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arxiv: 2309.04539 · v1 · pith:MA7MVELTnew · submitted 2023-09-08 · 🪐 quant-ph

R\'enyi-Holevo inequality from α-z-R\'enyi relative entropies

Diego G. Bussandri , Grzegorz Rajchel-Mieldzio\'c , Pedro W. Lamberti , Karol \.Zyczkowski This is my paper

Pith reviewed 2026-05-24 06:33 UTC · model grok-4.3

classification 🪐 quant-ph
keywords Rényi relative entropyHolevo inequalityα-mutual informationquantum channel capacityquantum distance measuresreliability function
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The pith

The α-z-Rényi relative entropies produce the Holevo-Rényi inequality and a bound on α-mutual information.

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

The paper introduces the family of α-z-Rényi relative entropies, which includes the ordinary Rényi relative entropy and the sandwiched version as special cases. These quantities are shown to meet the technical requirements of a generalized Holevo theorem that converts any suitable quantum distance measure into a single-letter upper bound on classical information transmission. The resulting Holevo-Rényi inequality supplies a concrete quantum bound on the α-mutual information. This bound is presented as a tool for analyzing the performance of memoryless quantum channels and the associated reliability functions.

Core claim

Through the introduction of the α-z-Rényi relative entropies, which comprise known relevant quantities such as the Rényi relative entropy and the sandwiched Rényi relative entropy, we establish the Holevo-Rényi inequality. This result leads to a quantum bound for the α-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.

What carries the argument

The α-z-Rényi relative entropies, used as quantum distance measures that satisfy the conditions of the generalized Holevo theorem.

If this is right

  • The α-mutual information between input and output of any quantum channel is upper-bounded by a quantity derived from the α-z-Rényi relative entropy.
  • Single-letter bounds become available for the reliability function of memoryless multi-letter channels.
  • The same construction recovers known Holevo-type bounds when the parameters α and z are chosen to recover the ordinary or sandwiched Rényi relative entropies.

Where Pith is reading between the lines

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

  • The family may be used to interpolate between existing bounds and produce tighter estimates for particular channels by tuning α and z.
  • Because the construction is parameter-dependent, it could be tested by checking whether the bound tightens or loosens monotonically with α for fixed z on standard channels such as the depolarizing channel.

Load-bearing premise

The α-z-Rényi relative entropies must qualify as valid quantum distance measures that satisfy the conditions required by the generalized Holevo theorem.

What would settle it

A concrete counter-example in which the α-z-Rényi relative entropy fails to produce a valid single-letter upper bound on the α-mutual information for some quantum channel.

Figures

Figures reproduced from arXiv: 2309.04539 by Diego G. Bussandri, Grzegorz Rajchel-Mieldzio\'c, Karol \.Zyczkowski, Pedro W. Lamberti.

Figure 1
Figure 1. Figure 1: Region plot of the parameter space M: Set of all pairs (α, z) ∈ R2 satisfying the data processing inequalities, Eq. (12). butions given by Eqs. (2) and (3), respectively: Dα(P||p × q) = 1 α − 1 log X x∈X ,y∈Y pM(x, y) α [p(x)qM(y)]1−α . (13) If we maximize this quantity over the set of possible measurements, we will obtain a R´enyi accessible information, which coincides with the usual accessible informati… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the bounds E(1), Eq. (29), E˜sq 0 (1), Eq. (30) and E q 0 (1), Eq. (33), for different pure-state binary channels, defined in Eq. (34). The bound E(1) is always greater than or equal to E˜sq 0 (1), and coincides with E q 0 (1) for all channels [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
read the original abstract

We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the $\alpha$-$z$-R\'enyi relative entropies, which comprise known relevant quantities such as the R\'enyi relative entropy and the sandwiched R\'enyi relative entropy, we establish the Holevo-R\'enyi inequality. This result leads to a quantum bound for the $\alpha$-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.

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

2 major / 2 minor

Summary. The paper introduces the two-parameter family of α-z-Rényi relative entropies (recovering the ordinary Rényi and sandwiched Rényi cases at special parameter values) and applies a generalized Holevo theorem to this family to obtain a single-letter Rényi-Holevo inequality. The resulting bound is then used to derive a quantum upper bound on the α-mutual information, with suggested implications for channel performance and reliability functions.

Significance. If the α-z family satisfies the monotonicity, joint convexity, and normalization conditions required by the generalized Holevo theorem over the full claimed parameter domain, the construction would supply a unified derivation of several known Rényi-type bounds and a systematic route to new single-letter inequalities. The approach of engineering distance measures to fit the hypotheses of the generalized theorem is technically interesting and could be extended to other information measures.

major comments (2)
  1. [Definition and properties of α-z-Rényi relative entropies] The central application of the generalized Holevo theorem (presumably in the section following the definition of the α-z quantities) requires that the α-z-Rényi relative entropies obey the data-processing inequality under arbitrary CPTP maps for every (α,z) in the stated domain. The manuscript defines the family via a two-parameter formula but supplies no independent verification or citation establishing monotonicity outside the regimes already known for the ordinary and sandwiched cases; this property is load-bearing for the claimed inequality.
  2. [Application to the generalized Holevo theorem] The abstract states that the new family 'comprise known relevant quantities' and thereby 'establish the Holevo-Rényi inequality,' yet the text does not appear to contain an explicit check that all required axioms (data-processing, normalization, and any convexity condition used by the generalized theorem) hold simultaneously across the full parameter range invoked for the α-mutual-information bound.
minor comments (2)
  1. Clarify the precise domain of (α,z) for which the inequality is asserted; the abstract does not list the restrictions.
  2. Add a short comparison table or paragraph contrasting the new bound with existing Rényi-Holevo results (e.g., those based solely on sandwiched entropy) to make the incremental contribution explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need for explicit verification of the key properties of the α-z-Rényi relative entropies. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Definition and properties of α-z-Rényi relative entropies] The central application of the generalized Holevo theorem (presumably in the section following the definition of the α-z quantities) requires that the α-z-Rényi relative entropies obey the data-processing inequality under arbitrary CPTP maps for every (α,z) in the stated domain. The manuscript defines the family via a two-parameter formula but supplies no independent verification or citation establishing monotonicity outside the regimes already known for the ordinary and sandwiched cases; this property is load-bearing for the claimed inequality.

    Authors: We agree that the manuscript does not contain an independent verification or additional citations for monotonicity (data-processing inequality) of the α-z family outside the already-established regimes for the ordinary and sandwiched Rényi cases. The definition is constructed so that it recovers those cases, where the property is known from the literature. For the general parameter domain used in the subsequent bounds, we will add an appendix or subsection that either sketches the monotonicity proof (following the standard techniques for the special cases) or supplies the relevant citations to recent results on the α-z family. This addresses the load-bearing requirement for applying the generalized Holevo theorem. revision: yes

  2. Referee: [Application to the generalized Holevo theorem] The abstract states that the new family 'comprise known relevant quantities' and thereby 'establish the Holevo-Rényi inequality,' yet the text does not appear to contain an explicit check that all required axioms (data-processing, normalization, and any convexity condition used by the generalized theorem) hold simultaneously across the full parameter range invoked for the α-mutual-information bound.

    Authors: The abstract emphasizes that the family includes the known quantities for which the axioms hold, thereby recovering the Holevo-Rényi inequality in those instances. We acknowledge, however, that an explicit, simultaneous verification of all required axioms (data-processing, normalization, and any convexity condition) across the full parameter range for the α-mutual-information bound is not present in the current text. In the revision we will insert a concise verification statement or table confirming that the axioms hold for the specific (α,z) values employed in the bound, ensuring the generalized theorem applies directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies general theorem to newly introduced family

full rationale

The paper defines the α-z-Rényi relative entropies as a two-parameter family recovering known cases at special points, then invokes a pre-existing generalized Holevo theorem (applicable to any quantum distance measure obeying data-processing, convexity, and normalization) to obtain the Rényi-Holevo bound. This constitutes an application rather than a self-referential loop: the result follows from the theorem once the new quantities are shown to meet its hypotheses. No quoted step reduces the output inequality to a fitted parameter, a self-citation chain, or a definition that already encodes the target bound. The derivation remains self-contained against external benchmarks once the monotonicity/convexity claims are established, whether inside the paper or via independent references.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review; the paper invokes a generalized Holevo theorem and introduces a new family of relative entropies whose properties are taken as given.

axioms (1)
  • domain assumption α-z-Rényi relative entropies satisfy the conditions of the generalized Holevo theorem for arbitrary quantum distance measures.
    Invoked to establish the inequality from the family of entropies.
invented entities (1)
  • α-z-Rényi relative entropies no independent evidence
    purpose: Parameterized family that unifies known Rényi quantities and enables derivation of the Holevo-Rényi inequality.
    Introduced in the abstract as the key tool; independent evidence not provided.

pith-pipeline@v0.9.0 · 5664 in / 1221 out tokens · 24208 ms · 2026-05-24T06:33:32.739432+00:00 · methodology

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Reference graph

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