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Quantum channel coding: Approximation algorithms and strong converse exponents

Aadil Oufkir, Mario Berta

TL;DR

This work develops approximation algorithms for entanglement-assisted quantum channel coding by leveraging non-signaling relaxations and the meta-converse SDP. It proves that NS and MC success probabilities are tightly connected, and it constructs rounding schemes that translate NS strategies into EA strategies, yielding multiplicative or additive guarantees depending on the setting. In the quantum-classical regime, EA and NS share the same strong-converse exponent, characterized by a sandwiched Rényi mutual information expression, and a parallel exponent equality is established for fully quantum channels via a multiplicative rounding argument grounded in De Finetti flattening. The results unify EA, NS, and MC from a one-shot perspective and provide an alternative pathway to strong-converse exponents, with implications for one-shot coding, hypothesis testing, and large deviation analyses in quantum communication.

Abstract

We study relaxations of entanglement-assisted quantum channel coding and establish that non-signaling assistance and a natural semi-definite programming relaxation\, -- \,termed meta-converse\, -- \,are equivalent in terms of success probabilities. We then present a rounding procedure that transforms any non-signaling-assisted strategy into an entanglement-assisted one and prove an approximation ratio of $(1 - e^{-1})$ in success probabilities for the special case of measurement channels. For fully quantum channels, we give a weaker (dimension dependent) approximation ratio, that is nevertheless still tight to characterize the strong converse exponent of entanglement-assisted channel coding [Li and Yao, IEEE Tran.~Inf.~Theory (2024)]. Our derivations leverage ideas from position-based coding, quantum decoupling theorems, the matrix Chernoff inequality, and input flattening techniques.

Quantum channel coding: Approximation algorithms and strong converse exponents

TL;DR

This work develops approximation algorithms for entanglement-assisted quantum channel coding by leveraging non-signaling relaxations and the meta-converse SDP. It proves that NS and MC success probabilities are tightly connected, and it constructs rounding schemes that translate NS strategies into EA strategies, yielding multiplicative or additive guarantees depending on the setting. In the quantum-classical regime, EA and NS share the same strong-converse exponent, characterized by a sandwiched Rényi mutual information expression, and a parallel exponent equality is established for fully quantum channels via a multiplicative rounding argument grounded in De Finetti flattening. The results unify EA, NS, and MC from a one-shot perspective and provide an alternative pathway to strong-converse exponents, with implications for one-shot coding, hypothesis testing, and large deviation analyses in quantum communication.

Abstract

We study relaxations of entanglement-assisted quantum channel coding and establish that non-signaling assistance and a natural semi-definite programming relaxation\, -- \,termed meta-converse\, -- \,are equivalent in terms of success probabilities. We then present a rounding procedure that transforms any non-signaling-assisted strategy into an entanglement-assisted one and prove an approximation ratio of in success probabilities for the special case of measurement channels. For fully quantum channels, we give a weaker (dimension dependent) approximation ratio, that is nevertheless still tight to characterize the strong converse exponent of entanglement-assisted channel coding [Li and Yao, IEEE Tran.~Inf.~Theory (2024)]. Our derivations leverage ideas from position-based coding, quantum decoupling theorems, the matrix Chernoff inequality, and input flattening techniques.

Paper Structure

This paper contains 22 sections, 23 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Overview of findings
  4. Non-signaling assistance and meta-converse.
  5. Quantum-classical setting.
  6. Quantum setting - Additive error.
  7. Quantum setting - Multiplicative error.
  8. Notation
  9. Quantum channel coding
  10. Entanglement-assisted channel coding
  11. Non-signaling assisted channel coding
  12. Non-signaling value and meta-converse
  13. Approximation algorithms in the quantum-classical setting
  14. Approximation algorithms in the quantum setting
  15. Approximation with an additive error
  16. ...and 7 more sections

Key Result

Proposition 1

Let $\mathcal{N}$ be a quantum channel and $M\ge 1$. We have that

Figures (1)

  • Figure 1: Illustration of an entanglement-assisted protocol achieving the performance of Proposition \ref{['prop:qq-rounding']}. Wavy lines: Represent the shared randomness. Thick lines: Represent the shared entanglement. Arrow: Indicates the time progression of the protocol. Depending on the message $m$ and the shared randomness, the system $A_{j_m}^{j_m}$ is swapped with $A'$. Then, the unitary channel $\mathcal{U}_m (\cdot)= U_m^\top(\cdot)\overline{U}_m$ (which depends on $m$ and the shared randomness) is applied before the transmission of the system $A'$ through the channel $\mathcal{N}_{A'\rightarrow B}$. In order to decode the message, a measurement (which depends on the shared randomness) is performed on the systems $B^1_1\cdots B_{v}^v B$ yielding the outcome $m'$.

Theorems & Definitions (32)

  • Proposition 1
  • Corollary 2
  • Proposition 3
  • Corollary 4
  • Proposition 5
  • Proposition 6
  • Corollary 7
  • Proposition 8
  • proof
  • Corollary 9
  • ...and 22 more