Classifying Density Matrices of 2 and 3 Qubit States Up To LU Equivalence

Abstract: In this paper we present a modified version of the proof given Jing-Yang-Zhao's paper titled "Local Unitary Equivalence of Quantum States and Simultaneous Orthogonal Equivalence," which established the correspondance between local unitary equivalence and simultaneous orthogonal equivalence of $2$-qubits. Our modified proof utilizes a hypermatrix algebra framework, and through this framework we are able to generalize this correspondence to $3$-qubits. Finally, we apply a generalization of Specht's criterion (first proved in "Specht's Criterion for Systems of Linear Mappings" by V. Futorney, R. A. Horn, and V. V. Sergeichuk) to reduce the problem of local unitary equivalence of $3$-qubits to checking trace identities and a few other easy-to-check properties. We also note that all of these results can be extended to $2$ and $3$ qudits if we relax the notion of LU equivalence to quasi-LU equivalence, as defined in the aforementioned paper by Jing et. al.