A General Purification Protocol with Imperfect State Preparation

Purification protocols take several copies of a mixed state and output a smaller number of copies with higher purity. Recently, a protocol based on the Swap test was shown to have optimal resource consumption, with the number of samples scaling proportional to the inverse of the error. In this work, we consider a more realistic scenario in which all the prepared states are noisy, including the auxiliary qubits used by the protocol. Here, we show that this generalization does not compromise convergence, with the protocol still converging for all non-extreme noise values. Moreover, we estimate the number of iterations and resources needed in the generalized scenario. Such an estimation allows us to address the optimality of the protocol with noisy state preparation and show that the protocol is no longer optimal.