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

• URL: http://arxiv.org/abs/2102.00312v4
• Date: Wed, 6 Jul 2022 07:54:17 GMT
• Title: Entanglement in bipartite quantum systems: Euclidean volume ratios and detectability by Bell inequalities
• Authors: A. Sauer, J. Z. Bern\'ad, H. J. Moreno, G. Alber
• Abstract summary: Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. A new numerical approach is developed to explore the typicality of entanglement and of its detectability by Bell inequalities. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Euclidean volume ratios between quantum states with positive partial transpose and all quantum states in bipartite systems are investigated. These ratios allow a quantitative exploration of the typicality of entanglement and of its detectability by Bell inequalities. For this purpose a new numerical approach is developed. It is based on the Peres-Horodecki criterion, on a characterization of the convex set of quantum states by inequalities resulting from Newton identities and from Descartes' rule of signs, and on a numerical approach involving the multiphase Monte Carlo method and the hit-and-run algorithm. This approach confirms not only recent analytical and numerical results on two-qubit, qubit--qutrit, and qubit--four-level qudit states but also allows for a numerically reliable numerical treatment of so far unexplored qutrit--qutrit states. Based on this numerical approach with the help of the Clauser-Horne-Shimony-Holt inequality and the Collins-Gisin inequality the degree of detectability of entanglement is investigated for two-qubit quantum states. It is investigated quantitatively to which extent a combined test of both Bell inequalities can increase the detectability of entanglement beyond what is achievable by each of these inequalities separately.

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