A Cycle Decomposition and Entropy Production for Circulant Quantum Markov Semigroups

Abstract

We propose a definition of cycle representation for Quantum Markov Semigroups (qms) and Quantum Entropy Production Rate (QEPR) in terms of the ρ-adjoint. We introduce the class of circulant qms, which admit non-equilibrium steady states but exhibit symmetries that allow us to compute explicitly the QEPR, gain a deeper insight into the notion of cycle decomposition and prove that quantum detailed balance holds if and only if the QEPR equals zero.

Structure of uniformly continuous quantum Markov semigroups

“…We begin by considering H = 0. This is a circulant QMS of those studied by Bolaños and Quezada in [9]. Let k = GCD(n, d) (Greatest Common Divisor) and let d = km.…”

Balance Between Quantum Markov Semigroups

“…In this section we use a very simple example based on the examples in [2, Section 6], [13], [28,Section 5] and [29,Subsection 7.1] to illustrate some of the ideas discussed in this paper. Our main reason for considering this example is that it is comparatively easy to manipulate mathematically.…”

Fermionic quantum detailed balance and entanglement

“…(11). The example is based on an example of the type discussed in [2, Section 6], [15], [ for all finite sequences (l 1 , ..., l n ) in L, for any n = 1, 2, 3, ..., where σ : L → L is a permutation of I. By this we mean that σ| I : I → I is a bijection, while σ| L\I is the identity map on the complement L\I of I.…”