Classification of five-qubit absolutely maximally entangled states

We classify the local unitary equivalence classes of absolutely maximally entangled (AME) states of five qubits. We show that every 5-qubit AME state is equivalent to a state within the unique ((5,2,3)) quantum error-correcting code $\mathcal{C}$, and that two such states are equivalent if and only if they are related by the action of a transversal gate of $\mathcal{C}$. Furthermore, we exhibit a set of three invariant polynomials that separates these equivalence classes. As auxiliary results, we construct a 3-uniform $n$-qubit state for even $n\geq 6$, determine the local symmetries of the 6-qubit AME state, and explain how these symmetries are related to the transversal gates of both the ((5,2,3)) and ((4,4,2)) codes. Additionally, we demonstrate that every 4-qubit pure code of distance 2 is equivalent to a subspace of a ((4,4,2)) code. Our approach leverages an embedding of the 4-qubit state space into the Lie algebra of $8\times 8$ skew-symmetric matrices, allowing us to apply results from Vinberg's theory of graded Lie algebras.