The map costs $D_{\alpha,\alpha-1}(\rho\|\sigma)+\log|\operatorname{spec}(\rho)|$, and this spectral-size correction grows only logarithmically in the number of copies, so it vanishes after regularization.

What would settle it

Take any $\alpha>2$ and a pair of full-support states $\rho,\sigma$, and compute $D_{\alpha,\alpha-1}(\rho\|\sigma)$ alongside $D_{\alpha,\alpha-1}(\Phi(\rho)\|\Phi(\sigma))$ for a trace-preserving channel $\Phi$ such as a dephasing map; a violation would falsify the imported data-processing inequality and with it the lower bound in Proposition 2.

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Extended reading notes

Core claim

The central claim is Theorem 1: for every pair of quantum states $\rho,\sigma$, every additive quantum relative entropy $D_\alpha$ that reduces to the classical R\'enyi relative entropy on commuting states satisfies $D_\alpha(\rho\|\sigma)\le\hat D_\alpha(\rho\|\sigma)$ for $\alpha\in[0,2]$ and $D_\alpha(\rho\|\sigma)\le D_{\alpha,\alpha-1}(\rho\|\sigma)$ for $\alpha\in(2,\infty]$, where $D_{\alpha,\alpha-1}$ is the $\alpha$-$z$ R\'enyi relative entropy at $z=\alpha-1$. The matching lower and upper bounds on the regularized prepared divergence, Propositions 2 and 5, identify this quantity exactly and establish additivity in the previously open range. Since $D_{\alpha,\alpha-1}$ itself satisfies data processing and additivity for $\alpha\ge2$, the theorem closes the maximal-extension problem.

Load-bearing premise

The load-bearing premise is that the divergence $D_{\alpha,\alpha-1}$ satisfies the data-processing inequality for every $\alpha\ge2$ on all pairs of full-support states; the paper imports this result from the literature rather than proving it.

Editorial extensions

If this is right

  • The maximal quantum extension of $D_\alpha$ is now known for every $\alpha\in[0,\infty]$: the geometric R\'enyi divergence for $\alpha\in[0,2]$ and $D_{\alpha,\alpha-1}$ for $\alpha>2$.
  • The regularized prepared R\'enyi divergence is additive for $\alpha>2$, so it is a genuine quantum relative entropy and can serve as a monotone in quantum resource theories.
  • The optimal asymptotic rate for transforming a classical dichotomy $(p,q)$ into a quantum dichotomy $(\rho,\sigma)$ is the minimum over both argument orderings of the ratios of these maximal divergences, as stated in Corollary 10.
  • An energy-incoherent state $p$ can be catalytically converted into a coherent state $\rho$ by a Gibbs-preserving operation exactly when the R\'enyi inequalities $D(p\|\gamma)\ge D(\rho\|\gamma)$ hold for every divergence in the completed family, including both orderings, as stated in Theorem 12.
  • At $\alpha=\infty$, the bound reduces to the max-relative entropy, so the characterization includes the sharp ordering by $D_{\max}$ in the limiting case.

Reading between the lines

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

  • The paper does not pursue it, but the same pinching-based preparation map should transfer to other resource theories whose monotone is a prepared divergence, because the spectral correction always vanishes under regularization.
  • A testable extension would allow the catalyst to be correlated with the system; the paper's criterion covers uncorrelated catalysts with approximate conversion, and it is open whether allowing correlation relaxes the inequalities in Theorem 12.
  • Because the maximal extension is now fixed, any new candidate quantum R\'enyi-like divergence for $\alpha>2$ can be screened simply by checking whether it lies between the sandwiched divergence and $D_{\alpha,\alpha-1}$ on a pair of test states.
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Editorial analysis

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

Referee Report

0 major / 4 minor

Summary. The paper completes the identification of the maximal quantum extension of the classical Rényi relative entropy. For α>2 it proves that every quantum relative entropy reducing to the classical Rényi relative entropy D_α on commuting states is bounded above by the α-z Rényi relative entropy D_{α,α−1} with z=α−1. The proof is via the prepared Rényi divergence: Proposition 2 gives the lower bound on the regularized prepared divergence from the data-processing inequality of D_{α,α−1}, while Lemma 4 and Proposition 5 construct an explicit preparation whose cost matches D_{α,α−1} up to a logarithmic spectral-size correction that vanishes under regularization. The paper then applies this result to large-sample and catalytic conversion of classical-to-quantum dichotomies, obtaining a rate formula and a characterization of catalytic coherence generation by Gibbs-preserving operations.

Significance. If the result holds, it closes a long-standing gap and provides the maximal element among quantum Rényi relative entropies for the entire range α≥0. The proof is constructive: the preparation map in Lemma 4 is explicit and asymptotically optimal, and the upper-bound argument is elementary and checkable. The applications show that the maximal divergences determine transformation rates and catalytic conversion criteria. The main caveat is that the converse bound in Proposition 2 imports the data-processing inequality for D_{α,α−1} at α≥2 from Zhang [43]; this is a standard external result, but it is genuinely load-bearing for Theorem 1.

minor comments (4)
  1. [§III, Proposition 2] The lower-bound proof is only a few lines and leans entirely on the data-processing inequality for D_{α,α−1} from [43]; please state the precise theorem and its support assumptions so the reader can verify that it applies to all pairs with supp(ρ)⊆supp(σ), including non-faithful states.
  2. [§IV, Lemma 4] In Eq. (66), the use of Lemma 13 is correct but terse; a one-sentence reminder that q=α−1 and that the pinching is with respect to the spectral projectors of ρ would make the comparison with Q_{α,α−1} easier to follow.
  3. [§V.C, Definition 11 and Theorem 12] The continuity step at the end of Theorem 12 is used to pass from ρ_ε to ρ; since the divergences in D are not all globally continuous on arbitrary pairs, the proof should explicitly state that continuity is used on faithful pairs and that the constructed states are full-rank.
  4. [§V.B, Corollary 10] The rate formula in Eq. (125) is undefined in the degenerate case p=q and ρ=σ, where ratios 0/0 occur; a convention should be added or the case excluded explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the maximal-extension proof is self-contained except for standard external DPI citations.

full rationale

The central claim (Theorem 1, Eq. 8) is established by proving that the regularized prepared Rényi relative entropy equals D_{α,α−1} for α>2. The lower bound (Prop. 2, Eq. 39) uses the data-processing inequality for D_{α,α−1}, cited to Zhang [43]; this is an external published theorem, not a result of the present paper, and it is not used to define the maximal extension. The upper bound (Lemma 4 and Prop. 5) is an explicit construction: it builds a feasible preparation map and bounds its cost by D_{α,α−1} plus a logarithmic spectral correction, using the pinching inequality, operator monotonicity of the inverse, and the Schatten contractivity of pinching proven in Lemma 13. These steps are algebraic and do not presuppose that D_{α,α−1} is maximal. The only self-citation is to the author's prior thesis for the conjecture being resolved, which is not load-bearing. No fitted parameters are renamed as predictions, and no uniqueness theorem is imported from the authors' own prior work. The application to catalytic coherence generation inherits the theorem but rests on the same independent DPI citations. Therefore there is no circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard definitions of quantum relative entropies, two external DPI results, and semiring theorems for the application. No free parameters are fitted and no new physical or mathematical entities are introduced; the quantum-majorization semiring is imported from [17].

assumptions (7)
  • domain assumption Quantum relative entropies are defined by the data-processing inequality and additivity under tensor products, and their classical restriction is the classical Rényi relative entropy.
    This defines the object bounded in Theorem 1 and is introduced in Section II.B.
  • standard math The α-z Rényi relative entropy D_{α,α−1} satisfies the data-processing inequality for all α≥2.
    Imported from Zhang [43] and used in Proposition 2 to obtain the lower bound on the regularized prepared relative entropy.
  • domain assumption Every additive quantum relative entropy extending the classical Rényi relative entropy lies between the regularized measured and regularized prepared relative entropies.
    Derived from DPI and additivity in Section II.C, this is the bridge that turns the prepared entropy computation into a maximality statement.
  • standard math Pinching inequality and Schatten contractivity of pinching as stated in Lemma 13.
    Proved in Appendix A and used in Lemma 4 for the achievability construction.
  • standard math Matsumoto's reverse-test variational form for the prepared f-divergence specializes to Lemma 3 for f(t)=t^α.
    External result cited and specialized in Section IV; it provides the variational formula for Q^P_α that underlies the construction.
  • standard math Fritz's Vergleichsstellensatz for preordered semirings, Theorem 8.6 in [12], and the semiring classification of classical Rényi entropies from [10].
    Imported in Section V.B and used to derive the transformation criteria in Theorems 9 and 12; not reproved in this paper.
  • domain assumption The divergences in D are continuous on pairs of faithful states.
    Used at the end of Theorem 12 to pass from the approximate states ρ_ε to the limit ρ.

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Pith. "Pith review of Maximal R\'enyi Relative Entropy for $\alpha>2$." pith.science (2026). https://pith.science/paper/U5OR4XQX

@misc{pith2026260801124,
  author       = {Pith},
  title        = {Pith review of: Maximal R\'enyi Relative Entropy for $\alpha>2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U5OR4XQX}},
  note         = {Machine review of arXiv:2608.01124}
}
abstract

Quantum relative entropies play a fundamental role in quantum information theory. In the classical setting, R\'enyi relative entropies constitute, up to linear combinations, the most general class of relative entropies, naturally motivating the search for their minimal and maximal quantum extensions. The minimal extension is known to be the reverse sandwiched R\'enyi relative entropy for $\alpha\in[0,1/2)$ and the sandwiched R\'enyi relative entropy for $\alpha\geq 1/2$. In contrast, the maximal extension had previously been identified only for $\alpha\in[0,2]$, where it is given by the geometric R\'enyi relative entropy. In this work, we complete this characterization by proving that for $\alpha>2$, the maximal extension is given by the $\alpha$-$z$ R\'enyi relative entropy with $z=\alpha-1$. As an application, we determine when an energy-incoherent state can be transformed into an energy-coherent state by a Gibbs-preserving operation assisted by an uncorrelated catalyst, thereby fully characterizing the coherence-generating power of this class of operations in the catalytic setting.

Figures

Figures reproduced from arXiv: 2608.01124 by the authors.

Figure 1
Figure 1. FIG. 1: Minimal and maximal relative entropies evaluated for the states [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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