Reliable Simulation of Quantum Channels

The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel assisted by unlimited shared entanglement is possible, if and only if, the classical communication cost is greater than or equal to the channel's entanglement-assisted capacity. In this letter, we are concerned with the performance of reliable reverse Shannon simulation of quantum channels. Our main result is an in-depth characterization of the reliability function, that is, the optimal rate under which the performance of channel simulation asymptotically approaches the perfect. In particular, we have determined the exact formula of the reliability function when the classical communication cost is not too high -- below a critical value. In the derivation, we have also obtained an achievability bound for the simulation of finite many copies of the channel, which is of realistic significance.