Asymptotically optimal purification and dilution of mixed qubit and Gaussian states

TL;DR

This paper develops asymptotically optimal methods for purifying and diluting mixed qubit states by translating the problem into Gaussian state manipulation using local asymptotic normality, providing comprehensive solutions for the trace norm metric.

Contribution

It introduces a novel approach to qubit state purification and dilution by leveraging Gaussian state techniques through local asymptotic normality, offering the first full solutions for these processes.

Findings

Optimal procedures for qubit purification and dilution are derived.

Solutions are provided for the Gaussian state attenuation and amplification problems.

The methods achieve asymptotic optimality under the trace norm figure of merit.

Abstract

Given an ensemble of mixed qubit states, it is possible to increase the purity of the constituent states using a procedure known as state purification. The reverse operation, which we refer to as dilution, reduces the level of purity present in the constituent states. In this paper we find asymptotically optimal procedures for purification and dilution of an ensemble of i.i.d. mixed qubit states, for some given input and output purities and an asymptotic output rate. Our solution involves using the statistical tool of local asymptotic normality, which recasts the qubit problem in terms of attenuation and amplification of a single displaced Gaussian state. Therefore, to obtain the qubit solutions, we must first solve the analogous problems in the Gaussian setup. We provide full solutions to all of the above, for the (global) trace norm figure of merit.