One-Shot Classical-Quantum Capacity and Hypothesis Testing (1007.5456v3)

Abstract: The one-shot classical capacity of a quantum channel quantifies the amount of classical information that can be transmitted through a single use of the channel such that the error probability is below a certain threshold. In this work, we show that this capacity is well approximated by a relative-entropy-type measure defined via hypothesis testing. Combined with a quantum version of Stein's lemma, our results give a conceptually simple proof of the well-known Holevo-Schumacher-Westmoreland theorem for the capacity of memoryless channels. More generally, we obtain tight capacity formulas for arbitrary (not necessarily memoryless) channels.

• The paper establishes a relationship between one-shot classical capacity and hypothesis testing relative entropy to simplify quantum channel performance analysis.
• It adapts quantum Stein's Lemma to streamline the proof of the Holevo-Schumacher-Westmoreland Theorem, extending capacity bounds beyond memoryless channels.
• The results offer practical insights for designing quantum communication protocols with finite channel uses, influencing both theory and applications.

One-Shot Classical-Quantum Capacity and Hypothesis Testing

The paper under review investigates the one-shot classical capacity of quantum channels and relates it to hypothesis testing measures, thereby providing novel insights into classical-quantum channel coding. The core contribution of this research is the establishment of a relationship between the one-shot capacity and a relative-entropy-type measure derived from hypothesis testing. This connection not only simplifies the understanding of channel capacity but extends the well-known asymptotic results to encompass arbitrary quantum channels.

Summary of Key Findings

The authors define the one-shot classical capacity for a quantum channel as the maximum amount of classical information that can be transmitted in a single use of the channel with error probability constrained by a threshold. The paper's principal result reveals that this capacity can be effectively approximated using a hypothesis testing relative entropy, denoted as $D_H^\epsilon(\rho\|\sigma)$. This relative entropy quantifies the difficulty of statistically distinguishing between two quantum states $\rho$ and $\sigma$, constrained by the type I error rate (related to the hypothesis testing criteria).

The research builds upon a quantum adaptation of Stein's Lemma, allowing the authors to offer a streamlined proof of the Holevo-Schumacher-Westmoreland Theorem, which addresses memoryless channels. Furthermore, the findings extend to determining tight capacity bounds for channels without the memorylessness constraint, making this a significant generalization.

Strong Numerical Results and Claims

The paper substantiates its claims with rigorous mathematical proofs, leveraging properties of the hypothesis testing relative entropy. Among the key properties of $D_H^\epsilon(\rho\|\sigma)$: it adheres to positivity, satisfies the Data Processing Inequality (DPI), and underpins a direct connection to quantum relative entropy. Moreover, the bounds on channel capacity demonstrated in the paper are notable for their simplicity and generality, showing compatibility both with existing theoretical frameworks and practical scenarios involving repeated channel use.

Implications and Future Directions

From a theoretical standpoint, this work seamlessly integrates hypothesis testing with the evaluation of one-shot capacities, directly impacting quantum Shannon theory. Practically, it offers a methodological blueprint for designing communication protocols that operate efficiently under quantum constraints with finite uses, which is particularly pertinent in burgeoning quantum computing and communication technologies.

Looking forward, this framework might inspire further exploration into the interplay between classical and quantum information theory, potentially uncovering newer bounds on communication in quantum networks or influencing the development of quantum error correction techniques. The findings suggest a deeper intrinsic relation between hypothesis testing metrics and entropic measures, inviting further research into this domain and its implications for advanced information-theoretic tasks.

In conclusion, the authors effectively extend classical-quantum information theory into more general and tangible scenarios with this comprehensive paper on one-shot classical-quantum capacity. Their integration of hypothesis testing as a tool emerges as a fundamental approach to tackling quantum channel capacity, paving the way for further innovative explorations in quantum information science.