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Exponents for Shared Randomness-Assisted Channel Simulation

Aadil Oufkir, Michael X. Cao, Hao-Chung Cheng, Mario Berta

TL;DR

The paper studies large-deviation exponents for channel simulation under non-signaling and shared randomness assistance, measured in worst-case total-variation distance. It derives exact error and strong converse exponents expressed as optimizations over Rényi channel mutual information $I_\alpha(W)$, revealing no critical rate and a tight rate-dependent trade-off for all $r$. The authors first solve the NS problem using a one-shot TVD formula, then establish a matching NS converse via the method of types and de Finetti reductions, and finally connect NS and SR via rounding techniques that preserve the exponents. They show entanglement does not change these exponents and provide finite-n refinements, delivering a comprehensive large-deviation understanding of randomness-assisted channel simulation with potential implications for broader channel interconversion tasks.

Abstract

We determine the exact error and strong converse exponents of shared randomness-assisted channel simulation in worst case total-variation distance. Namely, we find that these exponents can be written as simple optimizations over the Rényi channel mutual information. Strikingly, and in stark contrast to channel coding, there are no critical rates, allowing a tight characterization for arbitrary rates below and above the simulation capacity. We derive our results by asymptotically expanding the meta-converse for channel simulation [Cao {\it et al.}, IEEE Trans.~Inf.~Theory (2024)], which corresponds to non-signaling assisted codes. We prove this to be asymptotically tight by employing the approximation algorithms from [Berta {\it et al.}, Proc.~IEEE ISIT (2024)], which show how to round any non-signaling assisted strategy to a strategy that only uses shared randomness. Notably, this implies that any additional quantum entanglement-assistance does not change the error or the strong converse exponents.

Exponents for Shared Randomness-Assisted Channel Simulation

TL;DR

The paper studies large-deviation exponents for channel simulation under non-signaling and shared randomness assistance, measured in worst-case total-variation distance. It derives exact error and strong converse exponents expressed as optimizations over Rényi channel mutual information , revealing no critical rate and a tight rate-dependent trade-off for all . The authors first solve the NS problem using a one-shot TVD formula, then establish a matching NS converse via the method of types and de Finetti reductions, and finally connect NS and SR via rounding techniques that preserve the exponents. They show entanglement does not change these exponents and provide finite-n refinements, delivering a comprehensive large-deviation understanding of randomness-assisted channel simulation with potential implications for broader channel interconversion tasks.

Abstract

We determine the exact error and strong converse exponents of shared randomness-assisted channel simulation in worst case total-variation distance. Namely, we find that these exponents can be written as simple optimizations over the Rényi channel mutual information. Strikingly, and in stark contrast to channel coding, there are no critical rates, allowing a tight characterization for arbitrary rates below and above the simulation capacity. We derive our results by asymptotically expanding the meta-converse for channel simulation [Cao {\it et al.}, IEEE Trans.~Inf.~Theory (2024)], which corresponds to non-signaling assisted codes. We prove this to be asymptotically tight by employing the approximation algorithms from [Berta {\it et al.}, Proc.~IEEE ISIT (2024)], which show how to round any non-signaling assisted strategy to a strategy that only uses shared randomness. Notably, this implies that any additional quantum entanglement-assistance does not change the error or the strong converse exponents.

Paper Structure

This paper contains 17 sections, 24 theorems, 131 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Works
  3. Notations
  4. Channel Simulation and the Exact Distortion in TVD
  5. Error Exponent for Non-Signaling Channel Simulation
  6. Achievability for the Error Exponent
  7. Converse for the Error Exponent
  8. Strong Converse Exponent for Non-Signaling Channel Simulation
  9. Converse for the Strong Converse Exponent
  10. Achievability for the Strong Converse Exponent
  11. Exponents in the Shared Randomness-Assisted Scenario
  12. Error Exponent for Shared Randomness-Assisted Channel Simulation
  13. Strong Converse Exponent for Shared Randomness-Assisted Channel Simulation
  14. Conclusion
  15. Classical Post-Selection Lemma with Integral Representation
  16. ...and 2 more sections

Key Result

Proposition 1

Given a channel $W_{\mathsf{Y}|\mathsf{X}}\in\mathcal{P}(\mathcal{Y}|\mathcal{X})$ and a positive integer $M$, the minimal attainable distortion (measured in TVD) of non-signaling simulation codes for $W_{\mathsf{Y}|\mathsf{X}}$ with alphabet size at most $M$ can be expressed as where $(x)_+\coloneqq \max\{x,0\}$ for $x\in \mathbb{R}$.

Figures (1)

  • Figure 1: Various channel simulation schemes: (a) Non-assisted Channel Simulation; (b) Randomness-assisted Channel Simulation; (c) Non-signaling-assisted Channel Simulation. (This figure is copied from cao2024channel.)

Theorems & Definitions (48)

  • Proposition 1: Minimal distortion in TVD for non-signaling channel simulation
  • Lemma 2
  • proof
  • proof : Proof of Proposition \ref{['prop:epsilon:NS:exact']}
  • Proposition 3: Achievability for the Error Exponents
  • Remark 4
  • proof
  • Proposition 5
  • Lemma 6: Post-Selection Lemma / de Finetti reduction
  • Definition 7: Types
  • ...and 38 more