Quantum Non-Local Nonstabilizerness

Quantum entanglement and quantum nonstabilizerness are fundamental resources that characterize distinct aspects of a quantum state: entanglement reflects non-local correlations, while nonstabilizerness quantifies the deviation from stabilizer states. A quantum state becomes a valuable resource for applications like universal quantum computation only when both quantities are present. Here, we propose that quantum non-local nonstabilizerness (NN) serves as an effective measure of this combined resource, incorporating both entanglement and nonstabilizerness. We demonstrate that NN can be precisely computed for two-qubit pure states, where it is directly related to the entanglement spectrum. We then extend the definition of NN to mixed states and explore its presence in many-body quantum systems, revealing that the two-point NN decays according to a power law in critical states. Furthermore, we explore measurement-induced NN and uncover an intriguing phenomenon termed "nonstabilizerness swapping", analogous to entanglement swapping, wherein post-measurement NN decays more slowly than any pre-measurement correlations. Our results thus represent a pivotal step towards accurately quantifying the "quantumness" of a state and reveal the potential for manipulating this resource through measurements.