Locality of deep thermalisation through the lens of entanglement teleportation

This paper investigates the nature of locality in the phenomenon of deep thermalisation. Specifically, it asks whether a subsystem partitioned into two disjoint, causally disconnected regions can undergo deep thermalisation—a process where projective measurements on the environment force the subsystem into a universal state ensemble—and how the spatial separation of these regions affects this process.

To address this, the authors employ a hybrid framework combining unitary dynamics and projective measurements. They define a setup where two subregions, R1 and R2, are separated by a bath E. Because the unitary evolution is local, R1 and R2 remain causally disconnected. The researchers analyze the entanglement teleportation between these regions as a proxy for the onset of deep thermalisation. They utilize both numerical simulations of brickwork circuits and analytical modeling via random tensor networks (RTNs) to determine how the entanglement and thermalisation timescales depend on the distance between the subregions.

The study establishes a formal inequality showing that the timescales for deep thermalisation are bounded by the timescales of entanglement teleportation. In generic locally interacting systems, the researchers find that both deep thermalisation and entanglement teleportation exhibit an emergent locality, with timescales that grow logarithmically with the distance between the subregions. This logarithmic scaling is confirmed through numerical simulations and further supported by the RTN model, which demonstrates that for finite entanglement to develop between the disconnected regions, the circuit depth must scale with the system size. The paper identifies that genuine non-locality only occurs in exceptional circuits where measurement randomness is transmitted perfectly.

This work provides a rigorous link between quantum information theory and the dynamics of thermalisation. By demonstrating that deep thermalisation is not inherently non-local but rather constrained by the speed of information propagation (teleportation) through the environment, the authors clarify the limits of how quantum correlations can be established in measurement-induced dynamics. This has significant implications for understanding how quantum systems reach equilibrium and the role of locality in hybrid quantum circuits.