Uniqueness of SU(3) under Operational Distinguishability Constraints

We prove a classification-level result for compact connected Lie groups acting on C³. Under three independently motivated assumptions — a determinant constraint, the presence of a cyclic 3-symmetry, and operational distinguishability of chirally conjugate states — the only compact connected Lie group admitting a faithful irreducible unitary representation on C³ is SU (3). The operational distinguishability assumption is formulated via an Information Preservation Principle (IPP): states related by an antiunitary intertwiner must remain distinguishable within the physically admissible observable algebra, as measured by von Neumann relative entropy. We show that the IPP forces the representation to be complex (V not isomorphic to V-bar), eliminating SU (2) (whose 3-dimensional adjoint is real) and leaving SU (3) as the unique solution. The proof proceeds via the Peter-Weyl structure theorem, the Weyl dimension formula, and a center-quotient argument. A minimality corollary characterizes SU (3) as the minimal compact Lie group supporting a faithful irreducible unitary action on C³ together with a cyclic permutation symmetry and a genuinely complex representation.