A generalization of Schur-Weyl duality with applications in quantum estimation

TL;DR

This paper generalizes Schur-Weyl duality to enhance quantum estimation techniques, showing when unentangled measurements suffice and providing bounds on entanglement needed for optimal quantum state parameter estimation.

Contribution

It introduces a new generalized duality framework and applies it to quantum estimation, clarifying the role of entanglement and measurement strategies.

Findings

Unentangled measurements are sufficient when observables commute.

Provides bounds on entanglement needed for optimal estimation.

Generalized duality framework applicable to various quantum estimation problems.

Abstract

Schur-Weyl duality is a powerful tool in representation theory which has many applications to quantum information theory. We provide a generalization of this duality and demonstrate some of its applications. In particular, we use it to develop a general framework for the study of a family of quantum estimation problems wherein one is given n copies of an unknown quantum state according to some prior and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. In the case of pure states, we show that commutativity of the set of observables that define the estimation problem implies the sufficiency of unentangled measurements.