Uncertainty relations between quantum Fisher information and entanglement monotones

TL;DR

This paper establishes uncertainty relations linking quantum Fisher information to entanglement monotones, providing new bounds and insights into the role of high-dimensional entanglement in quantum metrology.

Contribution

It introduces a family of uncertainty relations that connect quantum Fisher information with entanglement monotones, filling a key gap in understanding their relationship.

Findings

Quantum Fisher information bounds bipartite entanglement monotones.

High-dimensional entanglement is necessary for multiparameter estimation.

The method extends to multipartite systems.

Abstract

Entanglement is widely regarded as an essential resource for a number of tasks and can in some cases be quantified by figures of merit related to those tasks. In quantum metrology, this is showcased by the connections between the quantum Fisher information (QFI), providing a bound to the precision, and multipartite entanglement quantifiers such as the entanglement depth. However, a connection between the QFI and entanglement monotones, i.e., functions that do not increase under Local Operations and Classical Communications, has so far remained elusive. In this work, we fill this gap by introducing a family of uncertainty relations that bound bipartite entanglement monotones from below via elements of a quantum Fisher information matrix. To further emphasize the significance of our results, we connect these relations to the achievable precision in multiparameter estimation. Considering a system split into two parts with arbitrary dimension, we also show that, while two-dimensional entanglement is sufficient to estimate a single parameter with maximal precision, genuine high-dimensional entanglement is required for multiparameter estimation. We conclude by illustrating how our method extends naturally to a multipartite splitting.