Fingerprints of classical memory in quantum hysteresis

We present a simple framework for classical and quantum ``memory’’ in which the Hamiltonian at time $t$ depends on past values of a control Hamiltonian through a causal kernel. This structure naturally describes finite-bandwidth or filtered control channels and provides a clean way to distinguish between memory in the control and genuine non-Markovian dynamics of the state. We focus on models where $H(t)=H_0+{\int }_{-\mathrm{\infty }}^{t}K(t-s)H_1(s)ds$, and illustrate the framework on single-qubit examples such as $H(t)={\sigma }_z+\mathrm{\Phi }(t){\sigma }_x$ with $\mathrm{\Phi }(t)={\int }_{-\mathrm{\infty }}^{t}K(t-s)u(s)ds$. We derive basic properties of such dynamics, discuss conditions for unitarity, give an equivalent time-local description for exponential kernels, and show explicitly how hysteresis arises in the response of a driven qubit.