Optimal weak measurements: Mixed States

TL;DR

This paper extends the analysis of optimal weak measurements from pure to mixed quantum states, demonstrating that mutually unbiased bases are essential for complete tomography and addressing issues in existing protocols.

Contribution

It generalizes the criteria for optimal weak measurements to mixed states and clarifies the role of mutually unbiased bases in quantum state tomography.

Findings

Weak measurements for mixed states are optimal when post-selected states are mutually unbiased.

Weak tomography effectively addresses previous limitations in state error analysis.

Similar conclusions are reached for different weak tomography protocols.

Abstract

In an earlier publication we had given an exhaustive analysis of the criteria for weak value measurements of pure states to be optimal in the sense considered by Wootters and Fields. We had proved, for arbitrary spin cases, that the measurements are optimal when the post-selected state is mutually unbiased wrt the eigenstates of the observable being measured.Here we extend the discussion to mixed states. For these, weak value measurements have several problems which we illustrate with the protocol proposed by Shengjun Wu. We discuss tomography of mixed states based on weak measurements and show that while the principal results of Wootters and Fields hold, namely, the set of observables needed for complete tomography are such that their eigenstates form a mutually unbiased bases, weak tomography removes a serious lacuna from the Wootters and Fields analysis i.e the need to consider only state averaged error volumes or information. We also consider another proposal for weak tomography of mixed states by Lundeen and Bamber, and reach similar conclusions about MUB.