Formulations of Quantum Thermodynamics and Applications in Open Systems

TL;DR

This work develops and contrasts two quantum-thermodynamic frameworks—entropy-based and ergotropy-based—for open, memory-possessing quantum systems. It introduces generalized non-Markovianity witnesses via heat flow and ergotropy changes, detailing how environment-induced work relates to ergotropy and passive-state transitions. The entropy-based approach uses monotonicity violations to signal memory effects, while the ergotropy-based formulation ties heat to passive-state changes and yields a nonnegative out-of-equilibrium temperature. Together, these approaches provide a unified view of memory, coherence, and energy exchange in quantum thermodynamics, with concrete results for qubits under amplitude-damping and phase-damping channels and implications for quantum thermal devices. The work highlights practical measures of non-Markovianity and establishes ergotropy as a central resource for energy extraction in open quantum systems, with potential impact on quantum batteries and thermal machines.

Abstract

Quantum thermodynamics has emerged as a central field for understanding how energy conversion processes occur in microscopic systems. In these systems, effects such as coherence, entanglement, and non-Markovianity play key roles. In this thesis, we explore different ways to describe quantum thermodynamics, using two main approaches: one based on entropy and the other on ergotropy. First, we introduce a generalized approach to quantify non-Markovianity through the breakdown of monotonicity in thermodynamic functions. In this context, the entropy-based heat flow serves as a practical tool to witness and measure quantum memory in unital maps that do not reverse the sign of the internal energy. Next, we analyze the dynamics of ergotropy in open qubits under both Markovian and non-Markovian evolutions. We identify phenomena such as freezing and sudden death of ergotropy, and we establish an analytical relation between the change in ergotropy and the environment-induced work . This provides a clear physical interpretation for the additional term in the first law in the entropy-based formulation. Finally, we propose an ergotropy-based thermodynamic formulation, in which heat is reinterpreted in terms of the change of the passive state associated with the density operator governing the quantum dynamics. This approach allows one to measure non-Markovianity of unital maps more generally and accurately than the entropy-based heat flow. This advantage comes from the direct link between heat and von Neumann entropy, a property ensured by invariance under passive transformations. Moreover, the out-of-equilibrium temperature naturally remains non-negative, similarly to equilibrium thermodynamics.

Figures (35)

• Figure 1: Representation of a qubit on the Bloch sphere. Reproduced from Nielsen and Chuang Nielsen-Book.
• Figure 2: (Color online). Representation of the evolution of an open system through a map $\Phi_{t,0}$, which transforms the state $\hat{\rho}_{s}(0)$ into $\hat{\rho}_{s}(t)$. The map includes environmental influences such as decoherence, dissipation, or noise. Adapted from Ref. Preskill2020.
• Figure 3: (Color online) Spontaneous emission process, where the excited state with energy $E_2$ decays to the ground state with energy $E_1$, releasing its energy into the environment. Adapted from Ref.Kaya2025
• Figure 4: (Color online) Schematic illustration of the AD for a single qubit, where the reservoir is at zero temperature. The blue curve shows the relaxation of the Bloch vector from $|1\rangle$ toward $|0\rangle$, while the red curve indicates the irreversible loss of coherence and energy into the environment, driving the system to its unique fixed point $|0\rangle$. Adapted from Ref.Arie2008
• Figure 5: (Color online) Schematic illustration of the GAD for a single qubit interacting with a finite temperature environment $T_e$. The red arrows represent irreversible loss of coherence and energy to the reservoir, while the trajectory inside the Bloch sphere describes the relaxation of the qubit state. Unlike the zero-temperature AD case (which relaxes to $|0\rangle$), here the relaxation leads the qubit to a thermal fixed point determined by $T_e$, $\hat{\rho}_s^{\, \rm th}$. Adapted from Ref.Renaud2011