Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916966429622272 |
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| author | Cheng, Hao-Chung Liu, Po-Chieh |
| author_facet | Cheng, Hao-Chung Liu, Po-Chieh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06232 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem Cheng, Hao-Chung Liu, Po-Chieh Quantum Physics Information Theory Mathematical Physics Functional Analysis 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. |
| title | Error Exponents for Quantum Packing Problems via An Operator Layer Cake Theorem |
| topic | Quantum Physics Information Theory Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2507.06232 |