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This paper resolves a 1998 conjecture by establishing a one-shot random coding bound for classical-quantum channels, thereby determining the optimal error exponent across infinite-dimensional Hilbert spaces. The authors introduce an operator layer cake theorem to prove that pretty-good measurements are operationally equivalent to randomized Holevo-Helstrom measurements, offering a rigorous explanation for their efficacy in packing problems. These theoretical advances provide foundational tools for analyzing reliability in fully quantum communication scenarios, including those with entanglement assistance and data compression tasks.
Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem
arXiv:2507.06232v4 Announce Type: replace 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.
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.
v4: some proofs simplified; v3: new added §3.1: Extension to Infinite Dimensions; v2: tables and references added
Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph); Functional Analysis (math.FA)
arXiv:2507.06232 [quant-ph]
(or arXiv:2507.06232v4 [quant-ph] for this version)
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