Optimal estimation of entanglement

TL;DR

This paper develops an optimal estimation method for quantifying entanglement in quantum systems using local quantum estimation theory, providing bounds on precision and insights into estimation efficiency for different state types.

Contribution

It derives the optimal observable for entanglement estimation and evaluates the quantum Fisher information for various bipartite states and measures, advancing the understanding of estimation limits.

Findings

Efficient estimation of large entanglement in discrete variables.

Inherent inefficiency in estimating weakly entangled states.

Effectiveness of entanglement estimation in Gaussian systems depends on the measure used.

Abstract

Entanglement does not correspond to any observable and its evaluation always corresponds to an estimation procedure where the amount of entanglement is inferred from the measurements of one or more proper observables. Here we address optimal estimation of entanglement in the framework of local quantum estimation theory and derive the optimal observable in terms of the symmetric logarithmic derivative. We evaluate the quantum Fisher information and, in turn, the ultimate bound to precision for several families of bipartite states, either for qubits or continuous variable systems, and for different measures of entanglement. We found that for discrete variables, entanglement may be efficiently estimated when it is large, whereas the estimation of weakly entangled states is an inherently inefficient procedure. For continuous variable Gaussian systems the effectiveness of entanglement estimation strongly depends on the chosen entanglement measure. Our analysis makes an important point of principle and may be relevant in the design of quantum information protocols based on the entanglement content of quantum states.