Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem

2025/07/08 by Hao–Chung Cheng, Cheng, Hao-Chung, Po-Chieh Liu +1 · 4 citations
Computer Science · Engineering · #Complexity and Algorithms in Graphs #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Information Theory (cs.IT) #Mathematical Physics (math-ph) #Quantum Information and Cryptography #Quantum Physics (quant-ph) #Wireless Communication Security Techniques

paper · pdf · doi:10.48550/arxiv.2507.06232

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this work, we prove a one-shot random coding bound for classical-quantum channel coding, a problem conjectured by Burnashev and Holevo in 1998. By choosing the optimal input distribution, the bound implies the optimal error exponent (i.e., the reliability function) of classical-quantum channels for rates above the critical rate, even in infinite-dimensional Hilbert spaces. Our result extends to various quantum packing-type problems, including classical communication over any fully quantum channel with or without entanglement-assistance, constant composition codes, and classical data compression with quantum side information via fixed-length or variable-length coding. Our technical ingredient is to establish an operator layer cake theorem - the directional derivative of an operator logarithm admits an integral representation of certain projections. This shows that a kind of pretty-good measurement is equivalent to a randomized Holevo-Helstrom measurement, which provides an operational explanation of why the pretty-good measurement is pretty good.

Citations

• Two-Parameter Rényi Information Quantities with Applications to Privacy Amplification and Soft Covering
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• Layer Cake Representations for Quantum Divergences
• On Araki-Type Trace Inequalities
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• Strong Converse for Privacy Amplification against Quantum Side Information
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• The χ ²χ2-divergence and mixing times of quantum Markov processes
• Security of Quantum Key Distribution
• Multiple operator integrals and higher operator derivatives
• Jensen's Operator Inequality
• Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem
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• On Reliability Function of Quantum Communication Channel
• The Capacity of Quantum Channel with General Signal States

Cited by

• The operator layer cake theorem is equivalent to Frenkel's integral formula
• Strong converse exponent of channel interconversion
• Sharp estimates of quantum covering problems via a novel trace inequality
• Layer Cake Representations for Quantum Divergences

Related

• Error exponent of activated non-signaling assisted classical-quantum channel coding
• Lower Bounds on Error Exponents via a New Quantum Decoder
• Random coding exponents galore via decoupling
• On the Error Exponent Distribution of Code Ensembles over Classical-Quantum Channels
• A random-coding based proof for the quantum coding theorem