Coarse-grained dynamics in quantum many-body systems using the maximum entropy principle

Starting from a coarse-grained map of a quantum many-body system, we construct the inverse map that assigns a microscopic state to a coarse-grained state based on the maximum entropy principle. Assuming unitary evolution in the microscopic system, we examine the resulting dynamics in the coarse-grained system using the assignment map. We investigate both a two-qubit system, with \swap\ and controlled-\textsc{not}\ gates, and $n$-qubit systems, configured either in an Ising spin chain or with all-to-all interactions. We demonstrate that these dynamics exhibit atypical quantum behavior, such as nonlinearity and non-Markovianity. Furthermore, we find that these dynamics depend on the initial coarse-grained state and establish conditions for general microscopic dynamics under which linearity is preserved. As the effective dynamics induced by our coarse-grained description of many-body quantum systems diverge from conventional quantum behavior, we anticipate that this approach could aid in describing the quantum-to-classical transition and provide deeper insights into the effects of coarse graining on quantum systems.