Optimal purification of a genericn-qudit state

Abstract

We propose a quantum algorithm for the purification of a generic mixed state $\ensuremath{\rho}$ 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 $|\ensuremath{\Psi}⟩$ exactly equals the number of degrees of freedom of $\ensuremath{\rho}$, and (iii) $|\ensuremath{\Psi}⟩$ is easily determined from the density matrix $\ensuremath{\rho}$. 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.