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

• URL: http://arxiv.org/abs/2507.06232v2
• Date: Mon, 28 Jul 2025 12:57:57 GMT
• Title: Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem
• Authors: Hao-Chung Cheng, Po-Chieh Liu,
• Abstract summary: We prove a one-shot random coding bound for classical-quantum channel coding.<n>Our result extends to various quantum packing-type problems.<n>This shows that a kind of pretty-good measurement is equivalent to a randomized Holevo-Helstrom measurement.
• Score: 6.675805308519987
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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, we recover the optimal error exponent (i.e., the reliability function) of classical-quantum channels for rates above the critical rate. 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.

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