Single-letter capacity formulas, which provide a concise expression for the capacity of a communication channel, are highly desirable in information theory as they offer insights into the fundamental limits of communication systems. In the context of quantum channels, obtaining single-letter capacity formulas for special classes of channels can be challenging but extremely valuable for understanding the communication capabilities of quantum systems.
To explore the feasibility of deriving single-letter capacity formulas for special classes of quantum channels, one can consider the following approaches:
Channel Symmetry: Exploit any inherent symmetries or structures in the quantum channel to simplify the capacity calculation. Symmetric channels often lend themselves to single-letter capacity expressions, where the capacity can be characterized by a single optimization problem.
Degraded Channels: For degraded quantum channels, where one channel is a degraded version of another, single-letter capacity formulas may be attainable. By leveraging the properties of degraded channels, one can potentially derive concise expressions for the capacity.
Additive Channels: Additive quantum channels, where the overall channel operation is a sum of independent channel operations, may allow for the derivation of single-letter capacity formulas. The linearity of additive channels can lead to simplified capacity expressions.
Special Quantum States: Utilize special quantum states or resources that simplify the capacity calculation. For example, entangled states with specific properties may enable the derivation of single-letter capacity formulas for certain classes of quantum channels.
Quantum Error Correction: Incorporate quantum error correction techniques to handle noise and imperfections in the channel. By designing efficient error correction codes tailored to the channel characteristics, one can potentially derive single-letter capacity formulas.
While obtaining single-letter capacity formulas for general quantum channels is a challenging task, focusing on special classes of channels with specific properties and structures can increase the likelihood of success. By leveraging the unique features of quantum communication systems and employing advanced mathematical techniques, researchers can work towards deriving concise and insightful single-letter capacity formulas for special classes of quantum channels.