Generators of Detailed Balance Quantum Markov Semigroups

TL;DR

This paper characterizes when the adjoint of a quantum Markov semigroup, with respect to certain scalar products, also forms a quantum Markov semigroup satisfying detailed balance, using the Lindblad generator operators.

Contribution

It provides a complete characterization of adjoint semigroups that are quantum Markov semigroups with detailed balance, in terms of Lindblad operators and scalar products.

Findings

Characterization of adjoint semigroups as quantum Markov semigroups.

Conditions for detailed balance in terms of Lindblad operators.

Analysis of adjoints with respect to different scalar products.

Abstract

For a quantum Markov semigroup $\T$ on the algebra $\B$ with a faithful invariant state $\rho$, we can define an adjoint $\widetilde{\T}$ with respect to the scalar product determined by $\rho$. In this paper, we solve the open problems of characterising adjoints $\widetilde{\T}$ that are also a quantum Markov semigroup and satisfy the detailed balance condition in terms of the operators $H,L_k$ in the Gorini Kossakowski Sudarshan Lindblad representation $\Ll(x)=i[H,x] - {1/2}\sum_k(L^*_kL_k x-2L^*_kxL_k + xL^*_kL_k)$ of the generator of $\T$. We study the adjoint semigroup with respect to both scalar products $<a,b> = \tr(\rho a^* b)$ and $<a,b> = \tr(\rho^{1/2} a^* \rho^{1/2}b)$.