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A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The paper proves that a powered fixed-point iteration computes the Petz-Augustin mean with linear convergence in the Thompson metric, yielding the first non-asymptotic guarantees for Petz capacity and Fisher-market equilibria.

desk verdict Solid non-asymptotic result for the Petz-Augustin mean; the Petz-capacity application overclaims by ignoring inner-solve error. read the letter →

arxiv 2502.06399 v2 pith:ZPMNLFPT submitted 2025-02-10 quant-ph cs.ITmath.ITmath.OC

Chun-Neng Chu , Wei-Fu Tseng , Yen-Huan Li This is my paper · ORCID
classification quant-phcs.ITmath.ITmath.OC MSC 81P4590C2591B5094A17 PACS 03.67.-a
keywords Petz-AugustinmeanPetz-RényidivergenceThompsonmetriclinearconvergencefixed-pointiterationPetzcapacityFishermarketCESutility
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 closes a gap in quantum information computation: the Petz-Augustin mean—the quantum state that minimizes a weighted sum of $n$ Petz-Rényi divergences—had no algorithm with a non-asymptotic convergence guarantee. The proposed fixed-point iteration, $Q_{t+1}=(\sum_j w[j] A_j^\alpha / \operatorname{Tr}[A_j^\alpha Q_t^{1-\alpha}])^{1/\alpha}$, contracts the powered variables $Q_t^{1-\alpha}$ by the factor $|1-1/\alpha|$ in each step, as measured in the Thompson metric. For $\alpha\in(1/2,1)\cup(1,\infty)$, the trace-normalized iterates therefore reach the mean $Q_\star$ at a linear rate $O(|1-1/\alpha|^T)$, with an explicit bound on the objective gap, at per-iteration cost $O(nd^2+d^3)$. The same iteration yields the first non-asymptotic algorithm for Petz capacity of order $\alpha\in(1/2,1)$, and, when all states commute, it becomes a tâtonnement dynamic that converges linearly to Fisher-market equilibrium prices.

What carries the argument

The workhorse is the corrected Thompson contraction operator $T_F(U)=(\sum_j w[j] A_j^\alpha/\operatorname{Tr}[A_j^\alpha U])^{(1-\alpha)/\alpha}$ on the positive-definite cone. The argument iterates in the powered variable $U=Q^{1-\alpha}$: Lemma 5.2 proves $d_T(T_F(V),T_F(U))\le |1-1/\alpha| d_T(V,U)$, Lemma 5.3 identifies the unique fixed point with $Q_\star^{1-\alpha}$, and Lemma 5.8 shows trace normalization changes the Thompson distance by at most a factor of two. The exponent $(1-\alpha)/\alpha$ lies in $(-1,1)$ for $\alpha\in(1/2,1)\cup(1,\infty)$, exactly the range in which the power-map inequality $d_T(U^r,V^r)\le |r|d_T(U,V)$ applies; this repair is what makes the linear contraction possible.

What would settle it

A numerical search over random positive-definite pairs $(U,V)$ and orders $\alpha\in(1/2,1)\cup(1,\infty)$ could settle Lemma 5.2 directly: if any pair satisfies $d_T(T_F(V),T_F(U))>|1-1/\alpha|\,d_T(V,U)$, the contraction step fails and Theorem 5.1 collapses. A cheaper surrogate is to run the iteration from two random initializations and check whether $d_T(Q_1^{1-\alpha},Q_2^{1-\alpha})$ is multiplied by at most $|1-1/\alpha|$ at every step.

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Extended reading notes

Core claim

The central claim is that the corrected operator $T_F(U)=(\sum_j w[j] A_j^\alpha/\operatorname{Tr}[A_j^\alpha U])^{(1-\alpha)/\alpha}$, viewed on positive-definite matrices, is a contraction with ratio $|1-1/\alpha|$ in the Thompson metric, and that its unique fixed point is $Q_\star^{1-\alpha}$, the powered Petz-Augustin mean. Hence the iteration $Q_{t+1}=T_F(Q_t^{1-\alpha})^{1/(1-\alpha)}$ satisfies $d_T(Q_\star^{1-\alpha}, Q_{T+1}^{1-\alpha}) \le |1-1/\alpha|^T d_T(Q_\star^{1-\alpha}, Q_1^{1-\alpha})$, and after trace normalization the objective gap obeys the bound in Theorem 5.1 with the same linear rate. The corrected operator matters because the naive update with exponents $1-\alpha$ and $1/\alpha$ fails to be order-preserving for $\alpha>2$, and the paper exhibits a pair of matrices for which the direct contraction bound is violated. Putting the exponent $(1-\alpha)/\alpha$ in the defining map keeps the matrix powers within the range where Thompson-metric power estimates apply, and for $\alpha>1$ the iterates also have non-increasing objective values.

Load-bearing premise

The load-bearing premise is that the sum of the given quantum states, $\sum_j A_j$, has full rank, so the minimizer and every iterate can be kept positive definite; when that sum is singular, the paper only suggests projecting onto a lower-dimensional subspace and does not analyze what the projection does to the convergence rate or the initialization.

Editorial extensions

If this is right

  • For any $\alpha\in(1/2,1)\cup(1,\infty)$, the Petz-Augustin mean can be computed to $\epsilon$ accuracy in the Thompson metric in $O(\log(1/\epsilon)/\log(1/|1-1/\alpha|))$ iterations, each costing $O(nd^2+d^3)$.
  • The Petz capacity of order $\alpha\in(1/2,1)$ is computable with objective error $O(\log(n)/T)$, giving numerical access to the classical-quantum channel-coding error-exponent bounds that call for $C_\alpha$.
  • In the commuting case, the algorithm is a tâtonnement dynamic that reaches Fisher-market equilibrium prices linearly for CES utilities with elasticity $\rho=1-1/\alpha\in(0,1)$, a regime where earlier tâtonnement-type guarantees were sublinear or incomparable.
  • For inhomogeneous Fisher markets with weak-gross-substitutes CES utilities and seller-held upper bounds $\hat\rho_i$, asynchronous price updates converge as $\hat\rho^T$ in the Thompson metric, with $\hat\rho=\max_i\hat\rho_i$.
  • For $\alpha>1$, objective values are non-increasing along the iterates, so the normalized iteration is simultaneously a monotone descent method.

Reading between the lines

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

  • The same repair—iterate on the powered variable so every matrix power in the map has exponent in $(-1,1)$—may yield non-asymptotic rates for other noncommutative divergence minimizers whose natural updates are not order-preserving.
  • Because the contraction ratio $|1-1/\alpha|$ tends to zero as $\alpha\to1$ while the constant $1/|\alpha-1|$ in the objective bound diverges, the worst-case bound near $\alpha=1$ is likely pessimistic; a refined analysis could separate the contraction rate from the objective-gap constant.
  • The asynchronous market update suggests a randomized per-coordinate version of the quantum iteration that would cut the $d^3$ matrix-power cost per step, but such a scheme is not analyzed in the paper.
  • Numerical divergence for $\alpha\le1/2$ indicates the proven range may be sharp; a useful test is to search for a modified fixed-point operator that contracts on the complementary interval.
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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

2 major / 4 minor

Summary. The paper studies the computation of the Petz-Augustin mean, defined as the minimizer of a weighted sum of Petz-Renyi divergences of order alpha over the set of quantum states. It proposes the fixed-point iteration (11), analyzes it through a corrected operator T_F on the positive-definite cone, and proves in Theorem 5.1 a linear convergence rate O(|1-1/alpha|^T) with respect to the Thompson metric for alpha in (1/2,1) union (1,infty), with initialization cost O(nd^3) and per-iteration cost O(nd^2+d^3). The paper then applies this algorithm to compute the Petz capacity of order alpha in (1/2,1), claiming an O(log(n)/T) rate for entropic mirror descent with approximate inner solves, and to compute equilibrium prices in CES Fisher markets, including an asynchronous tatonnement variant with rate O(hat_rho^T). The core contraction argument for the Petz-Augustin mean is coherent and detailed; the two application-level guarantees are less complete.

Significance. If the main theorem is correct, it is a genuinely useful algorithmic contribution: it appears to be the first non-asymptotic convergence guarantee for computing the Petz-Augustin mean, and the per-iteration cost is attractive compared with standard first-order methods. The use of the Thompson metric and the correction from the naive operator to T_F is a sound and non-obvious idea. The paper also gives a clear discussion of why the natural contractivity hypothesis fails, with a concrete counterexample. The Petz capacity and Fisher market applications are potentially significant, but as written the Petz capacity guarantee applies to an exact-gradient oracle rather than to the implemented approximate-gradient routine, and the Fisher market convergence proof in Section 7.3 contains a scaling error. These issues prevent the paper from being acceptable in its current form, although the central Petz-Augustin mean result appears sound and likely repairable.

major comments (2)
  1. [Section 6.2, Remark 6.1, Theorem 6.5] The advertised O(log(n)/T) guarantee for computing the Petz capacity is proved only for an exact-gradient oracle model. Theorem 6.5 applies Lemma 6.4 to the update (14) using the exact gradient nabla g(w_t), but the algorithm as described in Section 6.2 computes nabla g(w_t) and g(w_{t+1}) only to error epsilon by truncating the inner iteration (11) at T = O(log(1/epsilon)) steps (Remark 6.1). No perturbation analysis is provided that translates the inner-solve error into an outer suboptimality bound, and no total iteration or bit complexity is given. In particular, Lemma 6.2 only bounds the error of the approximate gradient in terms of the Thompson metric between Q_star(w_t) and the truncated iterate; it does not show that the approximate mirror-descent update stays within the regime in which Lemma 6.4 applies. Consequently, Theorem 6.5 as stated does not establish a non-asymptotic guarantee for the algorithm that is actually run.
  2. [Section 7.3, proof of Theorem 7.1] The coordinate-wise contraction argument in the proof of Theorem 7.1 contains a scaling error. In the displayed inequality after the definition of the update, replacing p_t by exp(d_T(p_star,p_t)) p_star gives a factor exp((1/(1-hat_rho_i) - 1/(1-rho_j)) d_T) in the numerator and exp(+rho_j/(1-rho_j) d_T) in the reciprocal denominator; the total exponent inside the sum is d_T/(1-hat_rho_i). Raising to the power 1-hat_rho_i therefore yields exp(d_T), not exp(hat_rho_i d_T). The same issue affects the lower bound. Thus the claimed inequality log(max(p_{t+1}[i]/p_star[i], p_star[i]/p_{t+1}[i])) <= hat_rho_i d_T(p_star,p_t) is not established by the proof as written, and the advertised tatonnement rate O(hat_rho^T) rests on this step. If a more refined argument is intended, it is not present in the manuscript.
minor comments (4)
  1. [Section 1] The full-rank assumption on sum_j A_j is stated, but the one-sentence reduction 'project all matrices onto a lower-dimensional subspace' is not analyzed. It would be helpful to spell out the support projection, the induced problem on that subspace, and how the initialization and convergence guarantees transfer; otherwise the reader cannot tell whether the singular case is fully covered.
  2. [Lemma 5.7 proof] In the proof of Lemma 5.7, the symbol sigma appears in the displayed equality 'Tr[TF(sigma^{1-alpha})^{alpha/(1-alpha)} TF(sigma^{1-alpha})]' without being defined; it should presumably be Q, the matrix introduced at the start of the proof.
  3. [Lemma C.2] In the definition of the operator L_2 within Lemma C.2, the summation index is written as m instead of n (the summation runs over k=1 to m, but the problem has n states). This should be corrected to avoid confusion.
  4. [Section 5.2.2] The fixed-point property for alpha in (1/2,1) is imported from Cheng et al. [2019, Proposition 2(b)] rather than proved in the paper. This is acceptable as a citation, but the paper should state more explicitly that the self-contained proof of Lemma 5.3 covers alpha>1 only, and that the lower range depends on the external result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 5.1's rate is derived from an external contraction argument and an independently cited fixed-point characterization, with no load-bearing self-citation.

full rationale

The core derivation is self-contained. Lemma 5.2 proves contractivity of TF directly from the Thompson metric order inequalities and the power-contraction lemma (Lemma 3.2), without assuming the target rate. Lemma 5.3 identifies the fixed point with the minimizer using Cheng et al. 2019 for alpha < 1, an external result by different authors, and an optimality-based argument for alpha > 1. Lemmas 5.8 and 5.10 convert Thompson-metric contraction into optimization-error bounds via trace normalization and trace monotonicity; none of these steps defines its output in terms of itself. The acknowledged algorithmic overlap with Cheng and Nakiboglu 2024a is disclosed, and their result is asymptotic and for alpha > 1 only, so it is not used to prove the new non-asymptotic rate. Self-citations such as Tsai et al. 2024, Wang et al. 2024, and You et al. 2022 appear only in related-work and complexity comparisons, not as load-bearing premises. The Section 6 Petz-capacity analysis invokes external results, including Lu et al. 2018 and the Cheng-Nakiboglu gradient formula; the gap between Theorem 6.5's exact-gradient model and Remark 6.1's approximate inner solves is a missing perturbation analysis, which is a correctness or completeness concern rather than a circular reduction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on standard matrix analysis and convex optimization lemmas, the domain assumption that the sum of the given states is full-rank, and, for alpha in (0,1), a fixed-point property imported from Cheng et al. 2019. No free parameters are fitted and no new physical or mathematical entities are introduced.

assumptions (4)
  • domain assumption The sum of the quantum states, sum_j A_j, is full-rank.
    Stated in Section 1. Guarantees existence and full-rankness of Q_star via Mosonyi and Ogawa 2021, and keeps all iterates positive definite so the Thompson metric and the fixed-point analysis apply. The paper only sketches a projection if this fails.
  • domain assumption Petz-Renyi divergence objective is finite only for full-rank Q, and the algorithm restricts to positive definite iterates.
    The objective F(Q) in (1) and the operator TF in (10) need positive denominators Tr[A_j^alpha U]; the domain is B_{d,++} as stated in Section 5.1.
  • standard math Standard matrix inequalities: Araki-Lieb-Thirring (Lemma 5.4), Holder (Lemma 5.5), Lieb-Ando concavity (Lemma C.3), and Thompson metric power bounds (Lemma 3.2).
    These are cited standard results used in the proof of Theorem 5.1 and in the relative smoothness lemma of Appendix C.
  • standard math For alpha in (0,1), the fixed-point property of Q_star^{1-alpha} under TF is taken from Cheng et al. 2019 Proposition 2(b).
    Used in the proof of Lemma 5.3 for the alpha in (1/2,1) case. This is a published prior result, not derived in the paper.

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Pith. "Pith review of A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean." pith.science (2026). https://pith.science/paper/ZPMNLFPT

@misc{pith2026250206399,
  author       = {Pith},
  title        = {Pith review of: A Linearly Convergent Algorithm for Computing the Petz-Augustin Mean},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZPMNLFPT}},
  note         = {Machine review of arXiv:2502.06399}
}
abstract

We study the computation of the Petz-Augustin mean of order $\alpha \in (0,1) \cup (1,\infty)$, defined as the minimizer of a weighted sum of $n$ Petz-R\'enyi divergences of order $\alpha$ over the set of $d$-by-$d$ quantum states, where the Petz-R\'enyi divergence is a quantum generalization of the classical R\'enyi divergence. We propose the first algorithm with a non-asymptotic convergence guarantee for solving this optimization problem. The iterates are guaranteed to converge to the Petz-Augustin mean at a linear rate of \( O\left( \lvert 1 - 1/\alpha \rvert^T \right) \) with respect to the Thompson metric for $\alpha\in(1/2,1)\cup(1,\infty)$, where \( T \) denotes the number of iterations. The algorithm has an initialization time complexity of $O\left(nd^3\right)$ and a per-iteration time complexity of $O\left(nd^2 + d^3\right)$. Two applications follow. First, we propose the first iterative method with a non-asymptotic convergence guarantee for computing the Petz capacity of order $\alpha\in(1/2,1)$, which generalizes the quantum channel capacity and characterizes the optimal error exponent in classical-quantum channel coding. Second, we establish that the Petz-Augustin mean of order $\alpha$, when all quantum states commute, is equivalent to the equilibrium prices in Fisher markets with constant elasticity of substitution (CES) utilities of common elasticity $\rho=1-1/\alpha$, and our proposed algorithm can be interpreted as a t\^{a}tonnement dynamic. We then extend the proposed algorithm to inhomogeneous Fisher markets, where buyers have different elasticities, and prove that it achieves a faster convergence rate compared to existing t\^{a}tonnement-type algorithms.

Figures

Figures reproduced from arXiv: 2502.06399 by the authors.

Figure 1
Figure 1. Approximate optimization error and iterate error versus the number of iterations for [PITH_FULL_IMAGE:figures/full_fig_p031_1.png] view at source ↗
Figure 2
Figure 2. Approximate optimization error and iterate error versus the number of iterations for [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗

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