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

arXiv commentv1 :: 14 pages, no figures. This work extends arXiv:2207.08899 to the case of general CQ channels. v2 :: See also independent work by Ke Li and Dong Yang: arXiv:2407.12403

References in corpus (25)

• Simple Proof of Security of the BB84 Quantum Key Distribution Protocol
• Unconditional Security Of Quantum Key Distribution Over Arbitrarily Long Distances
• The Uncertainty Principle in the Presence of Quantum Memory
• On quantum Renyi entropies: a new generalization and some properties
• The operational meaning of min- and max-entropy
• Quantum Information Processing with Finite Resources -- Mathematical Foundations
• Conjectured Strong Complementary Information Tradeoff
• Sandwiched Rényi Divergence Satisfies Data Processing Inequality
• Exponential decreasing rate of leaked information in universal random privacy amplification
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• Uncertainty relations from simple entropic properties
• Fundamental bound on the reliability of quantum information transmission
• On the strong converses for the quantum channel capacity theorems
• Strong converse exponent for classical-quantum channel coding
• Relating different quantum generalizations of the conditional Renyi entropy
• Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels
• Universal coding for classical-quantum channel
• Reliability function of general classical-quantum channel
• Optimal Quantization for Distribution Synthesis
• Non-Asymptotic Classical Data Compression with Quantum Side Information
• Tight Exponential Analysis for Smoothing the Max-Relative Entropy and for Quantum Privacy Amplification
• Precise evaluation of leaked information with universal2 privacy amplification in the presence of quantum attacker
• Duality of channels and codes
• Achievable error exponents of data compression with quantum side information and communication over symmetric classical-quantum channels
• Lower Bounds on Error Exponents via a New Quantum Decoder