Energy-Filtered Quantum States and the Emergence of Nonlocal Correlations.

Energy-filtered quantum states are promising candidates for efficiently simulating thermal states. We explore a protocol designed to transition a product state into an eigenstate located in the middle of the spectrum; this is achieved by gradually reducing its energy variance, which allows us to comprehensively understand the crossover phenomenon and the subsequent convergence toward thermal behavior. We introduce and discuss three energy-filtering regimes (short, medium, and long), and we interpret them as stages of thermalization.