Hermite张量的可分性与PSD分解
Separability of Hermitian Tensors and PSD Decompositions
摘要：
厄米特张量是埃尔米特矩阵的自然推广，但具有相当不同的性质。一个厄米特张量是可分的，如果它有一个只有正系数的厄米特分解，即，它是秩-1 PSD厄米特张量之和。本文研究了厄米张量可分性的判别问题。它相当于量子物理学中长期存在的量子可分性问题，即要求判断给定的量子态是否纠缠。我们将其表示为截断矩问题，并给出了一个半定松弛算法来求解。此外，我们还研究了可分厄米张量的PSD分解。当psd秩较低时，我们先将其展平为三阶张量，然后应用张量分解方法计算psd分解。我们证明了这种方法在psd秩较低的情况下效果很好。在计算中，这种平坦化方法可以检测更大尺寸的厄米张量的可分性。该方法是确定可分厄米张量psd秩的一个良好开端。
英文摘要：
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.
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