A subsystem interacting with an environment is described by a reduced density operator $\rho_S=\mathrm{Tr}_E(\rho_{SE})$. This generally leads to nonunitary dynamics, even if the composite system evolves unitarily.

Markovian evolution is often modeled by $$\frac{d\rho}{dt}=-\frac{i}{\hbar}[H,\rho]+\sum_k\left(L_k\rho L_k^\dagger-\frac12\{L_k^\dagger L_k,\rho\}\right).$$ This is the most general time-homogeneous generator of a completely positive trace-preserving semigroup.

Decoherence explains why certain pointer states are robust under monitoring by the environment. It does not by itself solve the measurement problem, but it clarifies the emergence of classical-looking alternatives.

Continuous monitoring can unravel a master equation into stochastic pure-state trajectories. This perspective is widely used in quantum optics, superconducting circuits, and feedback control.

When environmental memory matters, the dynamics can become non-Markovian, with information flowing back from environment to system. Such behavior is important in solid-state devices, structured reservoirs, and biological quantum effects.