Fisher-Hartwig Formula and XY Spin Chain

该奖项将支持对自旋模型中的纠缠的研究。在XY自旋链中，用块Toeplitz行列式表示几个自旋块的熵，然后用推广的Fisher-Hartwig公式来计算渐近性。该项目将同时考虑von Neumann和Renyi熵。还将对纠缠光谱进行评估。这个问题最近引起了人们的注意，需要进行严格的数学处理。在自旋模型的基态中，两个自旋块的纠缠是很重要的。在跟踪基态的其余部分之后，这两个块处于混合状态。衡量混合系统纠缠的一个合适的方法是负性。该项目将描述AKLT(阿弗莱克，利布，肯尼迪·塔卡西)自旋链中两个自旋块通过负性的纠缠。AKLT模型的基态称为价键固体(VBS)态。纠缠是量子系统中的一种特殊现象，在经典(宏观)情况下是不可能发生的。它是量子控制的重要资源，量子控制是包括量子计算机在内的量子设备的核心。这一点最近变得很重要，因为正在研究光学晶格的实验者现在可以建立相互作用的自旋模型，而这之前只从数学的角度进行研究。这是数学物理对量子信息处理目标的重大贡献，实验室的进展现在取决于进一步的理论研究。该奖项将支持对这种纠缠的数学分析，以指导原子-分子-光学实验。

This award will support a study of entanglement in spin models. In XY spin chains, the entropy of several blocks of spins will be represented as a block Toeplitz determinant, and then the generalized Fisher - Hartwig formula will be used to evaluate asymptotic. The project will consider both von Neumann and Renyi entropies. The entanglement spectrum also will be evaluated. This question has attracted attention recently and requires rigorous mathematical treatment. Entanglement of two blocks of spins in ground states of spin models is important. After tracing the rest of the ground state, these two blocks are left in a mixed state. An appropriate measure of entanglement of mixed systems is negativity. The project will describe entanglement of 2 blocks of spins in the AKLT (Affleck, Lieb, Kennedy Takasi) spin chain by means of negativity. The ground state of the AKLT model is known as valence-bond-solid (VBS) state. It plays an important role in quantum information, especially in measurement based quantum computation.Entanglement is a special phenomenon of quantum systems, which cannot happen in the classical (macroscopic) case. It is an important resource for quantum control, which is central for quantum devices, including quantum computers. This has recently become important, because experimentalist who are working with optical lattices can now build models of interacting spins, which had only previously been studied from the mathematical point of view. This is a substantial contribution of mathematical physics towards the goal of quantum information processing, and progress in the laboratory now depends on further theoretical study. The award will support the mathematical analysis of such entanglement, in order to guide atom-molecular-optics experiments.