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.

This chapter describes the results of the authors' research of automated legal guidance tools across the federal government, conducted over a five-year period from 2019 through 2023. The authors first began this study in preparation for a conference on tax law and artificial intelligence in 2019, and were able to expand it significantly, under the auspices of the Administrative Conference of the United States (ACUS), in 2021. ACUS is an independent US government agency charged with recommending improvements to administrative process and procedure. The goals of this study were to understand how federal agencies use automated legal guidance and to offer recommendations based on these findings. During their research, the authors examined the automated legal guidance activities of every US federal agency. This research found that agencies used automation extensively to offer guidance to the public, albeit with varying levels of sophistication and legal content. This chapter focuses on two well-developed forms of automated legal guidance currently employed by federal agencies: the US Citizenship Immigration Services’ “Emma” and the Internal Revenue Service’s “Interactive Tax Assistant.”