Classification of orthogonal arrays under local unitary equivalence

Determine precisely when two orthogonal arrays, viewed through their associated normalized positive equal-amplitude quantum states with distinct rows, are equivalent under local unitary transformations, beyond the Fourier-based equivalence constructions established for specific families.

Background

Orthogonal arrays with distinct rows define multipartite quantum states, and local unitary equivalence of those states induces an equivalence relation on the arrays. The paper develops sufficient conditions for Fourier-based local unitary equivalence, particularly for arrays generated by matrices over residue rings and for linear arrays associated with codes and their duals.

The general classification problem is not resolved by these constructions. The paper explicitly characterizes its results as partial solutions and notes that the constructions identify particular Fourier partners rather than all arrays related by arbitrary local unitary transformations. Thus, a complete criterion or classification for local unitary equivalence of orthogonal arrays remains open.