不变量理论的张量网络方法
Tensor Network Methods for Invariant Theory
摘要：
不变理论关注的是在给定群的作用下不发生变化的函数。在这里，我们交流的方法的基础上张量网络表示多项式局部酉不变量的量子态。这种图形方法为描述不变量的多项式方程提供了一种替代方法，这些方程通常包含大量系数提高到高幂的项。这种方法还使人们能够在研究多项式不变量时使用张量网络理论的已知方法（例如矩阵乘积状态分解）。作为我们的主要例子，我们考虑矩阵乘积状态的不变量。我们生成一个家庭的张量收缩，从而在一个完整的局部酉不变量，可以用来表达雷尼熵。我们发现，图形化的方法来表示不变量可以提供结构上的洞察收缩的不变量，以及一种替代的，有时更简单，手段来研究多项式不变量的量子态。此外，许多张量网络方法，如矩阵乘积状态，包含可以应用于不变量研究的优秀工具。
英文摘要：
Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state factorization) when studying polynomial invariants. As our main example, we consider invariants of matrix product states. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the Rényi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as matrix product states, contain excellent tools that can be applied in the study of invariants.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
