Estimating the expectation values of spin 1/2 observables with finite resources

TL;DR

This paper investigates optimal strategies for estimating the expectation values of spin-1/2 observables on qubits with limited resources, comparing separate and joint measurement approaches and analyzing their effectiveness.

Contribution

It provides a detailed comparison of joint versus separate measurement strategies for finite resources and introduces insights into their relative advantages for estimating multiple observables.

Findings

Joint measurements can outperform separate ones when estimating a single observable.

Using all measurement results for estimation favors individual measurements over joint ones.

The study extends to estimating three complementary observables, highlighting measurement strategy considerations.

Abstract

We examine the problem of estimating the expectation values of two observables when we have a finite number of copies of an unknown qubit state. Specifically we examine whether it is better to measure each of the observables separately on different copies or to perform a joint measurement of the observables on each copy. We find that joint measurements can sometimes provide an advantage over separate measurements, but only if we make estimates of an observable based solely on the results of measurements of that observable. If we instead use both sets of results to estimate each observable then we find that individual measurements will be better. Finally we consider estimating the expectation values of three complementary observables for an unknown qubit.