Confident entanglement detection via separable numerical range

TL;DR

This paper introduces a method for entanglement detection based on the geometry of separable and standard numerical ranges of measurements, providing analytical bounds and explicit volume calculations for two-qubit systems.

Contribution

It develops a geometric framework for entanglement detection using separable numerical ranges, including analytical bounds and explicit volume computations for specific quantum observables.

Findings

Separable numerical range can reliably detect entanglement when disjoint from measurement data confidence regions.

The volume ratio between separable and standard numerical ranges is bounded and cannot be arbitrarily small.

Explicit volume calculations are provided for two-qubit product observables, aiding practical measurement implementations.

Abstract

We investigate the joint (separable) numerical range of multiple measurements, i.e., the regions of expectation values accessible with (separable) quantum states for given observables. This not only enables efficient entanglement detection, but also sheds light on the geometry of the set of quantum states. More precisely, in an experiment, if the confidence region for the obtained data and the separable numerical range are disjoint, entanglement is reliably detected. Generically, the success of such an experiment is more likely the smaller the separable numerical range is compared to the standard numerical range of the observables measured. We quantify this relation using the ratio between these two volumes and show that it cannot be arbitrarily small, giving analytical bounds for any number of particles, local dimensions as well as number of measurements. Moreover, we explicitly compute the volume of separable and standard numerical range for two locally traceless two-qubit product observables, which are of particular interest as they are easier to measure in practice. Furthermore, we consider typical volume ratios for generic observables and extreme instances.