Quantitative characterization of several entanglement detection criteria

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 nonviolating 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 Rényi-entropy-based criteria are very ineffective compared to the positive partial transpose. In the case of the Rényi-entropy-based criterion, we show that the ratio of detectable entanglement increases with the order of the Rényi entropy.

Figure 1

Relations between various entanglement detection criteria. Arrows denote that a state meeting one criterion will also meet the criterion the arrow is pointing at. Dashed arrows indicate special cases when double implications are possible.

Figure 2

Volume ratios $R$ for the Rényi-entropy-based criterion with various $\alpha$. The curves belong to different families of qubit-qubit states.

Figure 3

Volume ratios $R$ between quantum states fulfilling Rényi-entropy-based criteria and all quantum states. For higher-dimensional systems only $\alpha =\infty$ is shown, as this is the most relevant case. For all systems except $2\times 4$ the statistical errors lie within the thickness of the plotted line.

Figure 4

Volume ratios $R$ between PPT and all quantum states for general systems of dimension $d$. For example, $d=35=2^2\times 3^2-1$ for qubit-qutrit systems. The statistical errors lie within the thickness of the dots except for $3\times 4$.