Distillation Protocols for Mixed States of Multilevel Qubits and the Quantum Renormalization Group

We study several properties of distillation protocols to purify multilevel qubit states (qudits) when applied to a certain family of initial mixed bipartite states. We find that it is possible to use qudits states to increase the stability region obtained with the flow equations to distill qubits. In particular, for qutrits we get the phase diagram of the distillation process with a rich structure of fixed points. We investigate the large-$D$ limit of qudits protocols and find an analytical solution in the continuum limit. The general solution of the distillation recursion relations is presented in an appendix. We stress the notion of weight amplification for distillation protocols as opposed to the quantum amplitude amplification that appears in the Grover algorithm. Likewise, we investigate the relations between quantum distillation and quantum renormalization processes.