🤖 AI Summary
This work investigates the error exponent of quantum channel simulation—the optimal exponential convergence rate at which simulation fidelity approaches unity as blocklength increases.
Method: Leveraging the channel purification distance and sandwiched Rényi divergences, we establish a unified analytical framework encompassing both asymptotic and finite-blocklength regimes.
Contributions/Results: We derive the first exact closed-form expression for the error exponent in the low-rate regime; prove that its upper and lower bounds coincide precisely at the critical rate, achieving a complete characterization; assign operational meaning to the 1st- and 2nd-order sandwiched Rényi information quantities; and obtain practical finite-blocklength achievability bounds. These results provide new theoretical tools and fundamental benchmarks for quantifying reliability and designing protocols in quantum communication.
📝 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 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.