Chapter 4 delves into the concept of majorization, an essential mathematical framework critical for understanding the intricate structures within quantum resource theories. At its core, majorization establishes a preorder relationship between probability vectors, revealing a structured approach to compare the dispersion or concentration of probabilistic distributions. The chapter systematically deconstructs majorization into comprehensible segments, incorporating axiomatic, constructive, and operational perspectives. It further explores the mathematical foundations of majorization through an in-depth look at doubly stochastic matrices and T-transforms. Additionally, the chapter examines various forms of majorization, including approximate, relative, trumping, catalytic, and conditional majorization. This exploration not only encompasses the theoretical facets but also offers practical insights derived from games of chance.

Chapter 14 transitions the focus to multipartite entanglement, a realm that broadens the discussion from bipartite systems to those involving multiple parties. This complex form of entanglement plays a crucial role in quantum computing, cryptography, and communication networks. The chapter introduces the foundational concepts of multipartite entanglement, including its characterization and the challenges associated with its classification. Significant attention is given to the classification of multipartite entangled states through SL-invariant polynomials, which provide tools for understanding the structure and properties of these states. Stochastic Local Operations and Classical Communication (SLOCC) are introduced as a means to classify entanglement. Furthermore, the chapter explores the entanglement of assistance and the monogamy of entanglement, two concepts that illustrate the limitations and potential for distributing entanglement among multiple parties. Through detailed explanations and examples in three and four qubits, this chapter offers insights into the intricate world of multipartite entanglement, revealing both its potential and challenges.

Chapter 12 provides an in-depth exploration of pure-state entanglement. It begins with a clear definition of quantum entanglement for pure states, emphasizing its critical role in quantum computing and communication. The chapter highlights various strategies for entanglement manipulation, encompassing deterministic, stochastic, and approximate methods. Quantification of bipartite entanglement is a key focus, with emphasis on entropy of entanglement and the Ky Fan norm-based entanglement monotones. Additionally, the chapter delves into entanglement catalysis and embezzlement of entanglement, presenting them as a nuanced nonintuitive phenomena that underscore the challenges of entanglement preservation during quantum operations. A notable aspect of this chapter is its connection between entanglement theory and the theory of majorization discussed in Chapter 4. Through a comprehensive treatment of these topics, the chapter equips readers with a robust understanding of the intricacies of pure-state entanglement theory.

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 2 serves as an introduction to the fundamental principles of quantum mechanics, focusing on closed systems. It begins with the historic Stern–Gerlach experiment, highlighting the discovery of quantum spin. The narrative then shifts to the mathematical framework of quantum mechanics, covering inner product spaces, Hilbert spaces, and linear operators. These concepts are crucial for understanding the behavior and manipulation of quantum states, the core of quantum information theory.

The chapter further explores the encoding of information in quantum states, emphasizing qubits, and discusses quantum measurements, revealing the probabilistic nature of quantum mechanics. Additionally, it addresses hidden variable models, offering insights into the deterministic versus probabilistic interpretations of quantum phenomena.

Unitary evolution and the Schrödinger equation are introduced as mechanisms for the time evolution of quantum states, showcasing the deterministic evolution in the absence of measurements. This section underscores the dynamic aspect of quantum systems, pivotal for advancements in quantum information theory.

Chapter 9 introduces the framework of static quantum resource theories, which provide a structured approach for studying different types of quantum resources like entanglement and coherence. The chapter begins by laying out the structure of quantum resource theories, defining what constitutes a quantum resource and how it can be quantified, manipulated, and converted. The text discusses the role of free states and free operations in resource theories, as they form the basis for comparing resources. It introduces state-based resource theories, which focus on the resource content of quantum states, and affine resource theories, which are used to study various interconversions of quantum resources. Resource witnesses, a key concept, are explored as tools to detect the presence of a resource within a quantum state.

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.

Chapter 11 delves into the manipulation of quantum resources, the core aspect of quantum resource theories that explore the transformation and conversion of quantum states within a given resource theory framework. The chapter introduces the generalized asymptotic equipartition property and the generalized quantum Stein’s lemma, both foundational to understanding the asymptotic behavior of quantum resources. These concepts pave the way for discussing the uniqueness of the Umegaki relative entropy in quantifying the efficiency of resource conversion processes. Furthermore, the text explores asymptotic interconversions, detailing the conditions and limits for converting one resource into another when multiple copies of quantum states are considered. This analysis is pivotal for establishing the reversible exchange rates between different resources in the asymptotic limit. By providing a comprehensive overview of resource manipulation strategies, the chapter equips readers with the theoretical tools needed for advanced study and research in quantum resource theories, emphasizing both the single-shot and asymptotic domains.

Chapter 10 delves into the quantification of quantum resources, an essential aspect of quantum resource theories that determines the value of quantum states for specific applications. It begins by defining resource measures and investigating their fundamental properties such as monotonicity under free operations and convexity. The chapter discusses distance-based resource measures, which quantify how far a given quantum state is from the set of free states. Such measures often utilize divergences and metrics explored in earlier chapters. Techniques to compute the relative entropy of a resource are also covered.

To refine resource measures, the chapter introduces the concept of smoothing, which considers small deviations from the ideal state to make the measures more robust against perturbations. This approach is crucial in single-shot scenarios where finite resources are available. Furthermore, the chapter examines resource monotones and support functions, offering a comprehensive framework for the theoretical and practical assessment of quantum resources.

Chapter 7 discusses quantum conditional entropy, extending the concept of conditional majorization and introducing the notion of negative quantum conditional entropy. The chapter starts with the basic definition of conditional entropy, exploring its key properties like monotonicity and additivity. It further delves into the concepts of conditional min- and max-entropies, emphasizing their roles in quantifying uncertainty in quantum states and their operational significance in quantum information theory.

The text presents conditional entropy as a measure sensitive to the effects of entanglement, showing that negative conditional entropy is a distinctive feature of quantum systems, contrasting with the classical domain where entropy values are nonnegative. This negativity is particularly pronounced in the context of maximally entangled states and is connected to the fundamental differences between classical and quantum information processing. Moreover, the chapter includes theorems and exercises to solidify understanding, like the invariance of conditional entropy under local isometric channels and its reduction to entropy for product states. It concludes by underscoring the inevitability of negative conditional entropy in quantum systems, a topic of both theoretical and practical importance in the quantum domain.

Chapter 8 explores the asymptotic regime of quantum information processing, beginning with quantum typicality, which illustrates the convergence of quantum states toward a typical form with increasing copies. This leads to the asymptotic equipartition property (AEP), indicating that with a high number of copies, probability vectors become uniformly distributed. The method of types is introduced next, a tool from classical information theory that classifies sequences based on their statistical properties. This is crucial for understanding the behavior of large quantum systems and has implications for quantum data compression. Advancing to quantum hypothesis testing, the chapter outlines efficient strategies for distinguishing between two quantum states through repeated measurements. Central to this is the Quantum Stein’s lemma, which asserts the exponential decline in the error probability of hypothesis testing as the sample size of quantum systems increases. The chapter highlights the deep interplay between typicality, statistical methods, and hypothesis testing, laying the groundwork for asymptotic interconversion of quantum resources.