Rank-1 Approximation for Entangled Multipartite Real Systems
Rank-1 Approximation for Entangled Multipartite Real Systems
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DOI：
10.1007/s10915-022-01805-y
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发表时间：
2022-03
影响因子：
2.5
通讯作者：
Matthew M. Lin;M. Chu
中科院分区：
文献类型：
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作者：
Matthew M. Lin;M. Chu
在一个系统中，多个部分按照一定的内在规则相互作用，是一种至关重要的自然现象。高维数组的纠缠及其分解的概念特别耐人寻味，因为它为数据处理和通信开辟了一种新的思维方式，其应用将是广泛和重要的。根据内部部分相互啮合的方式，存在具有不同特征的不同类型的纠缠。本文讨论了一个多部系统的近似，该系统的子系统由对称的秩1矩阵组成，这些矩阵通过Kronecker张量积纠缠。这种结构类似于量子力学中出现的结构，在量子力学中，混合态将由其最近的可分离态来近似，只是本文的讨论仅限于实值矩阵。与传统的低阶张量近似不同，由于Kronecker积的参与而增加的扭曲破坏了多线性，这使得问题变得更加困难。作为第一步，本文仅对秩1多部近似进行了探讨。将该问题转化为一个非线性特征值问题和一个非线性奇异值问题，分别用类幂迭代法和类SVD迭代法进行数值求解。这两类方法中的迭代都可以循环或非循环地实现。本文讨论了其动机、方案和收敛理论。初步的数值实验表明，与一些通用优化程序相比，这些方法是有效和高效的。
The interaction of multiple parts with each other within a system according to certain intrinsic rules is a crucial natural phenomenon. The notion of entanglement and its decomposition of high-dimensional arrays is particularly intriguing since it opens a new way of thinking in data processing and communication, of which the applications will be broad and significant. Depending on how the internal parts engage with each other, there are different types of entanglements with distinct characteristics. This paper concerns the approximation over a multipartite system whose subsystems consist of symmetric rank-1 matrices that are entangled via the Kronecker tensor product. Such a structure resembles that arising in quantum mechanics where a mixed state is to be approximated by its nearest separable state, except that the discussion in this paper is limited to real-valued matrices. Unlike the conventional low-rank tensor approximations, the added twist due to the involvement of the Kronecker product destroys the multi-linearity, which makes the problem harder. As a first step, this paper explores the rank-1 multipartite approximation only. Reformulated as a nonlinear eigenvalue problem and a nonlinear singular value problem, respectively, the problem can be tackled numerically by power-like iterative methods and SVD-like iterative methods. The iteration in both classes of methods can be implemented cyclically or acyclically. Motivations, schemes, and convergence theory are discussed in this paper. Preliminary numerical experiments suggest these methods are effective and efficient when compared with some general-purpose optimization packages.