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, requires a rate of classical communication equal to the channel's entanglement-assisted classical capacity. Here, we study the optimal speed at which the performance of channel simulation can exponentially approach the perfect, when the blocklength increases. This is known as the reliability function. We have determined the exact formula of the reliability function when the classical communication cost is not too high -- below a critical value. This enables us to obtain, for the first time, an operational interpretation to the channel's sandwiched R\'enyi mutual information of order from 1 to 2, since our formula of the reliability function is expressed as a transform of this quantity. 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.