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arxiv: 2511.10584 · v3 · submitted 2025-11-13 · 🪐 quant-ph

Finite-size quantum key distribution rates from R\'enyi entropies using conic optimization

Mariana Navarro , Andr\'es Gonz\'alez Lorente , Pablo V. Parellada , Carlos Pascual-Garc\'ia , Mateus Ara\'ujo This is my paper

Pith reviewed 2026-05-17 21:59 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum key distributionfinite-size effectsRenyi entropyconic optimizationsecurity proofsconditional entropynon-symmetric cones
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0 comments X

The pith

Non-symmetric conic optimization minimizes conditional Rényi entropy for finite-size QKD security proofs.

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

The paper introduces a computational method based on non-symmetric conic optimization to minimize conditional Rényi entropy in finite-size quantum key distribution security proofs. Earlier Rényi-based approaches offered tighter secret key bounds than von Neumann entropy methods but depended on unstable ad-hoc Frank-Wolfe algorithms limited to special cases. The new technique is shown to be fast, reliable, and applicable to arbitrary protocols, with performance improvements demonstrated on several examples. A reader would care because accurate finite-size analysis determines whether practical QKD systems can extract positive key rates under realistic device imperfections and data sizes.

Core claim

The authors develop a method based on non-symmetric conic optimization for solving the minimization of conditional Rényi entropy. This enables finite-size general security proofs for quantum key distribution that are more flexible and yield tighter bounds on the secret key rate than traditional formulations based on the von Neumann entropy. The technique is fast, reliable, and completely general, overcoming the instability and case-specific restrictions of prior Frank-Wolfe algorithms, as illustrated by improved results on several protocols.

What carries the argument

Non-symmetric conic optimization applied to the minimization of conditional Rényi entropy.

Load-bearing premise

The non-symmetric conic formulation exactly captures the minimization of conditional Rényi entropy without introducing numerical artifacts or relaxations that loosen the security bound.

What would settle it

Compute secret key rates for the BB84 protocol with known finite-size parameters using the new method and verify that the resulting rates are at least as high as those from prior stable methods while remaining below the asymptotic limit.

Figures

Figures reproduced from arXiv: 2511.10584 by Andr\'es Gonz\'alez Lorente, Carlos Pascual-Garc\'ia, Mariana Navarro, Mateus Ara\'ujo, Pablo V. Parellada.

Figure 1
Figure 1. Figure 1: Finite secret key rate for qubit BB84 protocol using the FastRényiQKD cone (solid lines), compared [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Finite secret key rate for qubit BB84 protocol using the FastRényiQKD cone for a different number [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Finite secret key generation rate for the MUB protocol using the FastRényiQKD [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Finite secret key generation rate for the MUB protocol using the RényiQKD [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Secret key generation rates for the DMCV protocol using the FastRényiQKD cone [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
read the original abstract

Finite-size general security proofs for quantum key distribution based on R\'enyi entropies have recently been introduced. These approaches are more flexible and provide tighter bounds on the secret key rate than traditional formulations based on the von Neumann entropy. However, deploying them requires minimizing the conditional R\'enyi entropy, a difficult optimization problem that has hitherto been tackled using ad-hoc techniques based on the Frank-Wolfe algorithm, which are unstable and can only handle particular cases. In this work, we introduce a method based on non-symmetric conic optimization for solving this problem. Our technique is fast, reliable, and completely general. We illustrate its performance on several protocols, whose results represent an improvement over the state of the art.

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 manuscript introduces a non-symmetric conic optimization method to minimize conditional Rényi entropies arising in finite-size security proofs for quantum key distribution. It positions the technique as faster, more reliable, and more general than prior ad-hoc Frank-Wolfe solvers, and reports improved key-rate bounds on several concrete protocols.

Significance. If the conic formulation is proven to recover the exact minimum of the conditional Rényi entropy without hidden relaxations or positive duality gaps, the work would supply a practical, general-purpose computational tool that tightens finite-size analyses beyond von Neumann entropy methods and broadens the range of protocols that can be treated rigorously.

major comments (2)
  1. [§3] §3 (Conic formulation): The central claim that the non-symmetric conic program exactly encodes the minimization of conditional Rényi entropy (without outer approximations or relaxations) is load-bearing for all subsequent security bounds, yet the manuscript provides only a sketch of equivalence rather than a full proof that the optimum is recovered with zero duality gap for the reported instances.
  2. [Numerical results] Numerical results section, Tables 1–3: Direct side-by-side comparisons with the Frank-Wolfe baseline on identical problem instances (including duality-gap certificates, solver tolerances, and stability under small perturbations) are absent; without them the asserted improvement in reliability and generality cannot be quantitatively verified.
minor comments (2)
  1. [§2] The notation for the non-symmetric cone and its dual could be introduced with a short explicit low-dimensional example to aid readers unfamiliar with conic programming.
  2. [Figures] Figure captions should explicitly state the numerical tolerance and solver used for each plotted curve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [§3] §3 (Conic formulation): The central claim that the non-symmetric conic program exactly encodes the minimization of conditional Rényi entropy (without outer approximations or relaxations) is load-bearing for all subsequent security bounds, yet the manuscript provides only a sketch of equivalence rather than a full proof that the optimum is recovered with zero duality gap for the reported instances.

    Authors: We agree that the current sketch of equivalence in Section 3 is insufficient to fully support the central claim. In the revised manuscript we will expand Section 3 (and add a dedicated appendix) with a complete, self-contained proof that the non-symmetric conic program recovers the exact minimum of the conditional Rényi entropy with zero duality gap for all problem instances reported in the paper. The proof will explicitly rule out hidden relaxations and will include the relevant strong-duality arguments under the conditions satisfied by the QKD instances considered. revision: yes

  2. Referee: [Numerical results] Numerical results section, Tables 1–3: Direct side-by-side comparisons with the Frank-Wolfe baseline on identical problem instances (including duality-gap certificates, solver tolerances, and stability under small perturbations) are absent; without them the asserted improvement in reliability and generality cannot be quantitatively verified.

    Authors: We agree that direct quantitative comparisons are necessary to substantiate the claimed improvements. In the revised manuscript we will augment the numerical results section with side-by-side tables that run both the conic solver and the Frank-Wolfe baseline on identical problem instances. These tables will report duality-gap certificates, the solver tolerances employed, and numerical stability results under small perturbations of the input data, thereby allowing direct verification of reliability and generality. revision: yes

Circularity Check

0 steps flagged

No significant circularity; numerical solver applied to independent optimization task

full rationale

The paper introduces a non-symmetric conic optimization technique to solve the minimization of conditional Rényi entropy, an existing problem in finite-size QKD security proofs. This is framed as an external, general-purpose numerical method rather than a derivation that reduces to fitted parameters, self-referential definitions, or load-bearing self-citations. No equations or steps in the provided abstract or description equate the output rates to the inputs by construction, and the approach remains independent of the specific key-rate values it computes. The derivation chain is self-contained as a solver for a pre-defined task.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract alone, the work relies on standard properties of Rényi entropies and convex optimization; no free parameters, ad-hoc axioms, or invented entities are visible.

axioms (1)
  • domain assumption Conditional Rényi entropy minimization can be cast as a non-symmetric conic program
    Invoked when the authors state that the optimization problem is solved via non-symmetric conic optimization.

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discussion (0)

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Forward citations

Cited by 4 Pith papers

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

  1. Rigorous Security Proofs for Practical Quantum Key Distribution

    quant-ph 2026-04 unverdicted novelty 7.0

    Rigorous security proofs for variable-length QKD, phase-error bounding with imperfect detectors, marginal-constrained entropy accumulation, and authentication reductions place practical QKD on firmer mathematical ground.

  2. Numerical security analysis for practical quantum key distribution

    quant-ph 2026-05 unverdicted novelty 6.0

    A numerical framework proves finite-key security for practical decoy-state QKD systems with transmitter and receiver imperfections including non-IID signals.

  3. Quantum Key Distribution with Imperfections: Recent Advances in Security Proofs

    quant-ph 2026-02 unverdicted novelty 2.0

    Overview of recent analytical and numerical advances in security proofs for QKD protocols that incorporate device imperfections to bridge theory and practice.

  4. Quantum Key Distribution with Imperfections: Recent Advances in Security Proofs

    quant-ph 2026-02 unverdicted novelty 1.0

    Overview of recent analytical and numerical developments in QKD security proofs that incorporate imperfections to re-establish security under realistic conditions.

Reference graph

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