Quantum Purity Amplification for Arbitrary Eigenstates and Multiple Outputs

TL;DR

This paper introduces a comprehensive solution for quantum purity amplification (QPA), enabling high-fidelity eigenstate copying from mixed states across arbitrary dimensions and spectra, with detailed performance laws and sample complexity bounds.

Contribution

It provides the first dimension-uniform sample complexity bounds for optimal QPA and develops a generalized Young diagram theory for nonasymptotic analysis.

Findings

Optimal QPA performance characterized across output regimes.

Input copies scale as O(m/(ff D_{k, ext{min}}^2)) for constant spectral gap.

Dimension-uniform guarantees established for QPA with tight sample complexity bounds.

Abstract

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 $D_{k,\mathrm{min}}$, achieving all-site error $\varepsilon$ requires a number of input copies independent of $d$ and scaling as $O(m/(\varepsilon D_{k,\mathrm{min}}^2))$. 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.