Extremal Magic States from Symmetric Lattices

Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this talk, I demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the E8, BW16 and E6 lattices into Hilbert space, we construct and classify stabiliser and maximal magic states for two-qubit, three-qubit and one-qutrit systems. In particular, this geometric approach allows us to construct, for the first time, closed-form expressions for the maximal magic states in the three-qubit and one-qutrit systems, and to conjecture their total counts. In the three-qubit case, we further classify the extremal magic states according to their entanglement structure. We also examine the distinctive behaviour of one-qutrit maximal magic states with respect to Clifford orbits. Our findings suggest that deep algebraic and geometric symmetries underlie the structure of extremal magic states.