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Optimality of meta-converse for channel simulation

Aadil Oufkir, Omar Fawzi, Mario Berta

TL;DR

A guarantee on the ratio of success probabilities of at least $(1-\frac{-1}{\mathbf{e}})$ for both the classical and the quantum setting is proved, which can be improved to $(1-\frac{1}{t})$ using $o$ (ln $(t)$) additional bits (qubits) of communication.

Abstract

We study the effect of shared non-signaling correlations for the problem of simulating a channel using noiseless communication in the one-shot setting. For classical channels, we show how to round any non-signaling-assisted simulation strategy--which corresponds to the natural linear programming meta-converse for channel simulation--to a strategy that only uses shared randomness. For quantum channels, we round any non-signaling-assisted simulation strategy to a strategy that only uses shared entanglement. Our main result is for classical and classical-quantum channels, for which we employ ideas from approximation algorithms to give a guarantee on the ratio of success probabilities of at least $(1-\mathrm{e}^{-1})$. We further show this ratio to be optimal for the purely classical case. It can be improved to $(1-t^{-1})$ using $O(\ln \ln(t))$ additional bits of communication.

Optimality of meta-converse for channel simulation

TL;DR

A guarantee on the ratio of success probabilities of at least for both the classical and the quantum setting is proved, which can be improved to using (ln ) additional bits (qubits) of communication.

Abstract

We study the effect of shared non-signaling correlations for the problem of simulating a channel using noiseless communication in the one-shot setting. For classical channels, we show how to round any non-signaling-assisted simulation strategy--which corresponds to the natural linear programming meta-converse for channel simulation--to a strategy that only uses shared randomness. For quantum channels, we round any non-signaling-assisted simulation strategy to a strategy that only uses shared entanglement. Our main result is for classical and classical-quantum channels, for which we employ ideas from approximation algorithms to give a guarantee on the ratio of success probabilities of at least . We further show this ratio to be optimal for the purely classical case. It can be improved to using additional bits of communication.

Paper Structure

This paper contains 28 sections, 16 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Overview of results
  4. Notation
  5. Simulation of classical channels
  6. Shared-randomness assistance
  7. Non-signaling assistance in the classical setting
  8. Rounding
  9. Shared randomness
  10. Encoding
  11. Decoding
  12. Tightness
  13. Non signaling assistance in the quantum setting
  14. Simulation of classical-quantum channels
  15. Entanglement-assistance
  16. ...and 13 more sections

Key Result

Proposition 2.1

Let $M,M'\in \mathbb{N}$ and $\widetilde{W}^{\rm{NS}}_{}$ be a feasible solution of the program classical-ns-program-sym of size $M$. Then, there exists a shared-randomness assisted strategy $\widetilde{W}^{\rm{SR}}_{}$ of size $M'$ such that

Theorems & Definitions (29)

  • Proposition 2.1
  • proof : Proof of Proposition \ref{['prop:meta-inequality-classical']}
  • Corollary 2.2
  • proof : Proof of Corollary \ref{['cor: succ-classical']}
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof : Proof of Lemma \ref{['app:optimal']}
  • Proposition 4.1
  • proof : Proof of Proposition \ref{['prop:meta-inequality-cq']}
  • ...and 19 more