No-Go Theorem for Generic Simulation of Qubit Channels with Finite Classical Resources

TL;DR

This paper proves that perfect simulation of ideal qubit channels with finite classical resources is impossible, but noisy channels can be simulated with finite communication, with resource requirements depending on the noise level.

Contribution

It establishes a fundamental no-go theorem for simulating ideal qubit channels classically, and explores conditions under which noisy channels can be simulated.

Findings

No finite classical communication can perfectly simulate a noiseless qubit channel.

Noisy qubit channels, like depolarizing channels, can be simulated with finite classical resources.

The amount of classical communication needed increases as the noise in the channel decreases.

Abstract

The mathematical framework of quantum theory, though fundamentally distinct from classical physics, raises the question of whether quantum processes can be efficiently simulated using classical resources. For instance, a sender (Alice) possessing the classical description of a qubit state can simulate the action of a qubit channel through finite classical communication with a receiver (Bob), enabling Bob to reproduce measurement statistics for any observable on the state. In this work, we contend that a more general simulation requires reproducing statistics of joint measurements, potentially involving entangled effects, on Alice's system and an additional system held by Bob, even when Bob's system state is unknown or entangled with a larger system. Within this broad framework, we prove that no finite amount of classical messaging, regardless of how many rounds are used or how large each message can be, can reproduce a perfect qubit channel, highlighting an inescapable barrier in quantum channel simulation with classical resources. We also establish that entangled effects crucially underlies this no-go result. However, for noisy qubit channels, such as those with depolarizing noise, we demonstrate that general simulation is achievable with finite communication. Notably, the required communication increases as the noise decreases, revealing an intricate relationship between the noise in the channel and the resources necessary for its classical simulation.