Faculty of Physics, University of Vienna, Währingerstrasse 17, 1090, Vienna, Austria.
Sci Rep. 2023 Feb 4;13(1):2037. doi: 10.1038/s41598-023-29211-w.
We compare the classification as entangled or separable of Bell diagonal bipartite qudits with positive partial transposition (PPT) and their properties for different dimensions. For dimension [Formula: see text], a form of entanglement exists that is hard to detect and called bound entanglement due to the fact that such entangled states cannot be used for entanglement distillation. Up to this date, no efficient solution is known to differentiate bound entangled from separable states. We address and compare this problem named separability problem for a family of bipartite Bell diagonal qudits with special algebraic and geometric structures and applications in quantum information processing tasks in different dimensions. Extending analytical and numerical methods and results for Bell diagonal qutrits ([Formula: see text]), we successfully classify more than [Formula: see text] of representative Bell diagonal PPT states for [Formula: see text]. Via those representative states we are able to estimate the volumes of separable and bound entangled states among PPT ququarts ([Formula: see text]). We find that at least [Formula: see text] of all PPT states are separable, [Formula: see text] bound entangled and for [Formula: see text] it remains unclear whether they are separable or bound entangled. Comparing the structure of bound entangled states and their detectors, we find considerable differences in the detection capabilities for different dimensions and relate those to differences of the Euclidean geometry for qutrits ([Formula: see text]) and ququarts ([Formula: see text]). Finally, using a detailed visual analysis of the set of separable and bound entangled Bell diagonal states in both dimensions, qualitative observations are made that allow to better distinguish bound entangled from separable states.
我们比较了具有正部分转置（PPT）的纠缠或可分离的 Bell 对角双量子比特 qudits 的分类及其在不同维度上的性质。对于维度[Formula: see text]，存在一种难以检测的纠缠形式，由于这种纠缠态不能用于纠缠蒸馏，因此称为束缚纠缠。迄今为止，对于如何区分束缚纠缠和可分离态，还没有有效的解决方案。我们针对一类具有特殊代数和几何结构的双 Bell 对角量子比特，并将其应用于不同维度的量子信息处理任务，解决并比较了这个名为分离性问题。扩展了 Bell 对角三体量子比特（[Formula: see text]）的分析和数值方法和结果，我们成功地对[Formula: see text]个代表性的 Bell 对角 PPT 态进行了分类。通过这些代表性的态，我们能够估计 PPT ququarts（[Formula: see text]）中的可分离和束缚纠缠态的体积。我们发现，至少[Formula: see text]的 PPT 态是可分离的，[Formula: see text]的是束缚纠缠的，对于[Formula: see text]，仍然不清楚它们是可分离的还是束缚纠缠的。比较束缚纠缠态及其探测器的结构，我们发现不同维度的检测能力存在相当大的差异，并将其与三体量子比特（[Formula: see text]）和四体量子比特（[Formula: see text]）的欧几里得几何差异联系起来。最后，通过对两个维度的可分离和束缚纠缠 Bell 对角态的集合进行详细的可视化分析，进行了定性观察，从而能够更好地区分束缚纠缠和可分离态。
