Thermodynamic formalism for continuous-time quantum Markov semigroups: the detailed balance condition, entropy, pressure and equilibrium quantum processes

TL;DR

This work extends classical thermodynamic formalism to continuous-time quantum Markov semigroups with detailed balance by introducing a Laplacian-derived entropy $h(\rho)$ from a fixed Laplacian $\mathcal{L}_0$ and a pressure functional $P_A$ for a Hermitian Hamiltonian $A$. Equilibrium density matrices $\rho_A$ are characterized through a nonlinear eigenvalue-like condition, linking $P_A(\rho_A)$ to an auxiliary eigenvalue $\kappa$ via $P_A(\rho_A) = \kappa - 2n$, and the associated generator $\mathcal{L}_A$ governs the equilibrium quantum Markov process. Under detailed balance, the Lindbladian has an explicit form in terms of ladder operators $V_{i,j}$ with weights, connecting to a quantum transfer operator framework and enabling a quantum-to-classical reduction by projecting onto energy eigenbasis. The results unify entropy, pressure, and equilibrium concepts in the quantum setting and reveal a precise relation between the quantum rate function and entropy, mirroring the classical theory and providing a pathway to quantum thermodynamic equilibrium.

Abstract

$M_n(\mathbb{C})$ denotes the set of $n$ by $n$ complex matrices. Consider continuous time quantum semigroups $\mathcal{P}_t= e^{t\, \mathcal{L}}$, $t \geq 0$, where $\mathcal{L}:M_n(\mathbb{C}) \to M_n(\mathbb{C})$ is the infinitesimal generator. If we assume that $\mathcal{L}(I)=0$, we will call $e^{t\, \mathcal{L}}$, $t \geq 0$ a quantum Markov semigroup. Given a stationary density matrix $ρ= ρ_{\mathcal{L}}$, for the quantum Markov semigroup $\mathcal{P}_t$, $t \geq 0$, we can define a continuous time stationary quantum Markov process, denoted by $X_t$, $t \geq 0.$ Given an {\it a priori} Laplacian operator $\mathcal{L}_0:M_n(\mathbb{C}) \to M_n(\mathbb{C})$, we will present a natural concept of entropy for a class of density matrices on $M_n(\mathbb{C})$. Given an Hermitian operator $A:\mathbb{C}^n\to \mathbb{C}^n$ (which plays the role of an Hamiltonian), we will study a version of the variational principle of pressure for $A$. A density matrix $ρ_A$ maximizing pressure will be called an equilibrium density matrix. From $ρ_A$ we will derive a new infinitesimal generator $\mathcal{L}_A$. Finally, the continuous time quantum Markov process defined by the semigroup $\mathcal{P}_t= e^{t\, \mathcal{L}_A}$, $t \geq 0$, and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian $A$. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian $A$.

Theorems & Definitions (39)

• Definition 1
• Theorem 2
• Definition 3
• Definition 4
• Definition 5
• Lemma 6
• proof
• Proposition 7
• Remark 1
• Lemma 8