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Reliability Function of Classical-Quantum Channels

Ke Li, Dong Yang

Abstract

We study the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Renyi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Renyi information.

Reliability Function of Classical-Quantum Channels

Abstract

We study the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Renyi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Renyi information.
Paper Structure (1 section, 11 theorems, 43 equations)

This paper contains 1 section, 11 theorems, 43 equations.

Table of Contents

  1. Appendices

Key Result

Theorem 2

Let $\mathcal{N}:\mathcal{X}\rightarrow\mathcal{S}(B)$ be a CQ channel and $r\geq 0$. For any $\alpha\in[\frac{1}{2},1]$ and any probability distribution $p$ on the input alphabet $\mathcal{X}$, we have

Theorems & Definitions (15)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Lemma 4: Renes Renes-exponent
  • Lemma 5
  • Corollary 6
  • Lemma 7: Hayashi-2CKR
  • Lemma 8: MH
  • Lemma 9: GW15
  • Lemma 10: Holevo
  • ...and 5 more