Entanglement governs early-time growth of randomness in projected ensembles

While quantum thermalization explains how isolated systems reach equilibrium at late times, the mechanisms governing the initial emergence of randomness in projected ensembles—where a subsystem is observed through measurements on its complement—remain poorly understood. This paper investigates the early-time dynamics of these ensembles, specifically asking how the growth of local randomness relates to bipartite entanglement and global spectral properties.

To quantify randomness, the authors use frame potentials, which measure the statistical distance between a projected ensemble and the Haar ensemble (the distribution of uniform random states). They derive analytical expressions for these potentials under arbitrary Hamiltonian evolution, assuming the initial state is a product state. They further specialize these results to unitarily invariant Hamiltonian ensembles, such as the Gaussian Unitary Ensemble (GUE), to connect local projected-ensemble dynamics with global spectral correlations via the spectral form factor.

• Entanglement-Randomness Relation: For any Hamiltonian, the early-time growth of the kth frame potential is determined entirely by the subsystem purity (and thus the bipartite entanglement) up to cubic order in time. This implies that the entanglement timescale sets the initial rate of randomness growth, independent of the measurement basis.
• Basis Independence: Because entanglement is basis-independent, the specific choice of measurement basis only influences the randomness growth at quartic order and higher, provided the initial state has full support in that basis.
• 2k to k Correspondence: For unitarily invariant ensembles, the authors establish a direct quantitative link between the kth frame potential of the projected ensemble and the 4k-point spectral form factor of the global system. This demonstrates that the global unitary dynamics at order 2k control the local randomness at order k.