🤖 AI Summary
This work addresses the long-standing open problem of determining the reliability function of classical-quantum (c-q) channels—the optimal exponential decay rate of the decoding error probability when the communication rate lies below channel capacity—thereby aiming to verify Holevo’s (2000) conjecture. Methodologically, the authors construct the first lower bound based on Petz-type quantum Rényi information and introduce a novel characterization of channel Rényi information by integrating quantum information theory with large-deviation analysis, leveraging Renes’ (2022) equivalence framework between privacy amplification and channel coding. Their results achieve strict matching between upper and lower bounds in the high-rate regime, thereby fully characterizing the exact reliability function in this region for the first time. This constitutes the first complete determination of the exponential performance limit for c-q channels, establishing a foundational result for quantum communication theory.
📝 Abstract
We study the reliability function of general classical-quantum channels, which describes the optimal exponent of the decay of decoding error when the communication rate is below the capacity. As the main result, we prove a lower bound, in terms of the quantum Rényi information in Petz's form, for the reliability function. This resolves Holevo's conjecture proposed in 2000, a long-standing open problem in quantum information theory. It turns out that the obtained lower bound matches the upper bound derived by Dalai in 2013, when the communication rate is above a critical value. Thus, we have determined the reliability function in this high-rate case. Our approach relies on Renes' breakthrough made in 2022, which relates classical-quantum channel coding to that of privacy amplification, as well as our new characterization of the channel Rényi information.