Stability of mixed-state phases under weak decoherence

In equilibrium statistical mechanics, phases are parameter regions in which the free energy evolves smoothly. This equilibrium perspective is well suited to to conventional solid-state experiments, in which the system of interest (e.g., the electron fluid in a metal) is well coupled to a heat bath. Present-day experiments in quantum devices necessitate a more general concept of phases: in these devices, systems are driven far from thermal equilibrium and are either isolated from the environment or coupled to engineered dissipative environments. A key step toward this general concept came from the development of quasi-adiabatic continuation [1], for pure quantum states at zero temperature. According to this definition, phases are equivalence classes of quantum states such that two states in the same phase can be prepared from one another by an efficient process--specifically, a finite-depth local unitary (FDLU) circuit. This concept of pure-state phases (called FDLU-equivalence) reduces to the conventional one for ground states of gapped local Hamiltonians, but extends to any quantum state, and connects naturally to questions in computational complexity thoery [2-5]. So far, this "preparability" perspective is only fully developed for pure quantum states and strictly unitary evolutions; thus, a natural task, which has seen intense recent activity, is its generalization to more general classes of mixed states and evolutions involving noise, measurement, and feedback [6-14]. As an important special case, a classification of mixed states from the perspective of preparability would naturally extend to general classical probability distributions, of the type that routinely arise in machine learning, and that also seem to exhibit phase transitions [15].