Open quantum system

An open quantum system interacts with degrees of freedom treated as an environment. If system and environment jointly have density matrix ${\rho }_{SE}$, the system state is the partial trace ${\rho }_S={\mathrm{Tr}}_E{\rho }_{SE}$. Even unitary joint evolution can change the system's purity because correlations with the environment are discarded.

For example, an interaction mapping $|0⟩|e⟩$ to $|0⟩|e_0⟩$ and $|1⟩|e⟩$ to $|1⟩|e_1⟩$ multiplies the system coherence by $⟨e_1|e_0⟩$. Distinguishable environment states suppress that coherence. A time-local memoryless master equation is an additional approximation, not part of the definition of an open system.

A quantum dynamical semigroup is a continuous family $({\Phi }_t{)}_{t\ge 0}$ of linear maps on density matrices satisfying ${\Phi }_0=I$ and ${\Phi }_{t+s}={\Phi }_t\circ {\Phi }_s$, with each map completely positive and trace preserving. The composition law expresses time-homogeneous evolution with no dependence on how the interval is divided.

In finite dimensions, continuity gives ${\Phi }_t=e^{t\mathcal{L}}$, whose generator has Lindblad equation form. For a qubit dephasing example,Off-diagonal decay factors multiply under composition, while populations stay fixed. The map is the mixture $(1+e^{-\gamma t})\rho /2+(1-e^{-\gamma t})Z\rho Z/2$, which verifies complete positivity. A semigroup need not have physical inverse maps at positive times: undoing this decay would generally amplify coherence beyond valid states.^[ref]