Chapter 17 delves into quantum thermodynamics, building on the concepts introduced in the resource theory of nonuniformity. This chapter focuses on thermal states and athermality as resources within the quantum domain, emphasizing the significance of Gibbs states and their role in quantum statistical mechanics. It outlines the operational framework for thermal operations, setting the stage for discussions on energy conservation and the second law of thermodynamics in quantum systems. A key aspect of the chapter is the exploration of quasi-classical athermality, illustrating how quantum states deviate from thermal equilibrium when the state of the system commutes with its Hamiltonian. In the fully quantum domain, the chapter introduces closed formulas for quantifying athermality, such as the athermality cost and distillable athermality, both in the single-shot and the asymptotic domains. These measures provide a quantitative understanding of the efficiency of thermal operations and the potential for work extraction or consumption.

Chapter 13 delves into the complex terrain of mixed-state entanglement, extending the discourse from pure-state entanglement to encompass the broader and more practical scenarios encountered in quantum systems. The chapter systematically explores the detection of entanglement in mixed states, introducing criteria and methods such as the Positive Partial Transpose (PPT) criterion and entanglement witnesses, which serve as diagnostic tools for identifying entanglement in a mixed quantum state. Furthermore, it addresses the quantification of entanglement in mixed states, discussing various measures like entanglement cost and distillable entanglement. These concepts highlight the operational aspects of entanglement, including its creation and extraction, within mixed-state frameworks. The chapter also introduces the notion of entanglement conversion distances, providing a quantitative approach to understanding the transformations between different entangled states.