Optimal purification of a generic n_qudit state

We propose a quantum algorithm for the purification of a generic mixed state rho of a n-qudit system by using an ancillary n-qudit system. The algorithm is optimal in that (i) the number of ancillary qudits cannot be reduced, (ii) the number of parameters which determine the purification state | Psi > exactly equals the number of degrees of freedom of rho , and (iii) | Psi > is easily determined from the density matrix rho . Moreover, we introduce a quantum circuit in which the quantum gates are unitary transformations acting on a 2n-qudit system. These transformations are determined by parameters that can be tuned to generate, once the ancillary qudits are disregarded, any given mixed n-qudit state.