Rank-1 Approximation for Entangled Multipartite Real Systems

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.

Automated summary

The paper presents a study on the approximation problem of a multipartite system with entangled symmetric rank-1 matrices via the Kronecker tensor product. The involvement of the Kronecker product makes the problem more challenging compared to conventional low-rank tensor approximations, but the paper proposes effective numerical iterative methods to tackle the problem.