Quantitative characterization of several entanglement detection criteria

TL;DR

This paper quantitatively compares various entanglement detection criteria for bipartite quantum systems, revealing their relative effectiveness and how certain criteria improve with specific parameters.

Contribution

It provides a numerical estimation of the volume ratios of states detected by different criteria, highlighting the superior effectiveness of the positive partial transpose criterion.

Findings

Positive partial transpose criterion is most effective.

Reduction, majorization, and Re9nyi-entropy criteria are less effective.

Detectable entanglement ratio increases with Re9nyi entropy order.

Abstract

Quantitative characterization of different entanglement detection criteria for bipartite systems is presented. We review the implication sequence of these criteria and then numerically estimate volume ratios between criteria non-violating quantum states and all quantum states. The numerical approach is based on the hit-and-run algorithm, which is applied to the convex set of all quantum states embedded into a Euclidean vector space of the Hilbert-Schmidt inner product. We demonstrate that reduction, majorization, and the R\'enyi-entropy-based criteria are very ineffective compared to the positive partial transpose. In the case of the R\'enyi-entropy-based criterion, we show that the ratio of detectable entanglement increases with the order of the R\'enyi entropy.