Estimation of quantum finite mixtures

TL;DR

This paper develops strategies to estimate the weights of quantum mixtures, providing optimal solutions in various scenarios and deriving explicit formulas for qubit systems, with Fisher information playing a key role.

Contribution

It introduces optimal estimation strategies for quantum mixture weights, including single-shot and asymptotic cases, with explicit formulas for qubits and a novel role for Fisher information.

Findings

Explicit formulas for error matrices in two-component qubit mixtures

Optimal strategies for single-shot and multiple-copy scenarios

Fisher information determines the minimum covariance matrix

Abstract

We consider the problem of determining the weights of a quantum ensemble. That is to say, given a quantum system that is in a set of possible known states according to an unknown probability law, we give strategies to estimate the individual probabilities, weights, or mixing proportions. Such strategies can be used to estimate the frequencies at which different independent signals are emitted by a source. They can also be used to estimate the weights of particular terms in a canonical decomposition of a quantum channel. The quality of these strategies is quantified by a covariance-type error matrix. According with this cost function, we give optimal strategies in both the single-shot and multiple-copy scenarios. The latter is also analyzed in the asymptotic limit of large number of copies. We give closed expressions of the error matrix for two-component quantum mixtures of qubit systems. The Fisher information plays an unusual role in the problem at hand, providing exact expressions of the minimum covariance matrix for any number of copies.