Alternating minimization for computing doubly minimized Petz Renyi mutual information

Laura Burri

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Alternating minimization for computing doubly minimized Petz Renyi mutual information

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Pith's one-line read Alternating minimization provably computes the doubly minimized Petz Rényi mutual information for all finite-dimensional quantum states.

desk verdict Solid, carefully proved convergence results for computing the doubly minimized Petz Rényi mutual information; the main dependency is on the author's own prior work, but the math here checks out and the paper deserves a serious referee. read the letter →

arxiv 2507.05205 v1 pith:MX2G7Z5N submitted 2025-07-07 quant-ph cs.ITmath.IT

Laura Burri This is my paper · ORCID

classification quant-phcs.ITmath.IT MSC 81P4594A1790C25

keywords quantuminformationtheoryPetzRényimutualalternatingminimizationHilbert'sprojectivemetricconvergenceratesdivergencebipartitestates

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 proves that the doubly minimized Petz Rényi mutual information, a one-parameter family of correlation measures defined as the smallest Petz divergence between a bipartite quantum state and a product state, can be computed by alternating minimization for every finite-dimensional bipartite state. For Rényi order $\alpha \in (1,2]$, the objective value after $n$ alternating rounds converges linearly, with error of order $|1 - 1/\alpha|^{2n}$; for $\alpha \in (\frac{1}{2},1)$, it converges sublinearly, with error $O(1/n)$. These are the first non-asymptotic convergence guarantees that apply to all quantum states, not only to classical-classical states. The paper turns the bounds into two explicit algorithms whose iteration counts can be fixed in advance from a desired error tolerance.

What carries the argument

Two separate mechanisms carry the argument. For $\alpha\in(1,2]$, the working tool is Hilbert's projective metric on the cone of positive semidefinite operators: each exact-minimizer update $N_{A\to B}$ and $N_{B\to A}$ is homogeneous and order-preserving or order-reversing in powers of the marginals, and a classical contraction theorem for positive linear maps bounds its Lipschitz constant by $\gamma = |1-1/\alpha|$; a full round therefore contracts by $\gamma^2$. For $\alpha\in(\frac{1}{2},1)$, the proof does not use the Hilbert metric. Instead it exploits the joint concavity of $Q_\alpha(\rho_{AB}\|\sigma_A\otimes\tau_B) = \mathrm{tr}[\rho_{AB}^{\alpha}(\sigma_A\otimes\tau_B)^{1-\alpha}]$, the uniqueness of the global minimizer, and a Rényi–Pinsker inequality to relate the gap $x_{n-1}-x_n$ to the remaining error, producing a recursion that solves to $O(1/n)$.

What would settle it

Take a concrete two-qubit state, such as a full-rank mixture of $\lvert 00\rangle$, $\lvert 11\rangle$, and a small noise term, and run the alternating iteration exactly for $\alpha\in(\frac{1}{2},1)$. If for any $n$ the inequality $|x_n-I_\alpha^{\downarrow\downarrow}| \le \max(\frac{3}{2} c_0^2, 2x_0)/n$ fails, the sublinear theorem is false; likewise, for $\alpha\in(1,2]$, computing the ratio $d_H(N_{A\to B}(\sigma), N_{A\to B}(\tilde\sigma))/d_H(\sigma,\tilde\sigma)$ for two random positive marginals and finding a value larger than $\gamma = |1-1/\alpha|$ would falsify the contraction theorem.

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Extended reading notes

Core claim

For any fixed $\rho_{AB}$, alternating minimization updates the marginal $\tau_B$ to the exact minimizer $\hat\tau_B = (\mathrm{tr}_A[\rho_{AB}^{\alpha}\sigma_A^{1-\alpha}])^{1/\alpha} / \mathrm{tr}[(\mathrm{tr}_A[\rho_{AB}^{\alpha}\sigma_A^{1-\alpha}])^{1/\alpha}]$, then updates $\sigma_A$ in the same way, and the sequence $x_n = D_\alpha(\rho_{AB}\|\sigma_A^{(n)}\otimes\tau_B^{(n)})$ decreases monotonically to the doubly minimized PRMI $I_\alpha^{\downarrow\downarrow}(A:B)_\rho$ for every $\alpha$ in $(\frac{1}{2},1)\cup(1,2]$. For $\alpha\in(1,2]$, the contraction argument in Hilbert's projective metric yields $|x_n - I_\alpha^{\downarrow\downarrow}| \le \frac{1}{\alpha-1}[\exp((\alpha-1)(1+\gamma)\gamma^{2n}d_H(\sigma_A^{(0)},\hat\sigma_A))-1]$ with $\gamma = 1 - 1/\alpha$, an explicit linear rate. For $\alpha\in(\frac{1}{2},1)$, the proof uses joint concavity of the trace function $Q_\alpha$ together with a Rényi–Pinsker inequality to obtain $|x_n - I_\alpha^{\downarrow\downarrow}| \le \max(\frac{3}{2} c_0^2, 2x_0)/n$, a sublinear $O(1/n)$ rate. The $\alpha=1$ case is degenerate and reaches the minimum in one step; for $\alpha\in(0,\frac{1}{2}]$, alternating minimization is shown not to converge to the global minimum in general.

Load-bearing premise

The premise that has to hold is that the surface being minimized curves upward in both marginal directions at once for orders between 1/2 and 1, and that the minimum point is unique; the paper imports both facts from its companion work, and the sublinear proof depends on them.

Editorial extensions

If this is right

For $\alpha\in(1,2]$, the objective error after $n$ full alternating rounds is at most $\frac{1}{\alpha-1}[\exp((\alpha-1)(1+\gamma)\gamma^{2n}d_H(\sigma_A^{(0)},\hat\sigma_A))-1]$, so the iteration count needed to reach a targeted precision is known before running the algorithm.
For $\alpha\in(\frac{1}{2},1)$, the same iteration reaches precision $\epsilon$ after a number of rounds no larger than $\max(\frac{3}{2} c_0^2, 2x_0)/\epsilon$, again giving a finite stopping rule.
The alternating sequence of states converges to the unique global minimizer of the PRMI optimization problem for every $\alpha\in(\frac{1}{2},1)\cup(1,2]$.
The algorithms output a value $x$ certified to satisfy $|x - I_\alpha^{\downarrow\downarrow}(A:B)_\rho| \le \epsilon_0$ for any prescribed $\epsilon_0>0$, for all quantum states in the stated range.

Reading between the lines

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

The Hilbert-metric technique for $\alpha\in(1,2]$ is the natural template to attack the sandwiched Rényi mutual information, whose partial minimizers are not explicit; finding such formulas would let the same contraction proof run for that family.
The sublinear $O(1/n)$ rate for $\alpha\in(\frac{1}{2},1)$ is likely conservative; numerical experiments on low-dimensional states could reveal much faster actual convergence, and a local strong-convexity analysis might upgrade the guarantee.
The $\alpha=\frac{1}{2}$ endpoint is connected to reflected-entropy-like measures mentioned in the introduction; the algorithm offers a numerical route to those quantities by taking a limit along $\alpha\in(\frac{1}{2},1)$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Jim,

The short version: this is a solid, carefully written paper that gives the first provable convergence guarantees for alternating minimization of the doubly minimized Petz Rényi mutual information for arbitrary finite-dimensional quantum states. For α∈(1,2] they get linear convergence of the objective value with rate |1−1/α|^{2n}, and for α∈(1/2,1) they get sublinear O(1/n). Previous work only handled classical-classical states, so this is a real step forward.

What I like: the proof structure is honest and complete. The contraction argument via Hilbert's projective metric and Birkhoff–Hopf for α∈(1,2] is clean, and the sublinear proof for α∈(1/2,1) is a genuinely different method, using joint convexity and Fréchet derivative bounds. The authors are upfront that they could not extend the classical linear-convergence proof to that range. The explicit constants are given, and the counterexample for α<1/2 is a nice sanity check.

The soft spot is the dependency, not the internal math. The sublinear result relies on joint concavity of Q_α and uniqueness of the global minimizer, both taken from the author's companion paper [17] and not re-derived. I read the argument and it is applied correctly; the stress-test note agrees. But for a theorem whose correctness hinges on an unpublished preprint by the same author, a referee should either verify [17] or require the author to include the needed statements as lemmas. That said, this is a normal citation pattern, not a circularity.

Minor quibble: the paper is titled “computing” but contains no numerical experiments. For a theory paper that is fine, but the algorithms are presented without any demonstration of practical behavior. The per-iteration cost is not discussed. That would be worth adding.

Who is this for: quantum information theorists working on Rényi quantities, especially anyone who needs to actually evaluate I_α↓↓ for a state. It deserves a serious referee. I would cite it and bring it to reading group if anyone is working on this. Recommend: accept with the usual request to clarify the dependency on [17] and maybe add a small numerical example.

Referee Report

1 major / 3 minor

Summary. The paper studies alternating minimization of the Petz divergence D_α(ρ_AB∥σ_A⊗τ_B) over states σ_A and τ_B, and proves convergence of the objective values to the doubly minimized Petz Rényi mutual information I_α^{↓↓}(A:B)_ρ. For α∈(1,2], it establishes a contraction of the alternating maps in Hilbert's projective metric with coefficient γ=|1−1/α| (Theorem 1), and from this derives linear convergence of the iterates (Corollary 2) and of the objective values at rate O(γ^{2n}) (Corollary 3), together with a termination-guaranteed algorithm. For α∈(1/2,1), it proves sublinear convergence of the objective values with an explicit O(1/n) bound (Theorem 4), again with an algorithm. The proofs are given in full in the appendices, and Appendix A recovers and extends the classical results of [22].

Significance. If the results are correct, they fill a genuine gap: no closed form is known for the doubly minimized PRMI, and prior non-asymptotic convergence analyses were limited to classical-classical states. The linear-rate result for α∈(1,2] is a clean quantum extension of the classical Hilbert-metric contraction argument, and the sublinear O(1/n) result for α∈(1/2,1) is the first quantum non-asymptotic guarantee in that parameter range. The paper is carefully written and provides explicit constants, finite-horizon termination statements, and complete appendix proofs; Remark 3 usefully identifies why the same approach fails for α<1/2. The main caveat is that the α<1 result depends on two load-bearing properties imported from the author's own unpublished preprint [17].

major comments (1)

[Section 3B, Appendix D, Eq. (D.32) and Eq. (2.11)] Theorem 4 is load-bearing on external results from the author's preprint [17] (arXiv:2406.01699): the inequality at (D.32) uses the joint convexity of f(σ_A,τ_B)=−Q_α(ρ_AB∥σ_A⊗τ_B), and the equality at (D.27)-(D.28) uses the uniqueness of the global minimizer stated in (2.11). Neither property is proved or stated as a precise theorem in this manuscript, and [17] is an unpublished preprint by the same author. Since the entire α∈(1/2,1) convergence claim rests on these facts, the manuscript should either include their statements with proofs or cite a peer-reviewed version; as it stands, Theorem 4 is conditional on [17]. The analogous fixed-point property used in (C.14)-(C.15) for the α∈(1,2] results is also cited from [17] and should be given the same treatment.

minor comments (3)

[Eq. (3.10)-(3.11)] The definition of δ in Theorem 1(b) is difficult to read because the line breaks separate the exponents from the operators; please typeset it as an explicit product of two operator norms with clear parentheses.
[Appendix A, Corollary 7] In Corollary 7(a) the condition "If α>1/2" appears inside a statement whose preamble fixes α∈[1/2,1)∪(1,∞); consider stating the corollary directly for α∈(1/2,1)∪(1,∞) to avoid ambiguity about the α=1/2 endpoint.
[Algorithm 1] Algorithm 1 refers to "c0 as in (C.65)", but (C.65) defines several quantities; please refer explicitly to Proposition 13 and its definition c0:=−2log min{min spec(σ̃_0), c_A}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence theorems are derived from contraction and convexity inputs, not equivalent to the conclusions.

full rationale

No significant circularity. The claimed main results (Corollary 3 and Theorem 4) are non-asymptotic bounds on |x_n - I^{↓↓}_α(A:B)_ρ|. For α∈(1,2], the proof derives a contraction in Hilbert's projective metric (Theorem 1) via Lemma 9 and the Birkhoff-Hopf theorem, then converts state convergence to objective-value convergence via an algebraic bound in Appendix C3; none of these steps assumes the conclusion. For α∈(1/2,1), the argument uses the joint convexity of f(σ_A,τ_B) = -Q_α(ρ_AB∥σ_A⊗τ_B) and uniqueness of the global minimizer, cited from the author's separate prior work [17]. Although [17] is a self-citation and is load-bearing for that range, it is a parameter-free theorem about the objective function, not about alternating-minimization convergence, and it is not justified by the present paper's results. Under the stated criteria this counts as independent support rather than circularity. The constants in all bounds are defined explicitly from ρ_AB, α, and the initialization; no fitted parameter is later reported as a prediction, and no definition is equivalent to the target quantity. The only caveat is a correctness risk: if the cited concavity/uniqueness result in [17] had a gap, Theorem 4 would fail. That is a correctness concern, not a circularity objection.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper's proofs rely on standard tools of convex optimization in quantum information (Hilbert's projective metric, Birkhoff-Hopf, Rényi Pinsker, Sibson identity) and on prior results by the same author [17] for convexity/concavity and uniqueness properties. These prior results are not re-derived, but they are cited with references. No new entities or free parameters are introduced.

assumptions (8)

standard math Finite-dimensional quantum mechanics: states are density operators on finite-dimensional Hilbert spaces, and all Hilbert spaces are over C.
Stated in Section 2A as the basic setting of the paper.
standard math Standard properties of Hilbert's projective metric, including scale invariance, projective definiteness, contraction properties for homogeneous order-preserving and order-reversing maps, and the Birkhoff-Hopf theorem.
Enumerated in Appendix B and cited to [26,27,33,34]; used in Lemma 9 and Theorem 1.
standard math Operator monotonicity of X^r for r∈[0,1] and operator anti-monotonicity for r∈[-1,0].
Used in Lemma 9(a) to bound dH(X^r, Y^r) in terms of dH(X,Y).
domain assumption Rényi Pinsker inequality: for α∈[1/2,1), (1−α)/(4α) ∥ρ^α−σ^α∥_{1/α}^2 ≤ 1−Qα(ρ∥σ).
Used in the proof of Theorem 4 at Eq. (D.42), cited to [25, Theorem 2.1].
domain assumption Quantum Sibson identity (2.7), giving the partial minimizer of the Petz divergence over τB.
Used in Remark 1 and in the proof of Theorem 4 at Eq. (D.21), cited to [13].
domain assumption Joint concavity of Qα(ρAB∥σA⊗τB) in (σA,τB) for α∈(1/2,1), equivalently joint convexity of f(σA,τB)=−Qα.
Used in the proof of Theorem 4 at Eq. (D.32), cited to prior work [17] by the same author; not re-derived.
domain assumption Uniqueness of the global minimizer of the doubly minimized PRMI for α∈(1/2,1] and its support properties.
Used in Corollary 2 and Remark 6 for α∈(1,2], and in Theorem 4 for α∈(1/2,1), cited to [17].
ad hoc to paper The support restriction WLOG ρA > 0 and ρB > 0 in the proof of Theorem 4, justified by restricting the Hilbert spaces to the supports.
Used in Section D2 to avoid zero eigenvalues when defining Fréchet derivatives; the proof states the same argument works on the support subspaces.

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Pith. "Pith review of Alternating minimization for computing doubly minimized Petz Renyi mutual information." pith.science (2026). https://pith.science/paper/MX2G7Z5N

@misc{pith2026250705205,
  author       = {Pith},
  title        = {Pith review of: Alternating minimization for computing doubly minimized Petz Renyi mutual information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MX2G7Z5N}},
  note         = {Machine review of arXiv:2507.05205}
}

abstract

The doubly minimized Petz Renyi mutual information (PRMI) of order $\alpha$ is defined as the minimization of the Petz divergence of order $\alpha$ of a fixed bipartite quantum state $\rho_{AB}$ relative to any product state $\sigma_A\otimes \tau_B$. To date, no closed-form expression for this measure has been found, necessitating the development of numerical methods for its computation. In this work, we show that alternating minimization over $\sigma_A$ and $\tau_B$ asymptotically converges to the doubly minimized PRMI for any $\alpha\in (\frac{1}{2},1)\cup (1,2]$, by proving linear convergence of the objective function values with respect to the number of iterations for $\alpha\in (1,2]$ and sublinear convergence for $\alpha\in (\frac{1}{2},1)$. Previous studies have only addressed the specific case where $\rho_{AB}$ is a classical-classical state, while our results hold for any quantum state $\rho_{AB}$.

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Works this paper leans on

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INTRODUCTION The mutual information is a measure that quantifies the amount of correlation in a bipartite quantum state. Over the past decade, a number of Rényi generalizations of the mutual informa- tion have been introduced, such as generalizations based on the sandwiched divergence [1–13] and generalizations based on the Petz divergence [9–17]. This pa...

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Notation We take “log” to refer to the natural logarithm

PRELIMINARIES A. Notation We take “log” to refer to the natural logarithm. The set of natural numbers strictly less than n ∈ N is denoted by[n] := {0, 1, . . . , n− 1}. All Hilbert spaces are assumed to be finite-dimensional and overC. The dimension of a Hilbert space A is denoted bydA. The tensor product of two Hilbert spaces,A and B, is denoted byA ⊗ B ...

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The following definition formalizes the alternating minimization problem under consideration

MAIN RESUL TS Problem formulation.Weareinterestedinwhetheralternatingminimizationof Dα(ρAB∥σA⊗τB) over states σA and τB converges to the doubly minimized PRMII ↓↓ α (A : B)ρ as the number of iterations tends to infinity, and if so, how fast this convergence occurs. The following definition formalizes the alternating minimization problem under consideratio...

work page
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Our main results address the non-asymptotic convergence of the objective function values after n iterations of alternating minimization, denoted byxn

CONCLUSION We have analyzed the convergence of alternating minimization ofDα(ρAB∥σA ⊗ τB) over states σA and τB to the doubly minimized PRMI I ↓↓ α (A : B)ρ for any fixed α ∈ ( 1 2 , 1) ∪ (1, 2] and ρAB ∈ S(AB). Our main results address the non-asymptotic convergence of the objective function values after n iterations of alternating minimization, denoted ...

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Probability mass functions Let X be a finite set

Notation for classical setting a. Probability mass functions Let X be a finite set. The support of a functionP : X →R is denoted assupp(P ) := {x ∈ X: P (x) ̸= 0}. For two functionsP, Q: X →R, P ≪ Q is true iffsupp(P ) ⊆ supp(Q). P ∼ Q is true iff supp(P ) = supp(Q). P ⊥ Q is true iffsupp(P ) ∩ supp(Q) =∅. 11 The set of PMFs overX is P(X ) := {P : X →[0, ...

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12 Definition 2(Alternating minimization: Iteration rule)

Linear convergence for α ∈ ( 1 2 , ∞) for classical setting Alternating minimization ofDα(PXY ∥QX RY ) over PMFsQX and RY proceeds as described in the following definition, see (A.3). 12 Definition 2(Alternating minimization: Iteration rule). For any givenα ∈ (0, ∞), PXY ∈ P(X × Y), we define the functions NX→Y : PPX ≪(X ) → P∼PY (Y), Q X 7→ (P x∈X PXY (x...

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Let α ∈ (0, 1) ∪ (1, ∞) and let Q, R ∈ P(X ) be such that Q ∼ R

Proof of Corollary 7 Lemma 8. Let α ∈ (0, 1) ∪ (1, ∞) and let Q, R ∈ P(X ) be such that Q ∼ R. Then, M (Q1−α/R1−α) ≥ 1. Proof. Case 1: α >1. M (Q1−α/R1−α) = max x∈supp(R) Q(x)1−α R(x)1−α (A.21) ≥ X x∈supp(R) R(x) Q(x)1−α R(x)1−α = X x∈supp(R) R(x)αQ(x)1−α (A.22) ≥   X x∈supp...
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(A.36) follows from the additivity of Hilbert’s projective metric with respect to the product of PMFs

(A.33) follows from Lemma 8. (A.36) follows from the additivity of Hilbert’s projective metric with respect to the product of PMFs. (A.37) follows from Corollary 6 (a). We conclude that for all n ∈ N 0 ≤ xn − I ↓↓ α (X : Y )P = 1 α − 1 log 1 +qn ˆq − 1 (A.38) 15 ≤ 1 α − 1 qn ˆ...
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All of the following hold

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By the fixed-point property of minimizers for the doubly minimized PRMI [17], ˆσA = NB→A ◦ NA→B(ˆσA) ∈ S≪ρA(A), (C.14) ˆτB = NA→B(ˆσA) ∈ S≪ρB (B)

Proof of Corollary 2 Proof. By the fixed-point property of minimizers for the doubly minimized PRMI [17], ˆσA = NB→A ◦ NA→B(ˆσA) ∈ S≪ρA(A), (C.14) ˆτB = NA→B(ˆσA) ∈ S≪ρB (B). (C.15) Since α ≥ 1, ˆσA ∈ SρA≪(A) and ˆτB ∈ SρB≪(B) [17]. We conclude thatˆσA ∼ ρA and ˆτB ∼ ρB. For a...
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Let α ∈ (1, ∞) and let σ, τ∈ S(A) be such thatσ ∼ τ

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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INTRODUCTION The mutual information is a measure that quantifies the amount of correlation in a bipartite quantum state. Over the past decade, a number of Rényi generalizations of the mutual informa- tion have been introduced, such as generalizations based on the sandwiched divergence [1–13] and generalizations based on the Petz divergence [9–17]. This pa...

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Notation We take “log” to refer to the natural logarithm

PRELIMINARIES A. Notation We take “log” to refer to the natural logarithm. The set of natural numbers strictly less than n ∈ N is denoted by[n] := {0, 1, . . . , n− 1}. All Hilbert spaces are assumed to be finite-dimensional and overC. The dimension of a Hilbert space A is denoted bydA. The tensor product of two Hilbert spaces,A and B, is denoted byA ⊗ B ...

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The following definition formalizes the alternating minimization problem under consideration

MAIN RESUL TS Problem formulation.Weareinterestedinwhetheralternatingminimizationof Dα(ρAB∥σA⊗τB) over states σA and τB converges to the doubly minimized PRMII ↓↓ α (A : B)ρ as the number of iterations tends to infinity, and if so, how fast this convergence occurs. The following definition formalizes the alternating minimization problem under consideratio...

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[6] [4]

Our main results address the non-asymptotic convergence of the objective function values after n iterations of alternating minimization, denoted byxn

CONCLUSION We have analyzed the convergence of alternating minimization ofDα(ρAB∥σA ⊗ τB) over states σA and τB to the doubly minimized PRMI I ↓↓ α (A : B)ρ for any fixed α ∈ ( 1 2 , 1) ∪ (1, 2] and ρAB ∈ S(AB). Our main results address the non-asymptotic convergence of the objective function values after n iterations of alternating minimization, denoted ...

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Probability mass functions Let X be a finite set

Notation for classical setting a. Probability mass functions Let X be a finite set. The support of a functionP : X →R is denoted assupp(P ) := {x ∈ X: P (x) ̸= 0}. For two functionsP, Q: X →R, P ≪ Q is true iffsupp(P ) ⊆ supp(Q). P ∼ Q is true iff supp(P ) = supp(Q). P ⊥ Q is true iffsupp(P ) ∩ supp(Q) =∅. 11 The set of PMFs overX is P(X ) := {P : X →[0, ...

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12 Definition 2(Alternating minimization: Iteration rule)

Linear convergence for α ∈ ( 1 2 , ∞) for classical setting Alternating minimization ofDα(PXY ∥QX RY ) over PMFsQX and RY proceeds as described in the following definition, see (A.3). 12 Definition 2(Alternating minimization: Iteration rule). For any givenα ∈ (0, ∞), PXY ∈ P(X × Y), we define the functions NX→Y : PPX ≪(X ) → P∼PY (Y), Q X 7→ (P x∈X PXY (x...

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