Quantum detailed balance and duality

Abstract

We develop quantum detailed balance on a finite-dimensional system in terms of elementary transitions, which are pure states on the composite space consisting of two copies of the system. This is in close analogy to detailed balance in classical mechanics described in terms of Markov chains. We show that this version of detailed balance may also be expressed in terms of quantum channels. acting on the states of the system by the Choi-Jamiolkowski duality. We then develop a more general setting of states and channels in a von Neumann algebraic framework and embed our theory of finite-dimensional elementary transition detailed balance into this framework. Lastly, we show that our version of quantum detailed balance is equivalent to the well-known standard quantum detailed balance with respect to a reversing operation in finite dimensions.