Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy (1306.1586v4)

Abstract: A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a "sandwiched" Renyi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.

• The paper establishes a strong converse theorem showing that if transmission rates exceed the classical capacity, the probability of successful decoding decays exponentially.
• The paper employs sandwiched Rényi relative entropy to rigorously bound the channel capacity by leveraging its monotonicity and subadditivity properties.
• The results clarify the performance limits of entanglement-breaking and Hadamard channels, offering both theoretical insights and practical guidance for quantum communication design.

Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels

Introduction

This paper presents a strong converse theorem for the classical capacity of both entanglement-breaking channels and their complementary Hadamard channels. The authors, Mark M. Wilde, Andreas Winter, and Dong Yang, leverage a novel quantity called the "sandwiched Rényi relative entropy" to establish their results. This work sheds light on the limitations and capabilities of quantum channels in transmitting classical information, reinforcing the understanding of classical capacity as a definitive boundary between achievable and unachievable communication rates.

Key Contributions

1. Strong Converse Theorem: The paper demonstrates that if communication rates exceed the classical capacity, the probability of successfully decoding a transmitted message decays exponentially to zero. This sharp contrast emphasizes the classical capacity as a boundary for communication rates over entanglement-breaking and Hadamard channels.
2. Sandwiched Rényi Relative Entropy: A pivotal concept introduced in this work is the sandwiched Rényi relative entropy, defined as a generalization of the traditional Rényi relative entropy. It possesses properties like monotonicity under quantum operations and reduces to von Neumann relative entropy in the limit.
3. Additivity and Subadditivity Results: The paper discusses conditions under which the sandwiched Rényi relative entropy maintains subadditivity for entanglement-breaking channels. This property is crucial for establishing a strong converse and understanding the behavior of channel capacities in quantum information theory.

Theoretical and Practical Implications

• Theoretical Insights: The paper's findings contribute to a deeper theoretical understanding of quantum information channels, particularly concerning their capacity to transmit classical information. The distinction between entanglement-breaking and Hadamard channels is clarified, and the role of subadditivity in determining capacity limits is highlighted.
• Practical Applications: These results have potential applications in quantum cryptography and quantum communication protocols. The insights into channel capacities can guide the design of more robust quantum communication systems, ensuring reliable information transfer.

Future Directions

The paper suggests several avenues for future exploration:

• Exploring Other Channel Types: Extending the analysis to other quantum channel classes to determine if similar strong converse properties hold.
• Further Investigation of Sandwiched Rényi Relative Entropy: Exploring additional properties and applications of this entropy measure could open up new areas of research in quantum information theory.
• Improving Communication Protocols: Applying these theoretical insights to develop enhanced communication protocols that optimize the use of quantum channels in practical scenarios.

Conclusion

This paper's articulation of a strong converse theorem for entanglement-breaking and Hadamard channels marks a significant contribution to quantum information theory. By employing the sandwiched Rényi relative entropy, the authors provide a robust framework for understanding quantum channel capacities, thereby enriching both the theoretical landscape and practical applications of quantum communication.