Efficient Pauli Truncation from Operator Stabilizer R\'enyi Entropy

arXiv:2611.07545v1 Announce Type: new Abstract: Operator stabilizer R\'enyi entropy (OSE) quantifies the magic of Heisenberg operators and serves as a diagnostic of their compressibility in the Pauli basis. We establish rigorous bounds showing when an operator can be efficiently approximated by Pauli truncation in terms of the scaling of its OSE. Specifically, we prove that volume-law scaling of OSE for $\alpha\geq1$ prevents efficient Pauli truncation from reproducing expectation values for all states at sufficiently small fixed error. On the other hand, logarithmic growth of OSE for $0<\alpha<1/2$ guarantees efficient Pauli truncation approximability uniformly over all states. A counterexample shows that this sufficient range cannot be extended on the basis of OSE scaling alone.