Long-time Quantum Scrambling and Generalized Tensor Product Structures

Much recent work has been devoted to the study of information scrambling in quantum systems. In this paper, we study the long-time properties of the algebraic out-of-time-order-correlator ("$\mathcal{A}$-OTOC") and derive an analytical expression for its long-time average under the non-resonance condition. The $\mathcal{A}$-OTOC quantifies quantum scrambling with respect to degrees of freedom described by an operator subalgebra $\mathcal{A}$, which is associated with a partitioning of the corresponding system into a generalized tensor product structure. Recently, the short-time growth of the $\mathcal{A}$-OTOC was proposed as a criterion to determine which partition arises naturally from the system's unitary dynamics. In this paper, we extend this program to the long-time regime where the long-time average of the $\mathcal{A}$-OTOC serves as the metric of subsystem emergence. Under this framework, natural system partitions are characterized by the tendency to minimally scramble information over long time scales. We consider several physical examples, ranging from quantum many-body systems and stabilizer codes to quantum reference frames, and perform the minimization of the $\mathcal{A}$-OTOC long-time average both analytically and numerically over relevant families of algebras. For simple cases subject to the non-resonant condition, minimal $\mathcal{A}$-OTOC long-time average is shown to be related to minimal entanglement of the Hamiltonian eigenstates across the emergent system partition. Finally, we conjecture and provide evidence for a general structure of the algebra that minimizes the average for non-resonant Hamiltonians.