Such stuff as magic is made on: compact operator algebra, stabilizer polytope and the structure of reduced density matrices in a Kitaev spin liquid

Quantum many-body systems are often characterized by entanglement, but magic (or non-stabilizerness) provides a complementary resource essential for universal quantum computation and a diagnostic for quantum criticality. This study investigates the structure of magic in local subsystems of the Kitaev honeycomb model at finite temperature, asking whether the onset of local magic reveals a physically meaningful, compact operator structure underlying the many-body physics.

Using the robustness of magic (RoM) as a diagnostic, the authors analyze the local reduced density matrices (rdms) of three, four, and six-site clusters in the Kitaev honeycomb model. By identifying the optimal magic witnesses at the temperature where magic first emerges, they construct a low-dimensional operator space that spans these witnesses. They then demonstrate that these operator spaces form finite-dimensional Euclidean Jordan algebras, which are naturally adapted to the symmetries of the Kitaev model.

The researchers find that the optimal magic witnesses are not arbitrary combinations of Pauli operators but are instead organized into compact, symmetry-adapted operator spaces. For the six-site hexagonal marginal, this space is defined by the symmetry-resolved plaquette algebra. Remarkably, these operator spaces are not only useful at the onset of magic but provide an extremely high-fidelity representation of the local thermal states across the entire temperature range. Furthermore, the authors prove that when the system's symmetries are realized exactly, the robustness of magic computed within this reduced operator space is identical to the full robustness of magic. This methodology also successfully captures other quantum resources, such as genuine multipartite entanglement, using the same compact description.

This work provides a powerful framework for reducing the complexity of characterizing quantum resources in many-body systems. By showing that the onset of magic signals the emergence of a physically meaningful, low-dimensional operator algebra, the authors offer a path to bypass the exponential costs typically associated with full state reconstruction. This approach is particularly relevant for modern quantum simulators, such as neutral-atom arrays, where local plaquette observables can be measured directly to probe the underlying many-body structure.