Reliable Simulation of Quantum Channels: The Error Exponent

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. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel’s sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.

I. Introduction

Any dynamical process in nature can be modeled as a quantum channel. The simulation of quantum channels is not only a fundamental problem in quantum communication [1], [2], [3], [4], [5], [6], [7], [8], [9], but has also attracted broad interest in the topics of quantum thermodynamics [10], [11] as well as general quantum resource theory [12], [13], [14]. Dual to the ordinary scenario of distilling perfect resources from noisy channels, channel simulation is the problem of generating a noisy channel by making use of noiseless resources.