Optimal purification of a generic n-qudit state

TL;DR

This paper introduces an optimal quantum algorithm for purifying any generic mixed state of an n-qudit system using an ancillary system, with a focus on efficiency and exact parameter matching.

Contribution

It presents a quantum algorithm that optimally purifies n-qudit states with minimal ancillary resources and a straightforward method to determine the purification state from the density matrix.

Findings

The algorithm uses an ancillary n-qudit system with no resource reduction possible.

The number of parameters matches the degrees of freedom of the original state.

The quantum circuit can generate any mixed n-qudit state through tunable unitary transformations.

Abstract

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.