2020/11/16 by Mareike Dressler, Dressler, Mareike, Jiawang Nie +3 · 2 citations
Computer Science · Mathematics · #FOS: Mathematics #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Quantum Computing Algorithms and Architecture #Tensor decomposition and applications #math.OC
paper · pdf · doi:10.48550/arxiv.2011.08132
26 pages
openalex publication_date 2020/11/16 · arxiv created 2021/08/09 · arxiv updated 2021/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Hermitian tensors are natural generalizations of Hermitian matrices, while possessing rather different properties. A Hermitian tensor is separable if it has a Hermitian decomposition with only positive coefficients, i.e., it is a sum of rank-1 psd Hermitian tensors. This paper studies how to detect separability of Hermitian tensors. It is equivalent to the long-standing quantum separability problem in quantum physics, which asks to tell if a given quantum state is entangled or not. We formulate this as a truncated moment problem and then provide a semidefinite relaxation algorithm to solve it. Moreover, we study psd decompositions of separable Hermitian tensors. When the psd rank is low, we first flatten them into cubic order tensors and then apply tensor decomposition methods to compute psd decompositions. We prove that this method works well if the psd rank is low. In computation, this flattening approach can detect separability for much larger sized Hermitian tensors. This method is a good start on determining psd ranks of separable Hermitian tensors.