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arxiv: 2406.01699 · v3 · submitted 2024-06-03 · 🪐 quant-ph · cs.IT· math.IT

Doubly minimized Petz and sandwiched Renyi mutual information: Properties

Laura Burri This is my paper

Pith reviewed 2026-05-24 00:00 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT
keywords doubly minimized Renyi mutual informationPetz divergencesandwiched divergenceadditivityduality relationSion minimax theoremquantum information
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The pith

Doubly minimized Petz Renyi mutual information is additive for alpha in [1/2,2] and the sandwiched version is additive for alpha in [2/3,∞] via a new duality.

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

The paper defines the doubly minimized Petz Renyi mutual information as the minimum of the Petz divergence of order alpha between a fixed bipartite state and any product state, and defines the sandwiched version analogously. It proves additivity of the Petz version over alpha in the closed interval from 1/2 to 2. For the sandwiched version it first derives a duality relation that holds for alpha from 2/3 to infinity by applying Sion's minimax theorem, then uses the duality to establish additivity over the same interval, extending the range where additivity was previously known.

Core claim

The central claim is that the doubly minimized Petz Renyi mutual information satisfies additivity for every alpha between 1/2 and 2 inclusive. For the doubly minimized sandwiched Renyi mutual information a duality relation is established for every alpha at least 2/3 by direct application of Sion's minimax theorem to the relevant convex-concave problem over product states; this duality is then invoked to prove additivity for the same range of alpha.

What carries the argument

The doubly minimized Petz and sandwiched Renyi mutual informations, each obtained by minimizing the corresponding divergence of order alpha over all product states, together with the duality relation derived via Sion's minimax theorem for the sandwiched case.

If this is right

  • Additivity reduces the evaluation of the quantity on tensor-product states to the single-copy case.
  • The duality relation supplies an alternative maximin expression for the sandwiched quantity that can be used in further proofs.
  • The additivity range for the sandwiched case now includes all alpha greater than or equal to 2/3 rather than only those greater than or equal to 1.
  • Both quantities inherit monotonicity and other divergence properties from the underlying Petz and sandwiched divergences under the stated parameter restrictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The remaining open interval (1/2, 2/3) for the sandwiched case indicates that a separate argument would still be needed to reach the full conjectured range down to 1/2.
  • Because the duality is obtained from a minimax theorem over product states, the same technique may apply to other families of quantum divergences whose minimization is likewise convex-concave.
  • Additivity of these quantities would simplify the analysis of their behavior under independent uses of a quantum channel or under repeated measurements on composite systems.

Load-bearing premise

Sion's minimax theorem applies directly to the convex-concave functions that appear when the minimization is restricted to product states, without extra regularity conditions that would fail for some quantum states.

What would settle it

An explicit bipartite quantum state together with a concrete alpha value at least 2/3 for which the value of the doubly minimized sandwiched Renyi mutual information on two copies fails to equal twice the value on one copy.

Figures

Figures reproduced from arXiv: 2406.01699 by Laura Burri.

Figure 1
Figure 1. Figure 1: Comparison of PRMIs for a pure state. Suppose [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of PRMIs for a copy-CC state. Suppose [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
read the original abstract

The doubly minimized Petz Renyi mutual information of order $\alpha$ is defined as the minimization of the Petz divergence of order $\alpha$ of a fixed bipartite quantum state relative to any product state. The doubly minimized sandwiched Renyi mutual information is defined analogously using the sandwiched divergence in place of the Petz divergence. In this work, we establish several properties of these two types of Renyi mutual information. In particular, for the Petz case, we prove additivity for $\alpha\in [1/2,2]$. For the sandwiched case, we establish a novel duality relation for $\alpha\in [2/3,\infty]$ via Sion's minimax theorem, and we subsequently use this duality relation to prove additivity for the same range of $\alpha$. Previously, additivity for the sandwiched case was known only for $\alpha\in [1,\infty]$, but it had been conjectured to hold for $\alpha\in [1/2,\infty]$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper defines the doubly minimized Petz Renyi mutual information (minimization of Petz divergence over product states) and the analogous doubly minimized sandwiched Renyi mutual information. It proves additivity of the Petz version for α ∈ [1/2, 2] and, for the sandwiched version, derives a novel duality relation for α ∈ [2/3, ∞] via Sion's minimax theorem, from which additivity follows in the same interval (extending the previously known range α ≥ 1).

Significance. If the central derivations hold, the results extend the parameter range for additivity of these Renyi mutual informations, which are relevant to quantum channel capacities, entanglement measures, and strong converse bounds. The explicit use of Sion's minimax to obtain the duality is a technical contribution; the manuscript supplies mathematical proofs rather than numerical fits.

major comments (1)
  1. [§ on duality via Sion's minimax] § on duality via Sion's minimax (the section establishing the novel duality relation for sandwiched Renyi mutual information): the application of Sion's theorem for α down to 2/3 requires explicit verification that the relevant functional (sandwiched divergence or mutual-information expression) is upper/lower semi-continuous in the respective variables over the compact convex set of product states. The manuscript must confirm these regularity conditions hold at the lower endpoint; without them the duality fails and the additivity claim for α ∈ [2/3, 1) does not follow.
minor comments (2)
  1. [Abstract and §1] The abstract and introduction should explicitly state the finite-dimensional assumption required for compactness of the product-state set when invoking Sion's theorem.
  2. [§2] Notation for the doubly minimized quantities should be introduced with a displayed equation immediately after the verbal definition to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for highlighting the need to verify the regularity conditions in the application of Sion's minimax theorem. We address the single major comment below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

  1. Referee: [§ on duality via Sion's minimax] § on duality via Sion's minimax (the section establishing the novel duality relation for sandwiched Renyi mutual information): the application of Sion's theorem for α down to 2/3 requires explicit verification that the relevant functional (sandwiched divergence or mutual-information expression) is upper/lower semi-continuous in the respective variables over the compact convex set of product states. The manuscript must confirm these regularity conditions hold at the lower endpoint; without them the duality fails and the additivity claim for α ∈ [2/3, 1) does not follow.

    Authors: We agree that an explicit verification of the upper/lower semi-continuity of the sandwiched Renyi divergence (and the associated mutual-information functional) over the compact convex set of product states is necessary to rigorously justify the application of Sion's minimax theorem down to α = 2/3. In the revised manuscript we will add a dedicated remark (or short appendix) confirming these conditions. The sandwiched Renyi divergence is jointly lower semi-continuous in its arguments for α ≥ 1/2 and upper semi-continuous for α > 1 in the relevant topologies; combined with compactness of the product-state set and convexity/concavity properties already established in the paper, this ensures the minimax equality holds on the closed interval [2/3, ∞]. We will explicitly check the endpoint α = 2/3 using the known continuity properties of the sandwiched divergence at that value. revision: yes

Circularity Check

0 steps flagged

No circularity: pure mathematical proofs of additivity via standard theorems

full rationale

The paper defines doubly minimized Petz and sandwiched Renyi mutual information directly from the respective divergences and then proves additivity properties for specified ranges of α. The Petz additivity proof and the sandwiched duality (via Sion's minimax) followed by additivity are self-contained derivations resting on the definitions plus the external Sion theorem; no fitted parameters, no self-citation chains, and no renaming of known results appear in the derivation chain. The work is therefore independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard convex analysis and quantum information definitions; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Sion's minimax theorem applies to the relevant functions on the set of product quantum states
    Invoked to obtain the duality relation for the sandwiched case

pith-pipeline@v0.9.0 · 5692 in / 1348 out tokens · 24447 ms · 2026-05-24T00:00:42.199390+00:00 · methodology

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Doubly minimized Petz and sandwiched Renyi mutual information: Operational interpretation from binary quantum state discrimination

    quant-ph 2024-06 unverdicted novelty 6.0

    The direct exponent in binary quantum state discrimination for correlation detection equals the doubly minimized Petz Renyi mutual information for alpha in (1/2,1), while the strong converse exponent equals the doubly...

Reference graph

Works this paper leans on

83 extracted references · 83 canonical work pages · cited by 1 Pith paper · 3 internal anchors

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    Proof of (a), (b), (c), (e), (f ), (g), (h), (i), (j), (k)

    Proof of Proposition 12 We prove the listed items not in alphabetical order, but in a d ifferent order. Proof of (a), (b), (c), (e), (f ), (g), (h), (i), (j), (k). These properties follow from the correspond- ing properties of the Petz divergence, see Proposition 2. In particular, (e) follows from the non- negativity of the Petz divergence because ρAB ̸⊥ σ...

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    These properties follow from the corresponding properties of the Petz divergence, see Proposition 2

    Proof of Proposition 13 Proof of (a), (e), (j), (n). These properties follow from the corresponding properties of the Petz divergence, see Proposition 2. In particular, (n) follows from the additivity and positiv e definiteness of the Petz divergence. Proof of (i). I ↓ 1 (ρAB∥σA) = D(ρAB∥σA ⊗ ρB) follows from ( 2.15). Now, suppose ρA ̸⊥ σA. Then exp(− I ↓ ...

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    This assertion follows from the symmetry of the definition of the non-minimized PRMI in ( 2.30) with respect to A and B

    Proof of Proposition 3 Proof of (a). This assertion follows from the symmetry of the definition of the non-minimized PRMI in ( 2.30) with respect to A and B. Proof of (b), (c), (d), (e), (f ), (h), (i), (j), (k), (l), (m). Since I ↑↑ α (A : B)ρ = I ↑ α(ρAB∥ρA), these properties follow from the corresponding properties of the non-minimized generalized PRMI,...

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    Since I ↑↓ α (A : B)ρ = I ↓ α(ρAB∥ρA), these properties follow from the corresponding propertie s of the minimized gener- alized PRMI, see Proposition 13

    Proof of Proposition 4 Proof of (a), (b), (c), (d), (e), (g), (h), (i), (j), (k), (l), (m), (n), (o), (r). Since I ↑↓ α (A : B)ρ = I ↓ α(ρAB∥ρA), these properties follow from the corresponding propertie s of the minimized gener- alized PRMI, see Proposition 13. Proof of (p). Let α ∈ (0, ∞ ). Case 1: α ∈ (0, 1) ∪ (1, ∞ ). Let β := 1 α . By Proposition 13 (...

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    Proof of Lemma 5 Proof. √ XA ⊗ √ YB + √ X ′ A ⊗ √ Y ′ B = (XA ⊗ 1B)#(1A ⊗ YB) + (X ′ A ⊗ 1B)#(1A ⊗ Y ′ B) (E.1) ≤ ((XA ⊗ 1B) + (X ′ A ⊗ 1B))#((1A ⊗ YB) + (1A ⊗ Y ′ B)) (E.2) = √ XA + X ′ A ⊗ √ YB + Y ′ B (E.3) The inequality follows from the subadditivity of the geomet ric operator mean

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    Let X, Y, Z ∈ L(A) be positive semidefinite and such that X ≤ Y

    Proof of Theorem 6 Lemma 14 (Saturation of operator inequality from saturation of trac e inequality) . Let X, Y, Z ∈ L(A) be positive semidefinite and such that X ≤ Y . Then all of the following hold. (a) tr[XZ ] ≤ tr[Y Z ]. (b) If tr[XZ ] = tr[ Y Z ] and Y ≪ Z, then X = Y . Proof. By spectral decomposition, Z = ∑ λ∈ spec(Z) λPλ, where Pλ ∈ L (A) denotes t...

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    We will now elucidate the notation for these derivatives

    Lemma for Theorem 7 (k) In order to prove the following lemma, we will use Fréchet der ivatives. We will now elucidate the notation for these derivatives. Consider BA := {XA ∈ L (A) : XA is self-adjoint } with ∥·∥∞ as 32 a Banach space over R. Similarly, consider BB := {YB ∈ L (B) : YB is self-adjoint } with ∥·∥∞ as a Banach space over R. Let U ⊆ B A be a...

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    This assertion follows from the symmetry of the definition of the doubly minimized PRMI in ( 2.32) with respect to A and B

    Proof of Theorem 7 Proof of (a). This assertion follows from the symmetry of the definition of the doubly minimized PRMI in ( 2.32) with respect to A and B. Proof of (b), (e), (f ), (h), (i), (n), (r). Since I ↓↓ α (A : B)ρ = inf σA∈S (A) I ↓ α(ρAB∥σA), these proper- ties follow from corresponding properties of the minimized generalized PRMI, see Propositi...

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    Lemmas for Theorem 8 Let us define the following function of µ ∈ [0, ∞ ) for any ρAB ∈ S (AB), n ∈ N>0. ˆα′ n,ρ(µ) = min T n AnBn ∈L (AnBn): 0≤ T n AnBn ≤ 1 {tr[ρ⊗ n AB(1 − T n AnBn)] : max σAn ∈S sym(A⊗ n), τBn ∈S (Bn) tr[σAn ⊗ τBnT n AnBn ] ≤ µ} (F.1) The following lemma describes some basic properties of the f unctions ˆαiid n,ρ, ˆαn,ρ, ˆαind n,ρ, ˆα′ n...

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    Proof of Theorem 8 First, we prove the bounds on R1/2. It follows from Theorem 7 (q) that I ↓↓ s (A : B)ρ is right- differentiable on s ∈ (0, 1). By the monotonicity of the doubly minimized PRMI in the Rényi order, see Theorem 7 (n), ∂ ∂s+ I ↓↓ s (A : B)ρ ≥ 0 ∀s ∈ (0, 1). (F.17) For any s ∈ (0, 1) and any fixed (σA, τB) ∈ arg min(σ′ A,τ ′ B )∈S (A)×S (B) Ds...

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    (F.43) We will now construct a counterexample for the consideratio n in Remark

    Example for Remark 4 Before we begin constructing a counterexample, let us note th at for any ρAB ∈ S (AB), n ∈ N>0, µ ∈ [0, ∞ ), ˆαind n,ρ(µ) = min T n AnBn ∈L (AnBn): 0≤ T n AnBn ≤ 1 {tr[ρ⊗ n AB(1 − T n AnBn)] : max m∈ N>0 max σ(i) An ∈S (An),τ (i) Bn ∈S (Bn), (pi)i∈ [m]∈ [0,1]× m: ∑ i∈ [m] pi=1 ∑ i∈ [m] pi tr[σ(i) An ⊗ τ (i) BnT n AnBn] ≤ µ}. (F.43) We...

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    Let p ∈ ( 1 2 , 1) and let ρAB := p|0, 0⟩ ⟨0, 0|AB + (1 − p)|1, 1⟩ ⟨1, 1|AB

    Suppose dA ≥ 2, dB ≥ 2, and let {|i⟩A}1 i=0, {|i⟩B}1 i=0 be orthonormal vectors in A, B. Let p ∈ ( 1 2 , 1) and let ρAB := p|0, 0⟩ ⟨0, 0|AB + (1 − p)|1, 1⟩ ⟨1, 1|AB . By Theorem 7 (v), R1/2 := I ↓↓ 1/2(A : B)ρ − 1 4 ∂ ∂s+ I ↓↓ s (A : B)ρ ⏐ ⏐ s= 1 2 = − log p − 1 4 (− 4 log max(p, 1 − p)) = 0 . (F.44) Consider now the left-hand side of ( 3.30) with ˆαn,ρ r...

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    Proof of Corollary 9 Proof. Let ρAB ∈ S (AB) and let R ∈ (−∞ , I(A : B)ρ). By Lemma 17 (b), 0 ≤ lim inf n→∞ ˆαn,ρ(e− nR) ≤ lim sup n→∞ ˆαn,ρ(e− nR). (F.47) We will now prove that limn→∞ ˆαn,ρ(e− nR) = 0 by cases. Case 1: R ∈ (0, I(A : B)ρ). By the proof of achievability for Theorem 8, see ( F.23), lim inf n→∞ − 1 n log ˆαn,ρ(e− nR) ≥ sup s∈ ( 1 2 ,1) 1 − ...

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    Let ρAB ∈ S (AB) and let R be defined as in (3.32)

    Proof of Corollary 10 Lemma 19. Let ρAB ∈ S (AB) and let R be defined as in (3.32). Let a, b ∈ [ 1 2 , 1] be such that a < b . Then, (a, b) → [0, ∞ ), s ↦→ I ↓↓ s (A : B)ρ is constant iff (a, b) → [0, ∞ ), s ↦→ R(s) is constant. Proof. Suppose (a, b) → [0, ∞ ), s ↦→ I ↓↓ s (A : B)ρ is constant, i.e., ∃c ∈ R such that I ↓↓ s (A : B)ρ = c for all s ∈ (a, b). ...

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    Let ρAB ∈ S (AB) be such that V (A : B)ρ ̸= 0

    Proof for Remark 5 Proof. Let ρAB ∈ S (AB) be such that V (A : B)ρ ̸= 0. By Theorem 7 (n), (p) it follows that there exists α0 ∈ ( 1 2 , 1) such that I ↓↓ α (A : B)ρ is strictly monotonically increasing for α ∈ [α0, 1]. Let R be defined as in ( 3.32). By Lemma 19 and Corollary 10, it follows that R(α) is strictly monotonically increasing for α ∈ [α0, 1]. T...

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