Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels

Abstract

In this paper, lower bounds on error probability in coding for discrete classical and classical-quantum channels are studied. The contribution of the paper goes in two main directions: i) extending classical bounds of Shannon, Gallager and Berlekamp to classical-quantum channels, and ii) proposing a new framework for lower bounding the probability of error of channels with a zero-error capacity in the low rate region. The relation between these two problems is revealed by showing that Lovász' bound on zero-error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical-quantum channels. A variation of Lovász' bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels. As a result of this study, connections between the Lovász theta function, the expurgated bound of Gallager, the cutoff rate of a classical channel and the sphere packing bound for classical-quantum channels are established.

Updated to published version + bug fixed in Figure 3

References in corpus (4)

• The Quantum Chernoff Bound
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function

Cited by in corpus (26)

• Moderate deviation analysis for classical communication over quantum channels
• Quantum Sphere-Packing Bounds with Polynomial Prefactors
• Applications of position-based coding to classical communication over quantum channels
• The Renyi Capacity and Center
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Encoding classical information into quantum resources
• Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions
• Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels
• Properties of Noncommutative Renyi and Augustin Information
• The Sphere Packing Bound For Memoryless Channels
• Simple and Tighter Derivation of Achievability for Classical Communication over Quantum Channels
• Spherical Cap Packing Asymptotics and Rank-Extreme Detection
• The Sphere Packing Bound via Augustin's Method
• On the Concavity of Auxiliary Function in Classical-Quantum Channels
• Observations on Graph Invariants with the Lovász $\vartheta$-Function
• Reliability Function of Classical-Quantum Channels
• Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels
• Reliability Function of Quantum Information Decoupling via the Sandwiched Rényi Divergence
• Reliable Simulation of Quantum Channels: the Error Exponent
• Tight lower bound on the error exponent of classical-quantum channels
• Lower Bounds on Error Exponents via a New Quantum Decoder
• Quantum $f$-divergences via Nussbaum-Szkoła Distributions and Applications to $f$-divergence Inequalities
• Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels
• Elias Bound for General Distances and Stable Sets in Edge-Weighted Graphs
• Quantum channel coding: Approximation algorithms and strong converse exponents
• On Decoder Ties for the Binary Symmetric Channel with Arbitrarily Distributed Input