Cloning Quantum Channels

For states, the no-cloning theorem is one of the first encounters in quantum information: N arbitrary unknown pure states cannot be perfectly copied onto M waiting registers. Relaxing the problem to allow for imperfect copying allows one to investigate the rate and quality of N-to-M cloning. Our first result is to prove that cloning states is actually equivalent to cloning a particular transformation---the trash-and-replace channel. State cloning turns out to be a special case of a more general problem. On this foundation, we build a unified framework that can address the most general cloning tasks. Our approach relies on higher-order quantum operations and we focus, in particular, on those that can super-replicate---achieve a quadratic cloning rate with vanishing error. We derive the strong converse on traditional state cloning and provide a novel bound on the linear replication rate. The quality measure of the clones matters: under some fidelities super-replication is possible, whilst under others it is not. Furthermore, we establish the necessary conditions for the super-replication of any family of channels: noisy phase gates can be super-replicated under bit flip noise, whereas the whole set of unitaries cannot, regardless of the noise model. We also prove that super-replication is not possible for classical noise channels nor the amplitude-damping channel. Finally, we present numerical methods to search for optimal cloning processes and establish a connection between channel cloning and Bayesian channel estimation.