Conceptual illustrations of quantum detailed balance in terms of elementary transitions

<p dir="ltr">These figures illustrate the the process of detailed balance in terms of elementary transitions.</p><p dir="ltr">Figure 1, illustrates the idea of using a composite Hilbert space to describe the time-evolution of a system. The first Hilbert space represents the original system at the current moment in time. The second copy in the tensor product represents the system at the next moment in time.</p><p dir="ltr">Figure 2, illustrates the use of pure states kappa_(i_,j₎ on the composite system as elementary transitions in the classical case. Here kappa_(i_,j) represents the transition from the pure state i to the pure state j. The term kappa_(j,i) represents the reverse elementary transition from j to i_{<em> </em>}<i>.</i></p><p dir="ltr">Figure 3, illustrates that for a system satisfying elementary transition detailed balance (ETDB), each elementary transition kappa_alpha will necessarily have a corresponding reverse elementary transition R(kappa_alpha). Likewise, the term kappa_beta = R(kappa_alpha) will have a corresponding reverse, namely R(kappa_beta) = kappa_alpha. This takes into account that the operator R is its own inverse.</p>