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.”

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.

This chapter explores how automated legal guidance helps both federal agencies and members of the public. It outlines several specific benefits, including administrative efficiency, communication of complex law in plain language, transparency regarding agency interpretations of the law, internal and external consistency regarding agency communications, and public engagement with the law.

This chapter identifies and explores a central feature of automated legal guidance: “simplexity.” As this chapter introduces this term, simplexity occurs when the government presents clear and simple explanations of the law without highlighting its underlying complexity or reducing this complexity through formal legal changes. Automated legal guidance inherently relies on simplexity as a result of the tension between the complexity of the law and the need of agencies to explain the law in simple terms. In creating the law, the federal government must address complex problems, and it often does so by creating legislation that is replete with errors, ambiguities, and problems. This disconnect between complex federal law and agencies’ need to explain the law to the public in simple and understandable ways forces agencies to rely on simplexity. Automated legal guidance only exacerbates the need for simplexity, because when individuals use automated online tools offered by government agencies, they expect the explanations to be even simpler, more straightforward, and easier to apply than would be the case if they were relying upon written agency publications.

The chapter begins with localizations, including a topological proof of Las Vergnas’s characterization of localizations. Adjoints and their relationship to extensions are discussed. The final part of the chapter discusses intersection properties, particularly on the Euclidean property and non-Euclidean oriented matroids.

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.