Chapter 5 delves into divergences and distance measures, which are crucial for comparing quantum states. It begins with classical divergences such as the Kullback–Leibler and Jensen–Shannon, then advances to their quantum counterparts, discussing their optimal characteristics. Influenced by quantum resource theories, these quantum extensions provide foundational insights into the robust tools of resource theories. The chapter concentrates on particular divergences that serve as true metrics, including the trace distance and a variant of the fidelity, and explores the concept of distance between subnormalized states, which is essential in the context of quantum measurements. It emphasizes the purified distance, a useful tool for understanding the entanglement cost of quantum systems, setting the stage for further exploration in later chapters. The chapter offers a mathematically approachable survey of these measures, underscoring their practical importance in quantum information theory.

This chapter sets forth how government agencies are using artificial intelligence to automate their delivery of legal guidance to the public. The chapter first explores how many federal agencies have a duty not only to enforce the law but also to serve the public, including by explaining the law and helping the public understand how it applies. Agencies must contend with expectations that they will provide customer service experiences akin to those provided by the private sector. At the same time, government agencies lack sufficient resources. The complexity of statutes and regulations significantly compounds this challenge for agencies. As this chapter illustrates, the federal government has begun using virtual assistants, chatbots, and related technology to respond to tens of millions of inquiries from the public about the application of the law.

Introduction to the oriented matroid abstraction of convex polytopes is the goal of this chapter. The chapter shows that the existence of a polar of an oriented matroid polytope is closely related to the existence of an adjoint. The Lawrence construction is introduced and used to produce interesting examples.

This chapter introduces quantum resource theories (QRTs), tracing their evolution and key principles, starting from physics’ quest to unify distinct phenomena into a single framework. It highlights the unification of electricity and magnetism as a pivotal advancement, setting a precedent for QRTs in quantum information science. Quantum resource theories categorize physical system attributes as “resources,” notably transforming the role of quantum entanglement from mere theoretical interest to a crucial element in quantum communication and computation.

The chapter further describes the book’s layout and educational strategy, designed to offer a comprehensive understanding of QRTs. It explores the application of quantum resources in fields like quantum computing and thermodynamics, presenting a unique viewpoint on subjects such as entropy and nonlocality. Emphasizing on axiomatic beginning followed by practical uses, the book serves as a vital resource for both beginners and experts in quantum information science, preparing readers to navigate the complex terrain of QRTs and highlighting their potential to advance quantum science and technology.

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.

This chapter illuminates some of the hidden costs of the federal agencies’ use of automated legal guidance to explain the law to the public. It highlights the following features of these tools: they make statements that deviate from the formal law; they fail to provide notice to users about the accuracy and legal value of their statements; and they induce reliance in ways that impose inequitable burdens among different user populations. The chapter also considers how policymakers should weigh these costs against the benefits of automated legal guidance when contemplating whether to adopt, or increase, agencies’ use of these tools.

Some basic properties and operations, including poset structure of covector sets and flats, are provided in this chapter.

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.