We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. Such problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the decision problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problem over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d>3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.

翻译：我们研究了d维数组（即张量）在正交群、酉群与辛群等经典群自然作用下的同构问题的复杂度。此类问题自然出现在统计数据分析与量子信息领域中。我们探讨两类复杂度理论问题：第一，针对固定作用类型（同构、共轭等），将经典群上的同构问题复杂度与一般线性群上的对应问题建立关联；第二，针对固定群类型（正交、酉或辛），比较不同作用决策问题的复杂度。主要结果如下：首先，对于作用于三维数组的正交群与辛群，其同构问题可归约至一般线性群上的对应问题。其次，对于正交群与酉群，三维数组上的五种自然作用同构问题在多项式时间内等价，且对于任意固定d>3，d阶张量同构问题可归约至三阶张量同构问题。对酉群而言，该结论表明：任意d方量子态的LOCC分类难度不低于三方量子态的LOCC分类。最后，我们证明图同构问题可归约至正交群与酉群上的张量同构问题。