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Channel coding against quantum jammers via minimax

Michael X. Cao, Yongsheng Yao, Mario Berta

TL;DR

This work introduces a minimax framework based on Sion's theorem to analyze classical communication over fully quantum arbitrarily varying channels (FQAVCs) with quantum jammers, removing the restriction to finite jammer dimension. By formulating codes as convex joint-input-output maps and leveraging one-shot bounds alongside a generalized AEP, the authors derive capacity characterizations that match the corresponding compound-channel capacities under entanglement-assisted and shared-randomness-assisted scenarios, even when the adversary system is infinite-dimensional. The key results show $C_{fqavc}^{EA}(\mathcal{N}_{AE\to B}) = \sup_{\rho_{A'}} \inf_{\sigma_E} I(A':B)_{(id_{A'}\otimes \mathcal{N}_{AE\to B})(\rho_{A'A}\otimes \sigma_E)}$ and analogous SR-capacities, with extensions to CQ-FQAVCs and QQ-FQAVCs; the approach also lays groundwork for quantum capacities. Overall, the minimax method provides a universal, dimension-agnostic tool for robust quantum channel coding under adversarial interference and suggests several open problems in capacity dichotomy and higher-order analyses.

Abstract

We introduce a minimax approach for characterizing the capacities of fully quantum arbitrarily varying channels (FQAVCs) under different shared resource models. In contrast to previous methods, our technique avoids de Finetti-type reductions, allowing us to treat quantum jammers with infinite-dimensional systems. Consequently, we show that the entanglement-assisted and shared-randomness-assisted capacities of FQAVCs match those of the corresponding compound channels, even in the presence of general quantum adversaries.

Channel coding against quantum jammers via minimax

TL;DR

This work introduces a minimax framework based on Sion's theorem to analyze classical communication over fully quantum arbitrarily varying channels (FQAVCs) with quantum jammers, removing the restriction to finite jammer dimension. By formulating codes as convex joint-input-output maps and leveraging one-shot bounds alongside a generalized AEP, the authors derive capacity characterizations that match the corresponding compound-channel capacities under entanglement-assisted and shared-randomness-assisted scenarios, even when the adversary system is infinite-dimensional. The key results show and analogous SR-capacities, with extensions to CQ-FQAVCs and QQ-FQAVCs; the approach also lays groundwork for quantum capacities. Overall, the minimax method provides a universal, dimension-agnostic tool for robust quantum channel coding under adversarial interference and suggests several open problems in capacity dichotomy and higher-order analyses.

Abstract

We introduce a minimax approach for characterizing the capacities of fully quantum arbitrarily varying channels (FQAVCs) under different shared resource models. In contrast to previous methods, our technique avoids de Finetti-type reductions, allowing us to treat quantum jammers with infinite-dimensional systems. Consequently, we show that the entanglement-assisted and shared-randomness-assisted capacities of FQAVCs match those of the corresponding compound channels, even in the presence of general quantum adversaries.
Paper Structure (19 sections, 4 theorems, 50 equations, 1 figure, 1 table)

This paper contains 19 sections, 4 theorems, 50 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The minimax approach
  4. CPTP on classical-quantum mixed systems.
  5. One-shot largest coding size within given average error.
  6. Codes as joint-input-output maps.
  7. Entanglement-assisted classical capacity of FQAVCs
  8. Converse bound
  9. Achievability bound
  10. Shared-randomness-assisted classical capacities of FQAVCs
  11. SR-assisted capacity of CQ-FQAVCs
  12. Strong converse.
  13. Achievability.
  14. SR-assisted capacity of (QQ-)FQAVCs
  15. Weak converse.
  16. ...and 4 more sections

Key Result

theorem 1

Given a quantum channel $\mathcal{N}_{\sys{AE}\to\sys{B}}$ with adversary system $\sys{E}$, it holds that for all $\epsilon\in(0,1)$.

Figures (1)

  • Figure 1: Point-to-point communication with an interfering adversary, where in both figures $\mathcal{E}$ is the encoder, $\mathcal{D}$ is the decoder, the adversary (Eve) controls the systems $\syss{E}_1^n$. The goal for Alice and Bob is to minimize the probability that $\hat{m}\neq m$ without knowing Eve's messages.

Theorems & Definitions (7)

  • theorem 1
  • proof : Proof of Theorem \ref{['thm:minimax']}
  • proposition 1
  • proof
  • proposition 2
  • lemma 1: Generalized AEP fang2024generalized
  • proof : Proof of Proposition \ref{['prop:achievability:ea:fqavc']}