We show how local bounded interactions in an unbounded Hamiltonian lead to eigenfunctions with favorable low-rank properties. To this end, we utilize ideas from quantum entanglement of multi-particle spin systems. We begin by analyzing the connection between entropy area laws and low-rank approximability. The characterization for 1D chains such as Matrix Product States (MPS) / Tensor Trains (TT) is rather extensive though incomplete. We then show that a Nearest Neighbor Interaction (NNI) Hamiltonian has eigenfunctions that are approximately separable in a certain sense. Under a further assumption on the approximand, we show that this implies a constant entropy bound. To the best of our knowledge, this work is the first analysis of low-rank approximability for unbounded Hamiltonians. Moreover, it extends previous results on entanglement entropy area laws to unbounded operators. The assumptions include a variety of self-adjoint operators and have a physical interpretation. The weak points are the aforementioned assumption on the approximand and that the validity is limited to MPS/TT formats.
翻译:我们展示了在无约束的汉密尔顿山中,局部的封闭性相互作用如何导致偏好的低质属性的偏差。 为此,我们利用了多粒子旋转系统数量缠绕的量子缠绕思想。 我们首先分析了英特罗比区域法和低级别相近性之间的联系。 我们最先分析的是, 1D 链条, 如母体产品国(MPS) / Tonsor train (TTT) 的特性相当广泛, 虽然不完整。 然后我们又显示,近邻界面互动(NNI) 汉密尔顿丹(NNI) 的功能有某种可能在某些意义上可以分离的偏差。 根据对近似点的进一步假设, 我们显示这意味着一个恒定的酶捆绑。 据我们所知, 这项工作是对未受约束的汉密尔密尔顿人低级别相近的相近似性分析。 此外, 它将先前关于缠绕性昆虫区域法的结果推广到未受约束的操作者。 假设包括各种自相连接的操作者, 并有物理解释。 弱点是上述假设的安特/ 格式。