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We study the complexity of isomorphism problems for d-way arrays, or tensors,under natural actions by classical groups such as orthogonal, unitary, andsymplectic groups. Such problems arise naturally in statistical data analysisand quantum information. We study two types of complexity-theoretic questions.First, for a fixed action type (isomorphism, conjugacy, etc.), we relate thecomplexity of the isomorphism problem over a classical group to that over thegeneral linear group. Second, for a fixed group type (orthogonal, unitary, orsymplectic), we compare the complexity of the decision problems for differentactions.  Our main results are as follows. First, for orthogonal and symplectic groupsacting on 3-way arrays, the isomorphism problems reduce to the correspondingproblem over the general linear group. Second, for orthogonal and unitarygroups, the isomorphism problems of five natural actions on 3-way arrays arepolynomial-time equivalent, and the d-tensor isomorphism problem reduces to the3-tensor isomorphism problem for any fixed d>3. For unitary groups, thepreceding result implies that LOCC classification of tripartite quantum statesis at least as difficult as LOCC classification of d-partite quantum states forany d. Lastly, we also show that the graph isomorphism problem reduces to thetensor isomorphism problem over orthogonal and unitary groups.

On the complexity of isomorphism problems for tensors, groups, and polynomials III: actions by classical groups