Abstract
Achievability in information theory refers to demonstrating a coding strategy that accomplishes a prescribed performance benchmark for the underlying task. In quantum information theory, the crafted Hayashi-Nagaoka operator inequality is an essential technique in proving a wealth of one-shot achievability bounds since it effectively resembles a union bound in various problems. In this work, we show that the so-called pretty-good measurement naturally plays a role as the union bound as well. A judicious application of it considerably simplifies the derivation of one-shot achievability for classical-quantum channel coding via an elegant three-line proof. The proposed analysis enjoys the following favorable features. (i) The established one-shot bound admits a closed-form expression as in the celebrated Holevo-Helstrom theorem. Namely, the average error probability of sending messages through a classical-quantum channel is upper bounded by the minimum error of distinguishing the joint channel input-output state against decoupled product states. (ii) Our bound directly yields asymptotic achievability results in the large deviation, small deviation, and moderate deviation regimes in a unified manner. (iii) The coefficients incurred in applying the Hayashi-Nagaoka operator inequality or the quantum union bound are no longer needed. Hence, the derived one-shot bound sharpens existing results relying on the Hayashi-Nagaoka operator inequality. In particular, we obtain the tightest achievable -one-shot capacity for classical communication over quantum channels heretofore, improving the third-order coding rate in the asymptotic scenario. (iv) Our result holds for infinite-dimensional Hilbert space. (v) The proposed method applies to deriving one-shot achievability for classical data compression with quantum side information, entanglement-assisted classical communication over quantum channels, and various quantum network information-processing protocols.
- Received 28 April 2023
- Revised 20 September 2023
- Accepted 16 October 2023
DOI:https://doi.org/10.1103/PRXQuantum.4.040330
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
In quantum information science, transmitting classical information over a noisy quantum channel is a foundational task. To protect the transmitted messages against potential quantum noise, we have developed a simple, effective, and physically implementable coding strategy based on the so-called pretty-good measurement. Our approach improves upon existing error bounds and does not rely on structural hypotheses of the underlying quantum channel, such as independent and identical noises. The derived one-shot capacities for unassisted and entanglement-assisted classical communication over quantum channels are both the tightest heretofore. We have also extended our coding methods and analysis to cover various quantum network information-processing protocols in a unified manner.
The established error bounds are tighter than previously known, and the derivation is simpler, technically innovative, and more conceptually intuitive. The strength, simplicity, and variability of the derived error characterizations make the proposed coding and analysis likely a textbook approach when studying classical communication in one-shot and asymptotical quantum information theory. We anticipate this fundamental technique to yield further applications in other fields of quantum information and technology.
