Quantum Purity Amplification for Arbitrary Eigenstates and Multiple Outputs

Quantum purity amplification (QPA) is the task of coherently transforming n copies of a mixed state into high-fidelity copies of a chosen eigenstate. We solve QPA in the general setting of n input copies, m output copies, arbitrary target eigenstates, arbitrary local dimension d, and generic input spectra. We characterize the optimal channel and derive its all-site and one-site performance laws across output regimes. For the asymptotic analysis, we use a path-graph parametrization to show that, when the target eigenvalue has a constant spectral gap D_k,min, achieving all-site error varepsilon requires a number of input copies independent of d and scaling as O\(m/(varepsilon D_k,min²\)). When m/n approaches a constant, the performance exhibits phase-like regimes, which we characterize explicitly. For the nonasymptotic analysis, we develop a theory of generalized Young diagrams that yields tight sample complexity bounds and provides the first dimension-uniform guarantee for optimal QPA. We also provide asymptotically efficient implementations of the optimal protocol. Together, these results establish QPA as a rigorous example of coherent quantum information processing with dimension-uniform sample complexity, supplying the technical foundation for the coherent-incoherent separation developed in the companion work.