REVIEW 2 major objections 5 minor 15 references
This paper proves that the variational objectives behind measured relative entropy and measured Rényi relative entropy are β-smooth and γ-strongly concave/convex, so accelerated projected gradient ascent/descent computes them to precision ε
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
Measured relative entropies can be computed by Nesterov accelerated gradient descent/ascent because their variational objective functions are smooth and strongly convex/concave.
T0 review reviewed 2026-08-03 challenge →
load-bearing objection A mostly careful paper on Nesterov acceleration for measured relative entropies, with a real sign error in the alpha>1 branch that is easy to fix but invalidates Algorithm 15 as printed. the 2 major comments →
Accelerated optimization of measured relative entropies
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
Core claim
The central discovery is that the variational functions behind measured relative entropy and measured Rényi relative entropy are not merely convex or concave but strongly so, with explicit constants. For h_ρ,σ(ω) = Tr[(ln ω)ρ] + 1 − Tr[ωσ], the paper establishes that the Hessian superoperator has all eigenvalues in [−β, −γ] on the operator interval containing the optimal ω, where the interval endpoints are max-relative-entropy bounds for ρ and σ, and where β is proportional to λ_max(ρ) while γ is proportional to λ_min(ρ). Similar explicit eigenvalue bounds are proved for the Rényi objectives g^α and h^α for every α ∈ (0,1) ∪ (1,∞), with signs matching convexity for α < 1 and concavity for α
What carries the argument
The load-bearing machinery is the Hessian superoperator of the variational objective, viewed as a self-adjoint superoperator on Hilbert–Schmidt space. The paper evaluates its eigenvalues variationally, bounding them between −β and −γ (for concave objectives) or between γ and β (for convex objectives) on the operator interval containing the optimal ω. Combined with the matrix gradient formulas written in terms of divided differences or convergent integral representations, this reduces computation of a quantum distinguishability measure to a standard accelerated projected gradient method: repeated gradient updates, projection by eigenvalue clamping, and momentum steps. The eigenvalue bounds gi
Load-bearing premise
The entire analysis assumes ρ and σ have strictly positive eigenvalues; if either state is rank-deficient, the strong-convexity parameter γ collapses to zero and the accelerated O(√κ log(1/ε)) guarantee and stopping test cease to apply.
What would settle it
Take a rank-deficient pair such as ρ = |0⟩⟨0| and σ = (1−ε)|0⟩⟨0| + ε|1⟩⟨1|, compute the Hessian superoperator of h_ρ,σ at the optimal interval, and check whether its largest eigenvalue is ≤ −γ for any positive γ from Corollary 7; equivalently, run the paper’s Algorithm 8 on that pair and test whether the stopping rule ||∇h|| ≤ √(2γε) ever triggers with γ > 0. If γ is forced to zero, the accelerated iteration bound cannot hold for that input.
If this is right
- Measured relative entropy and measured Rényi relative entropy become computable by first-order methods with O(√κ log(1/ε)) iterations, each requiring O(d³) time for d-dimensional states.
- Memory use drops to O(d²), since only a constant number of d×d matrices need to be stored during the gradient updates.
- For well-conditioned states, the new approach is asymptotically faster than the earlier semi-definite programming route because it avoids the precision-dependent matrix inequalities and lifted matrix dimensions of SDP formulations.
- The stopping rule ||∇f|| ≤ √(2γε) gives a certificate that the objective value is within ε of the optimum.
- All Rényi parameter regimes are covered: α ∈ (0, 1/2] via convex minimization of g^α, and α ∈ (1/2, 1) ∪ (1, ∞) via convex or concave optimization of h^α.
Where Pith is reading between the lines
- A natural extension the paper leaves open is to channel versions of measured relative entropy; the same smoothness/strong-convexity framework could plausibly carry over if a comparable variational structure exists there.
- Since the γ parameters vanish for rank-deficient states, practitioners may need to add a small positive perturbation to make ρ and σ positive definite; one could likely track how the resulting approximation error scales with the perturbation through the same β/γ bounds.
- The divided-difference and integral formulas for gradients and Hessians may transfer to other quantum divergences and to quantum Fisher information matrices, which share similar functional-calculus structure.
- Numerically comparing the paper’s β/γ bounds with empirically computed Hessian spectra on random well-conditioned pairs would reveal whether the quoted condition numbers are tight or conservative; a tighter γ would directly speed up the iteration bound.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the variational formulas for measured relative entropy D_M(ρ∥σ) and measured Rényi relative entropy D_M^α(ρ∥σ) given in (3.2) and (4.3). The authors compute explicit matrix gradients and Hessian superoperators for the objectives h_{ρ,σ}, g^α_{ρ,σ}, and h^α_{ρ,σ}, and then prove β-smoothness and γ-strong convexity/concavity on operator intervals that contain the optimizer (Corollaries 7 and 13). Based on these properties, they propose Nesterov accelerated projected gradient descent/ascent algorithms (Algorithms 8, 14, and 15) and claim O(√κ log(1/ε)) iterations, O(d^3√κ log(1/ε)) total cost, and O(d^2) memory. The paper also compares these costs with the earlier SDP approach [HW24]. The analysis is carried out under the standing assumption that ρ and σ are positive definite.
Significance. If the main claims are correct, the paper provides a significant and useful advance: it gives explicit, parameter-free smoothness and strong-convexity/concavity parameters for the variational objectives of measured relative entropies, and it shows how to use them with Nesterov acceleration. The derivations in Sections 3–4 and the appendices are detailed and largely self-contained, and no fitted parameters are introduced. The cost comparison with the prior SDP algorithm is concrete. However, the α>1 branch of Corollary 13 and Algorithm 15 contains a sign error that makes that branch non-executable as printed, and the advertised domain of the algorithms is more general than the full-rank setting in which the bounds are proven.
major comments (2)
- [§4.3 (Corollary 13) and §4.4.2 (Algorithm 15)] For α>1, the factor (1−α)/α is negative. Corollary 13 defines β_α and γ_α in (4.109)–(4.110) with this negative factor, whereas Lemma 12, Eqs. (4.64)–(4.66), shows that the Hessian of h^α has all eigenvalues in the interval [−λmax(ρ)(α−1)/α · ..., −λmin(ρ)(α−1)/α · ...]. Thus for α>1 the function h^α is γ-strongly concave with a positive parameter γ = λmin(ρ)((α−1)/α) ||σ^{-1/2}ρσ^{-1/2}||^{-(α+1)}, not with the negative γ_α printed in Corollary 13. As stated, the stopping test √(2γ_α ε) in Algorithm 15 is not real, and the update (4.120) with β_α<0 moves in the wrong direction for a concave objective. The claimed O(√κ log(1/ε)) iteration complexity for α>1 is therefore not executable as written. The fix is local—replace (1−α)/α by (α−1)/α in the α>1 branch, and make explicit that the α∈(1/2,1) branch is strongly convex while the α>1 branch is strongly concave—but it must be corrected be
- [Sections 3 and 4 (scope of the main claim)] The paper's abstract and introduction state that the algorithms compute the measured relative entropies of quantum states, without qualifying that the states must be positive definite. In the body, the analysis assumes ρ and σ are positive definite, and all strong-convexity/concavity parameters are proportional to λ_min(ρ) or λ_min(σ); the condition numbers κ, κ_α similarly contain λmax/λmin. For rank-deficient states, these parameters collapse, the stopping criterion ||∇f|| ≤ √(2γε) degenerates, and the O(√κ log(1/ε)) guarantee is vacuous. This is a domain limitation rather than an internal inconsistency, but it should be stated prominently in the abstract and conclusions, and the paper should either restrict the advertised scope to full-rank states or discuss a perturbation/regularization strategy.
minor comments (5)
- [Lemma 11, proof around Eq. (4.53)] In the proof of the Hessian of h^α, the line after (4.52) writes the t-power as t^{α/(α−1)}; it should be t^{(α−1)/α}. Also, the sentence after (4.53) says 'we conclude (4.44)' but for h^α the correct target is (4.45). These are typesetting/typo issues but should be fixed.
- [Algorithm 15, Eq. (4.120)] The update uses '1/β' without a subscript; it should be '1/β_α' to match the notation of Corollary 13.
- [Corollary 13, wording] The phrase 'γα-strongly convex / concave' conflates two distinct regimes. For α∈(1/2,1), γα>0 and h^α is strongly convex; for α>1, after the correction above, γα>0 and h^α is strongly concave. Please state the two cases separately.
- [Corollary 7 and related interval notation] Corollary 7 writes the interval as ω∈(ℓI, uI) with open parentheses, but Lemma 6 and the optimization algorithms use the closed interval I=[ℓ,u]. Since the endpoint operators ω=ℓI and ω=uI are positive definite and satisfy the bounds, the closed interval is the intended one; please make the notation consistent.
- [Lemmas 2 and 9, references to [SGW25]] The text says that some gradient formulas were 'recently presented in [SGW25]', which is an in-preparation reference. Since the formulas are derived in the present paper, this dependence is not circular, but if the properties are independently claimed, a public reference or a statement that they are rederived here would be helpful.
Circularity Check
No significant circularity: the smoothness/strong-concavity claims and algorithms are derived in-paper from external variational formulas, not assumed from fitted inputs or self-citations.
full rationale
The variational expressions (3.2)-(3.3) and (4.3)-(4.5) are quoted from [BFT17], an external prior theorem. The paper's new content is the derivation of the gradients (Lemmas 2, 9), Hessians (Lemmas 5, 11), eigenvalue bounds (Lemmas 6, 12), and consequent beta/gamma parameters (Corollaries 7, 13). These are computed directly from the objectives rather than fitted to the target values. Algorithms 8, 14, and 15 are standard Nesterov methods applied once beta and gamma are established; the convergence guarantees use external textbook results ([Bub15]), not the paper's own conclusions. The references to the authors' in-preparation [SGW25] for 'some properties' are explicitly for completeness and the paper supplies its own proofs; [Wil25] is used only for a standard integral identity, which is parameter-free and independent of the measured-entropy target. The rank-definiteness restriction and any alpha>1 sign issue are correctness/domain concerns, not circularity: they do not make an output equal to an input by construction. No fitted parameter is renamed as a prediction, and no uniqueness or ansatz is imported from the authors' prior work as a load-bearing premise.
Axiom & Free-Parameter Ledger
axioms (4)
- domain assumption Variational formulas from BFT17: D_M(ρ||σ) = sup_{ω>0} h_{ρ,σ}(ω) with h = Tr[ρ ln ω] + 1 − Tr[σω]; similarly for measured Rényi (4.3)–(4.5).
- domain assumption ρ and σ are positive definite, so λ_min(ρ), λ_min(σ) > 0; the strong-convexity/concavity constants γ are proportional to these minimum eigenvalues.
- standard math Nesterov accelerated projected gradient theorem (Bubeck 2015, Thm 3.18) gives O(√κ log(1/ε)) complexity and the gradient-norm stopping condition.
- standard math Matrix derivative rules and integral representations in Appendix A (e.g., (A.25), (A.27), (B.5), (B.25)) used to derive gradients and Hessians.
Cite this review
Pith. "Pith review of Accelerated optimization of measured relative entropies." pith.science (2026). https://pith.science/paper/BFJN7U5U
@misc{pith2026251117976,
author = {Pith},
title = {Pith review of: Accelerated optimization of measured relative entropies},
year = {2026},
howpublished = {\url{https://pith.science/paper/BFJN7U5U}},
note = {Machine review of arXiv:2511.17976}
}
abstract
The measured relative entropy and measured R\'enyi relative entropy are quantifiers of the distinguishability of two quantum states $\rho$ and $\sigma$. They are defined as the maximum classical relative entropy or R\'enyi relative entropy realizable by performing a measurement on $\rho$ and $\sigma$, and they have interpretations in terms of asymptotic quantum hypothesis testing. Crucially, they can be rewritten in terms of variational formulas involving the optimization of a concave or convex objective function over the set of positive definite operators. In this paper, we establish foundational properties of these objective functions by analyzing their matrix gradients and Hessian superoperators; namely, we prove that these objective functions are $\beta$-smooth and $\gamma$-strongly convex / concave, where $\beta$ and $\gamma$ depend on the max-relative entropies of $\rho$ and $\sigma$. A practical consequence of these properties is that we can conduct Nesterov accelerated projected gradient descent / ascent, a well known classical optimization technique, to calculate the measured relative entropy and measured R\'enyi relative entropy to arbitrary precision. These algorithms are generally more memory efficient than our previous algorithms based on semi-definite optimization [Huang and Wilde, arXiv:2406.19060], and for well conditioned states $\rho$ and $\sigma$, these algorithms are notably faster.
Reference graph
Works this paper leans on
-
[4]
arXiv:1512.03401. URL:https://www.sciencedirect. com/science/article/pii/S0024379516304852,doi:10.1016/j.laa.2016. 10.012. [FSP19] Hamza Fawzi, James Saunderson, and Pablo A. Parrilo. Semidefinite approx- imations of the matrix logarithm.Foundations of Computational Mathemat- ics, 19(2):259–296,
Pith/arXiv arXiv 2016
-
[7]
doi:10.1103/PhysRevA.109.032431. [Hel02] Christoph Helmberg. Semidefinite programming.European Jour- nal of Operational Research, 137(3):461–482,
-
[10]
org/abs/2406.19060,arXiv:2406.19060
URL:https://arxiv. org/abs/2406.19060,arXiv:2406.19060. [Löf22] Johan Löfberg. YALMIP: Modeling and optimization in MATLAB.SoftwareX, 17:100918,
-
[11]
URL:https://arxiv.org/abs/1412.3676, arXiv:1412.3676. [MN19] Jan R. Magnus and Heinz Neudecker.Matrix Differential Calculus with Applica- tions in Statistics and Econometrics. John Wiley & Sons, Chichester, 3 edition, 2019.doi:10.1002/9781119541219. [MO15] Milán Mosonyi and Tomohiro Ogawa. Quantum hypothesis testing and the operational interpretation of t...
Pith/arXiv arXiv 2019
-
[13]
33 [SGW25] Sreejith Sreekumar, Ziv Goldfeld, and Mark M
URL:https://link.aps.org/doi/10.1103/PhysRevLett.103.160504,doi: 10.1103/PhysRevLett.103.160504. 33 [SGW25] Sreejith Sreekumar, Ziv Goldfeld, and Mark M. Wilde. Performance guarantees for quantum neural estimation of entropies
-
[1983]
Springer, 2004.doi:10.1007/ 978-1-4419-8853-9
[Nes04] Yurii Nesterov.Introductory Lectures on Convex Optimization: A Basic Course, volume 87 ofApplied Optimization. Springer, 2004.doi:10.1007/ 978-1-4419-8853-9. [NN94] Yurii Nesterov and Arkadi Nemirovskii.Interior-Point Polynomial Algorithms in Convex Programming. Studies in Applied Mathematics. SIAM, 1994.doi: 10.1137/1.9781611970791. [Pet07] Dénes...
-
[2001]
[VB96] Lieven Vandenberghe and Stephen Boyd
doi:10.1017/S096249290100005X. [VB96] Lieven Vandenberghe and Stephen Boyd. Semidefinite programming.SIAM Review, 38(1):49–95, 1996.doi:10.1137/1038003. [Wat18] John Watrous.The Theory of Quantum Information. Cambridge University Press, 2018.doi:10.1017/9781316848142. [Wil25] Mark M. Wilde. Quantum Fisher information matrices from Rényi relative en- tropies,
-
[2002]
[Hia21] Fumio Hiai.Quantumf-divergences in von Neumann Algebras: Reversibility of Quantum Operations
URL:https: //www.sciencedirect.com/science/article/pii/S0377221701001436,doi: 10.1016/S0377-2217(01)00143-6. [Hia21] Fumio Hiai.Quantumf-divergences in von Neumann Algebras: Reversibility of Quantum Operations. Mathematical Physics Studies. Springer Singapore,
-
[2009]
31 [DB16] Steven Diamond and Stephen Boyd
arXiv:0803.2770.doi:10.1109/TIT.2009.2018325. 31 [DB16] Steven Diamond and Stephen Boyd. CVXPY: A Python-embedded model- ing language for convex optimization.Journal of Machine Learning Research, 17(83):1–5,
Pith/arXiv arXiv 2009
-
[2016]
[Don86] Matthew J. Donald. On the relative entropy.Communications in Mathematical Physics, 105(1):13–34, 1986.doi:10.1007/BF01212339. [FS17] Hamza Fawzi and James Saunderson. Lieb’s concavity theorem, matrix geomet- ric means, and semidefinite optimization.Linear Algebra and its Applications, 513:240–263,
-
[2017]
doi.org/10.1007/s11005-017-0990-7,arXiv:1512.02615v2,doi:10.1007/ s11005-017-0990-7
URL:http://dx. doi.org/10.1007/s11005-017-0990-7,arXiv:1512.02615v2,doi:10.1007/ s11005-017-0990-7. [BT09] Amir Beck and Marc Teboulle. A fast iterative shrinkage-thresholding algorithm for linear inverse problems.SIAM Journal on Imaging Sciences, 2(1):183–202, 2009.doi:10.1137/080716542. [Bub15] Sébastien Bubeck. Convex optimization: Algorithms and compl...
Pith/arXiv arXiv 2009
-
[2019]
[Fuc96] Christopher Fuchs.Distinguishability and Accessible Information in Quantum Theory
Package cvxquad athttps://github.com/hfawzi/ cvxquad.doi:10.1007/s10208-018-9385-0. [Fuc96] Christopher Fuchs.Distinguishability and Accessible Information in Quantum Theory. PhD thesis, University of New Mexico, December 1996.arXiv: quant-ph/9601020. [GB14] Michael Grant and Stephen Boyd. CVX: Matlab software for disciplined convex programming.http://cvx...
Pith/arXiv arXiv 1996
-
[2021]
[HM17] Fumio Hiai and Milán Mosonyi
doi:10.1007/978-981-33-4199-9. [HM17] Fumio Hiai and Milán Mosonyi. Different quantumf-divergences and the reversibility of quantum operations.Reviews in Mathematical Physics, 29(07):1750023, 2017.arXiv:1604.03089,doi:10.1142/S0129055X17500234. 32 [HRVW96] Christoph Helmberg, Franz Rendl, Robert J. Vanderbei, and Henry Wolkow- icz. An interior-point metho...
Pith/arXiv arXiv 2017
-
[2024]
URL:https://arxiv.org/abs/2301. 11235,arXiv:2301.11235. [GPSW24] Ziv Goldfeld, Dhrumil Patel, Sreejith Sreekumar, and Mark M. Wilde. Quantum neural estimation of entropies.Physical Review A, 109:032431, March
-
[2025]
URL:https://arxiv.org/abs/2510.02218,arXiv:2510.02218. A Derivatives of scalar functions with respect to matrices In this appendix, we review the theory of matrix derivatives and calculate matrix derivatives of various functions that play a role in our paper. LetXbe ad×dmatrix. Letf(X)be a scalar function of a matrix. Then the matrix derivative (in numera...
This paper was first reviewed by deepseek-v4-flash on August 3, 2026.
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