Reliable Simulation of Quantum Channels: the Error Exponent

TL;DR

This paper investigates the error exponent in quantum channel simulation, providing bounds and an exact formula in the low-rate case, linking it to sandwiched Rényi information and offering finite-blocklength insights.

Contribution

It derives bounds and an exact formula for the quantum channel simulation error exponent, connecting it to sandwiched Rényi information and including finite-blocklength bounds.

Findings

Exact error exponent formula in the low-rate regime

Bounds for the error exponent based on channel purified distance

Finite-blocklength achievability bound for quantum channel simulation

Abstract

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 R\'enyi 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.