Approximate Degradable Quantum Channels (1412.0980v3)

Abstract: Degradable quantum channels are an important class of completely positive trace-preserving maps. Among other properties, they offer a single-letter formula for the quantum and the private classical capacity and are characterized by the fact that a complementary channel can be obtained from the channel by applying a degrading channel. In this work we introduce the concept of approximate degradable channels, which satisfy this condition up to some finite $\varepsilon\geq0$. That is, there exists a degrading channel which upon composition with the channel is $\varepsilon$-close in the diamond norm to the complementary channel. We show that for any fixed channel the smallest such $\varepsilon$ can be efficiently determined via a semidefinite program. Moreover, these approximate degradable channels also approximately inherit all other properties of degradable channels. As an application, we derive improved upper bounds to the quantum and private classical capacity for certain channels of interest in quantum communication.

• The paper introduces ε-degradable channels, extending traditional degradable channels with a quantifiable closeness parameter ε.
• It employs semidefinite programming to efficiently compute the smallest ε, establishing tight upper bounds for channel capacities.
• The study bridges theoretical insights and practical applications, broadening the understanding of near-degradability in quantum communications.

Approximate Degradable Quantum Channels: An Analysis

This paper presents a comprehensive paper on the concept of approximate degradable quantum channels, extending the well-established category of degradable channels in quantum information theory. Degradable channels are known for their favorable mathematical properties, offering a single-letter characterization of quantum and private classical capacities. The paper's focus is on channels that approximately meet the criteria for degradability, quantified by a small parameter $\varepsilon$ which indicates closeness to a perfectly degradable state.

Main Contributions

The authors introduce the notion of $\varepsilon$-degradable channels, where the degradability condition is satisfied up to a finite $\varepsilon\geq0$. They demonstrate that for any fixed channel, the smallest such $\varepsilon$ can be efficiently computed using a semidefinite program (SDP). This approach allows for the characterization of channels that are not exactly degradable but retain most beneficial properties of degradable channels within an approximate framework.

Numerical Methodology and Results

The SDP-based method offers a practical approach to evaluate the smallest $\varepsilon$ for approximate degradability. This quantification helps establish effective upper bounds for the quantum and private classical capacities of certain quantum channels. The paper provides examples of depolarizing channels, among others, showcasing the method's applicability and effectiveness in deriving tight upper bounds.

Theoretical Implications

The introduction of approximate degradable channels broadens the class of channels for which capacities can be effectively bounded and computed. This work provides insights into the structure and characteristics of quantum channels that are close to being degradable, offering a practical bridge between purely theoretical constructs and real-world applications where perfect degradability might be unattainable.

Future Directions

This research invites further exploration into the relationship between approximate degradability and other channel capacities, such as entanglement-breaking or privacy-based capacities. Additionally, potential extensions might explore how these concepts could integrate with recent developments in AI, particularly in optimizing quantum communication protocols where near-degradability could be a significant factor. Further work could also investigate the dual concept of approximate anti-degradability, which was touched upon by the authors.

Conclusion

The paper provides a crucial expansion of the degradable channel concept, offering a robust framework for understanding and utilizing approximate degradability in quantum communication. The introduction of a computationally efficient method for determining this property underscores the relevance of the paper in both theoretical and applied quantum information science.