量子力学的同调工具
Homological Tools for the Quantum Mechanic
摘要：
本文介绍了基于“多体纠缠能用同调代数检测吗？“我们引入与多体密度态相关的上链复合物，其上同调检测可因子化性。这种上链复合物的$k$ th上同调分量产生$(k+1)$ -体算子的元组，由于状态的不可因式分解性，这些元组是非局部相关的。相关的庞加莱多项式在局部可逆线性变换（分解为张量积的自同构）下不变。这些复合物可以被认为是实现互信息作为欧拉特征的一个步骤。我们激发的定义的“状态指数”相关联的多体状态：一个三参数的功能，这是：不变的局部可逆变换下，良好的行为下的状态张量积，和多体之间的互信息和整数值的欧拉特征，我们的复合物。我们计算了多体W态和GHZ态的上同调和态指数。本文中的方法是针对有限维量子力学的从业者，虽然开发的机器推广远远超出。某些结果适用于无限维，并应推广到量子场论的背景下。为了弥补篇幅过长，在导言部分提供了详细的摘要。
英文摘要：
This paper is an introduction to work motivated by the question "can multipartite entanglement be detected by homological algebra?" We introduce cochain complexes associated to multipartite density states whose cohomology detects factorizability. The $k$th cohomology components of such cochain complexes produce tuples of $(k+1)$-body operators that are non-locally correlated due to the non-factorizability of the state. Associated Poincare polynomials are invariant under local invertible linear transformations (automorphisms that decompose as tensor products). These complexes can be considered as a step toward realizing mutual information as an Euler characteristic. We motivate the definition of the "state index" associated to a multipartite state: a three-parameter function which is: invariant under local invertible transformations, well-behaved under tensor products of states, and interpolates between multipartite mutual information and the integer-valued Euler characteristics of our complexes. We compute cohomologies and state indices of multipartite W and GHZ states. The approach in this paper is directed toward practitioners of finite-dimensional quantum mechanics, although the machinery developed generalizes far beyond. Some results are applicable in infinite dimensions and should be generalizable to the context of quantum field theory. In order to compensate for the long length, a detailed summary is provided in the introduction section.
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