Quantum Mechanics of Open Systems and Stochastic Maps

The time dependence is completely described by the unitary matrix U(t, t0), or equivalently, by the hermitian Hamiltonian matrix H(t). This type of evolution is natural, but it is not the most general nor adequate for many applications such as relaxation phenomena and irreversibility. Quantum systems can rarely be thought of as being in total isolation. A system which is interacting with its surroundings is called an open system. Boltzmann’s collision formula is a well known example; as is the classical Newton’s Law of Cooling. A quantum mechanical example would be a metastable quantum state that can undergo decay. In experiments it is not possible to completely isolate a quantum system from its environment. For a quantum system, a level of probabilistic description arises in the specification of the density matrix. The equation of motion for the density matrix may be also stochastic, rather than Hamiltonian. This would be the case if the system is ‘open’ with outside influences on its evolution. These systems deal with temporal changes that are not unitary and cannot be evolved with a Hamiltonian scheme. For a bipartite system ρ ⊗ ρ governed by a Hamiltonian H(t), one can calculate the reduced evolution of one part: