Quantum Scrambling Mereology under Long-Time Dynamics

The recent work of Zanardi et al. associated each possible partition of a quantum system with an operational subalgebra and proposed that the short-time growth of the algebraic out-of-time-correlator ("A-OTOC") is a suitable criterion to determine which partition arises naturally from a system's unitary dynamics. We extend this work to the long-time regime. Specifically, we consider the long-time average of the A-OTOC ("A-OTOC LTA") as our metric of subsystem emergence; under this framework, natural system partitions are characterized by the tendency to, on average, minimally scramble information over long time scales. We derive an analytic expression for the A-OTOC LTA under the non-resonance condition (NRC). This is then applied to several examples in which we perform minimization of the A-OTOC LTA both analytically and numerically over relevant families of algebras. For simple cases subject to the NRC, minimal A-OTOC LTA is shown to be related to minimal entanglement of the Hamiltonian eigenstates over the emergent system partition. Finally, we conjecture and provide evidence for a general structure of the algebra which minimizes the LTA for a given NRC Hamiltonian.