论文标题
Hermitian张量和PSD分解的可分离性
Separability of Hermitian Tensors and PSD Decompositions
论文作者
论文摘要
Hermitian张量是对遗传学矩阵的自然概括，同时具有相当不同的特性。如果Hermitian张量只有一个正系系数的Hermitian分解，即它是Rank-1 PSD Hermitian Tensors的总和。本文研究了如何检测Hermitian张量的可分离性。它等同于量子物理学中长期存在的量子可分离性问题，该物理学要求确定给定的量子状态是否纠缠。我们将其作为截断的力矩问题提出，然后提供半芬矿弛豫算法来解决它。此外，我们研究了可分离的Hermitian张量的PSD分解。当PSD级别较低时，我们首先将它们变成立方顺序张量，然后将张量分解方法应用于计算PSD分解。我们证明，如果PSD等级较低，则该方法效果很好。在计算中，这种扁平化的方法可以检测到更大尺寸的Hermitian张量的可分离性。这种方法是确定可分离的赫尔米亚张张量的PSD等级的一个良好开始。
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.
