Bridging Entanglement and Magic Resources within Operator Space

Local-operator entanglement (LOE) dictates the complexity of simulating Heisenberg evolution using tensor network methods, {and bears witness to many-body chaos for local dynamics}. We show that LOE is also sensitive to how non-Clifford a unitary is: its magic resources. In particular, we prove that LOE is always upper-bound by three distinct magic monotones: $T$-count, unitary nullity, and operator stabilizer R\'enyi entropy. Moreover, in the average case for large, random circuits, LOE and magic monotones approximately coincide. Our results imply that an operator evolution that is expensive to simulate using tensor network methods must also be inefficient using both stabilizer and Pauli truncation methods. {In terms of a previous conjecture on the characteristic scaling of LOE, our results also mean that non-integrable spin chains cannot be simulated classically}. Entanglement in operator space therefore measures a unified picture of non-classical resources, in stark contrast to the Schr\"odinger picture.