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Tight lower bound on the error exponent of classical-quantum channels

Joseph M. Renes

TL;DR

A lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels is shown which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels.

Abstract

A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel $W$ and rate $R$: the constant $E(W,R)$ which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate $R$ to communicate over ever more (memoryless) instances of a given channel $W$. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai's sphere-packing upper bound [IEEE TIT 59, 8027 (2013)] for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem [IEEE TIT 46, 2256 (2000)]. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification [CMP 333, 335 (2015)]. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager's method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [IEEE TIT 69, 1680 (2023)], at least for linear randomness extractors.

Tight lower bound on the error exponent of classical-quantum channels

TL;DR

A lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels is shown which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels.

Abstract

A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel and rate : the constant which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate to communicate over ever more (memoryless) instances of a given channel . Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai's sphere-packing upper bound [IEEE TIT 59, 8027 (2013)] for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem [IEEE TIT 46, 2256 (2000)]. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification [CMP 333, 335 (2015)]. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager's method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. [IEEE TIT 67, 902 (2021)]. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi [IEEE TIT 69, 1680 (2023)], at least for linear randomness extractors.
Paper Structure (11 sections, 6 theorems, 59 equations)

This paper contains 11 sections, 6 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Mathematical setup
  3. Entropies
  4. Entropy duality
  5. Duality of linear functions
  6. Main result: Lower bound on the error exponent of CQ channels
  7. One-shot error probability bound
  8. Improved bounds for compression and privacy amplification
  9. Discussion
  10. Duality statements
  11. One-shot privacy amplification bound

Key Result

Theorem 3.1

For an arbitrary CQ channel $W_{B|Y}$ whose input $Y$ comes from a discrete alphabet $\mathcal{Y}$ and whose output is a density operator on a quantum system $B$ with finite-dimensional state space, and any rate $R\geq 0$,

Theorems & Definitions (11)

  • Theorem 3.1
  • proof
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Lemma A.1
  • proof
  • Theorem A.2
  • proof
  • Theorem B.1: Theorem 1 hayashi_precise_2015
  • ...and 1 more