Large Deviation Analysis for the Reverse Shannon Theorem

Abstract: Channel simulation is to simulate a noisy channel using noiseless channels with unlimited shared randomness. This can be interpreted as the reverse problem to Shannon's noisy coding theorem. In contrast to previous works, our approach employs R\'enyi divergence (with the parameter $\alpha\in(0,\infty)$) to measure the level of approximation. Specifically, we obtain the reverse Shannon theorem under the R\'enyi divergence, which characterizes the R\'enyi simulation rate, the minimum communication cost rate required for the R\'enyi divergence vanishing asymptotically. We also investigate the behaviors of the R\'enyi divergence when the communication cost rate is above or below the R\'enyi simulation rate. When the communication cost rate is above the R\'enyi simulation rate, we provide a complete characterization of the convergence exponent, called the reliability function. When the communication cost rate is below the R\'enyi simulation rate, we determine the linear increasing rate for the R\'enyi divergence with parameter $\alpha\in(0,\infty]$, which implies the strong converse exponent for the $\alpha$-order fidelity.