Quantum detailed balance via elementary transitions

Abstract

Quantum detailed balance is formulated in terms of elementary transitions, in close analogy to detailed balance in a classical Markov chain on a finite set of points. An elementary transition is taken to be a pure state of two copies of the quantum system, as a quantum analogue of an ordered pair of classical points representing a classical transition from the first to the second point. This form of quantum detailed balance is shown to be equivalent to standard quantum detailed balance with respect to a reversing operation, thus providing a new conceptual foundation for the latter. Aspects of parity in quantum detailed balance are clarified in the process. The connection with the Accardi-Cecchini dual and the KMS dual (or Petz recovery map) is also elucidated.

POPULAR SUMMARY : A simple form of equilibrium for a process in a classical system is the situation where, at any time, the probability of a transition from any state of the system to another during this process, is the same as the probability for the opposite transition. This needs to be true for all pairs of states of the system. This equilibrium is called detailed balance. In such a classical setup, the available states are fixed. In a quantum system, however, the relevant states can conceivably depend on the observable being considered. Should one focus on a specific set of states, say energy states? Or be more general? Can one express the equality of probabilities in the two directions in a way directly analogous to the classical case? Or are we forced to approach the problem in some indirect way? Our strategy is simply to focus on the transitions, classically the pairs of states, rather than the states themselves, to open a clear path to the quantum case.