Confident entanglement detection via separable numerical range

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.

21 pages, 8 figures, 4 tables

References in corpus (6)

• Entanglement detection
• Determining lower bounds on a measure of multipartite entanglement from few local observables
• Ultrafine Entanglement Witnessing
• Entangled states close to the maximally mixed state
• Iterations of nonlinear entanglement witnesses
• Formulas for Rational-Valued Separability Probabilities of Random Induced Generalized Two-Qubit States