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Euclidean volume ratios between quantum states with positive partialtranspose and all quantum states in bipartite systems are investigated. Theseratios allow a quantitative exploration of the typicality of entanglement andof its detectability by Bell inequalities. For this purpose a new numericalapproach is developed. It is based on the Peres-Horodecki criterion, on acharacterization of the convex set of quantum states by inequalities resultingfrom Newton identities and from Descartes' rule of signs, and on a numericalapproach involving the multiphase Monte Carlo method and the hit-and-runalgorithm. This approach confirms not only recent analytical and numericalresults on two-qubit, qubit--qutrit, and qubit--four-level qudit states butalso allows for a numerically reliable numerical treatment of so far unexploredqutrit--qutrit states. Based on this numerical approach with the help of theClauser-Horne-Shimony-Holt inequality and the Collins-Gisin inequality thedegree of detectability of entanglement is investigated for two-qubit quantumstates. It is investigated quantitatively to which extent a combined test ofboth Bell inequalities can increase the detectability of entanglement beyondwhat is achievable by each of these inequalities separately.

Entanglement in bipartite quantum systems: Euclidean volume ratios and detectability by Bell inequalities