Krebs Robin, Gachechiladze Mariami
Department of Computer Science, Technical University of Darmstadt, Germany.
Phys Rev Lett. 2024 May 31;132(22):220203. doi: 10.1103/PhysRevLett.132.220203.
A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
深入理解量子纠缠对于推进量子技术至关重要。纠缠强度可以通过计算纠缠的自由度数量来量化，这会得出一个称为施密特数的量。一个特殊的挑战是确定在部分转置下仍为正定的量子态（PPT）中的纠缠强度，这类态也被视为不可提纯态。找到具有高施密特数的PPT态已成为一个数学和计算上的挑战。在本信函中，我们引入了有效的分析工具来计算一类称为网格态的二分态的施密特数。我们的方法改进了具有高施密特数的PPT态的已知最佳界限。最值得注意的是，我们在五维系统中构造了一个施密特数为3的PPT态，并为奇数维d的系统构造了一族施密特数为(d + 1)/2的态，这代表了在局部维度中施密特数的已知最佳标度。此外，这些态具有有趣的几何性质，我们利用这些性质构造了不可分解的纠缠见证者。
