Equivalence of 2-tensors and 3-tensors under Local Unitary and General Linear Groups

In quantum information, pure states correspond to tensors. Two states equivalent under local unitary group or local general linear group can be used for the same quantum-information tasks, and it is thus an essential problem to determine whether two tensors are equivalent. In this paper, we determine the two sorts of equivalence for 2-tensors in terms of the Schmidt decomposition and singular value decomposition of matrices. We also extend the equivalence to some 3-tensors, proving that Schmidt decomposition is unachievable for 3-tensors. However, we present a method to express 3-tensors using the logic behind Schmidt decomposition. Furthermore, we generalized the scenario to higher dimensions and discussed the characteristic behaviors of different groups in tensor transformations.