$2 \times N$网格张量网络态的几何性质
On the geometry of Tensor Network States of $2 \times N$ Grids
摘要：
我们讨论了一类张量网络态的几何结构，在物理学文献中称为投影纠缠对态。我们提供了初步的结果对一个问题的Verstraete和Rizzi关于张量网络状态的$M \times N$网格，我们部分回答了这个问题的$2 \times N$网格。我们还研究了$2 \times N$网格坐在一个环面，并提供初步的结果，了解Zalberki封闭的一组张量网络状态与此图。最后，我们给出了明确的张量，提供最佳的（使用目前可用的方法）边界上的通用张量在张量网络状态的$2 \times N$网格的秩的界限。
英文摘要：
We discuss the geometry of a class of tensor network states, called projected entangled pair states in the Physics literature. We provide initial results towards a question of Verstraete and Rizzi regarding the tensor network state of an $M \times N$ grid; we partially answer the question for a $2 \times N$ grid. We also study the $2 \times N$ grid sitting on a torus and provide initial results towards understanding the Zariski closure of the set of tensor network states associated to this graph. Finally, we give explicit tensors that provide optimal (using currently available methods) bounds on the border rank of a generic tensor in the tensor network state of the $2 \times N$ grid.
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