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 the 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 the separability for much larger-sized Hermitian tensors. This method is a good starting point to determine psd ranks of separable Hermitian tensors.

Hermitian 张量是 Hermitian 矩阵的自然推广，同时具有相当不同的性质。如果 Hermitian 张量具有仅具有正系数的 Hermitian 分解，则它是可分离的，即它是 rank-1 psd Hermitian 张量的总和。本文研究了如何检测厄米张量的可分离性。它相当于量子物理学中长期存在的量子可分离性问题，它要求判断给定的量子态是否纠缠。我们将其表述为截断矩问题，然后提供半定松弛算法来解决它。此外，我们研究了可分离厄米张量的 psd 分解。当 psd 等级较低时，我们首先将它们展平为三次阶张量，然后应用张量分解方法来计算 psd 分解。我们证明，如果 psd 等级较低，此方法效果很好。在计算中，这种扁平化方法可以检测更大尺寸的 Hermitian 张量的可分离性。此方法是确定可分离厄米张量的 psd 秩的良好起点。