The rate of optimal purification procedures

Purification is a process in which decoherence is partially reversed by usi ng several input systems which have been subject to the same noise. The pur ity of the outputs generally increases with the number of input systems, an d decreases with the number of required output systems. We construct the op timal quantum operations for this task, and discuss their asymptotic behavi our as the number of inputs goes to infinity. The rate at which output syst ems may be generated. depends crucially on the type of purity requirement. If one tests the purity of the output systems one at a time, the rate is in finite : this fidelity may be made to approach 1, while at the same time th e number of outputs goes to infinity arbitrarily fast. On the other hand, i f one also requires the correlations between outputs to decrease, the rate is zero: if fidelity with the pure product state is to go to 1, the number of outputs per input goes to zero. However, if only a fidelity close to 1 i s required, the optimal purifier achieves a positive rate, which we compute .