Entanglement entropy scaling laws from fluctuations of nonconserved quantities

Entanglement patterns reveal essential information on many-body states and provide a way to classify quantum phases of matter. However, experimental studies of many-body entanglement remain scarce due to their unscalable nature. The present Letter aims to mitigate this theoretical and experimental divide by introducing reduced fluctuations of observables, consisting of a sum of on-site operators, as a scalable experimental probe of the entanglement entropy. Specifically, we illustrate by density matrix renormalization group calculations in spin chains that the reduced fluctuations exhibit the same size scaling properties as the entanglement entropy. Generalizing previous observations restricted to special systems with conserved quantities, our work introduces an experimentally feasible protocol to extract entanglement scaling laws.