Extremal Magic States from Symmetric Lattices

TL;DR

This paper reveals a geometric connection between symmetric lattices and quantum magic states, enabling new classifications and explicit formulas for extremal states in multi-qubit and qutrit systems, highlighting underlying symmetries.

Contribution

It introduces a novel geometric approach linking symmetric lattices to quantum magic states, providing explicit formulas and classifications for extremal states in complex quantum systems.

Findings

Constructed maximal magic states from $E_8$, $BW_{16}$, and $E_6$ lattices.

Classified extremal magic states by entanglement and Clifford orbits.

Provided closed-form expressions and conjectures for state counts.

Abstract

Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this paper, we demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the $E_8$, $BW_{16}$, and $E_6$ 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.