Stabilizer Rényi entropies (SREs) probe the non-stabilizerness (or “magic”) of many-body systems and quantum computers. Here, we introduce the mutual von-Neumann SRE and magic capacity, which can be efficiently computed in time O (N³) O (Nχ3) for matrix product states (MPSs) of bond dimension χ. We find that mutual SRE characterizes the critical point of ground states of the transverse-field Ising model, independently of the chosen local basis. Then, we relate the magic capacity to the anti-flatness of the Pauli spectrum, which quantifies the complexity of computing SREs. The magic capacity characterizes transitions in the ground state of the Heisenberg and Ising model, randomness of Clifford+T circuits, and distinguishes typical and atypical states. Finally, we make progress on numerical techniques: we design two improved Monte-Carlo algorithms to compute the mutual 2 2 -SRE, overcoming limitations of previous approaches based on local update. We also give improved statevector simulation methods for Bell sampling and SREs with O (8^N/2) O (8N/2) time and O (2N) O (2N) memory, which we demonstrate for 24 24 qubits. Our work uncovers improved approaches to study the complexity of quantum many-body systems.
Tarabunga et al. (2025) studied this question.