Comparison of entanglement detection criteria for mixtures of states in 2 x 2 and 3 x 3 quantum systems

Abstract

We explore two entanglement detection methods, purity, and the Peres-Horodecki (PPT) criterion, and compare their detection ranges for mixtures of families of states in different dimensions of Hilbert spaces. Specifically, this paper is motivated to explore the advantages of PPT in Hilbert spaces of lower dimension and its shortcomings as we expand the study to higher dimensions. We were able to show that the PPT entanglement criterion is superior to purity in detecting entanglement for mixtures of Bell states in H² ⊗ H², as shown by graphical representation and numerical integration. However, as we apply both PPT and purity for a mixture of states in H³ ⊗ H³, we see that there are states detected through purity that were undetected by PPT, showing that it is no longer sufficient in detecting all possible entangled states for Hilbert spaces of higher dimensions.