A discussion of realization spaces, including an example of an oriented matroid with disconnected extension space, is provided. In the later part, a proof of the Universality Theorem and a discussion of some of its consequences follows.

Chapter 3 explores open quantum systems, emphasizing their interactions with environments, unlike isolated closed systems. It introduces the concept of generalized measurements and mixed quantum states, reflecting the complex scenarios arising from these interactions. The chapter utilizes Positive Operator Valued Measures (POVMs) to describe generalized measurements, broadening the conventional approach to quantum measurements.

A significant focus is on the evolution of open systems through quantum channels, which illustrate the transfer or transformation of quantum information amid noise and external disturbances. This section underpins the dynamics open systems exhibit, critical for understanding quantum computing and information processing in realistic settings.

Through practical examples, the chapter elucidates how environmental factors influence quantum information, vital for applications in quantum technologies. It aims to equip readers with foundational knowledge of open quantum systems, highlighting their importance in the broader context of quantum mechanics.

This chapter roots the authors' insights about automated legal guidance in a broader examination of why and how to address the democracy deficit in administrative law. As this chapter contemplates the future of agency communications, it also explores in greater detail the possibility that technological developments may allow government agencies not only to explain the law to the public using automated tools but also to automate the legal compliance obligations of individuals. While automated legal compliance raises serious concerns, recent examples reveal that it may soon become a powerful tool that agencies can apply broadly under the justifications of administrative efficiency. As this chapter argues, the lessons learned from our study of automated legal guidance are critical to maintaining values like transparency and legitimacy, as automated compliance expands as a result of perceived benefits like efficiency.

The Conclusion emphasizes the growing importance of automated legal guidance tools across government agencies. It crystalizes the insight that automated legal guidance tools reflect a trade-off between government agencies representing the law accurately and presenting it in accessible and understandable terms. While automated legal guidance tools enable agencies to reach more members of the public and provide them quick and easy explanations of the law, these quick and easy explanations sometimes obscure what the law actually is. The Conclusion acknowledges and accepts the importance of automated legal guidance to the future of governance, and, especially in light of this acknowledgement, recommends that legislators and agency officials adopt the policy recommendations presented in this book.

This chapter explores ways in which administrative law fails to address problems raised by automated legal guidance. Administrative law requires notice-and-comment procedures for so-called legislative rules, or rules that bind agencies and the public. Other, less binding agency statements regarding the law, including, for instance, statements that offer an agency’s interpretation of the law or its enforcement policy, are subject to lesser procedural requirements. This chapter examines how this blind spot in the administrative law framework mirrors a broader democracy deficit in administrative law. Strikingly, this area of law, the purpose of which is to mandate that administrative agencies act in certain ways to protect the public, simply fails to address the pervasive, and impactful, ways that agencies often communicate law to people through the types of informal explanations found in automated legal guidance. As this chapter argues, administrative law reflects a bias toward sophisticated parties, rather than the general public.

The chapters addresses the various axiomatizations and the equivalences between them and presents an introduction to the Plucker relations. The chapter finishes with some discussion of nonrealizable oriented matroids and the impossibility of an excluded minor characterization.

The geometric motivation for the theory is combinatorial data associated with matrices, vector arrangements, hyperplane arrangements, and subspaces of real vector spaces. Interpretations of this data are given in terms of linear algebra, discrete geometry, and the Plucker embedding of the Grassmannian. Elementary proofs of cryptomorphisms for realizable oriented matroids are provided. The chapter finishes with an application of Gale Diagrams.

The chapters provides a survey on the topology of various posets of oriented matroids analogous to various topological spaces, including extension spaces, combinatorial Grassmannians, and combinatorial flag spaces. A general framework for interpreting maps from spaces to posets is laid down, by way of McCord’s Theorem and the Semi-algebraic Triangulation Theorem. The chapter includes a discussion of the (now-disproved) extension space conjecture and of the various results on the topology of the MacPhersonian.

Chapter 15 extensively examines the resource theory of asymmetry, focusing on the significance of asymmetry as a quantum resource, particularly in situations lacking a shared reference frame. The chapter begins by identifying the foundational elements, such as free states and operations within this theory, emphasizing their role in alignment of quantum reference frames. A significant part of the discussion revolves around the quantification of asymmetry, utilizing measures like the Fisher information and Wigner–Yanase–Dyson skew information to assess the degree of asymmetry in quantum states. The concept of G-twirling is introduced as a method to achieve symmetric states, serving as a key technique in analyzing and understanding asymmetry. Moreover, the chapter explores how asymmetry can enhance tasks like parameter estimation, leveraging the maximum likelihood method to improve precision.

As Chapter 4 demonstrated, automated legal guidance often enables the government to present complex law as though it is simple without actually engaging in simplification of the underlying law. While this approach offers advantages in terms of administrative efficiency and ease of use by the public, it also causes the government to present the law as simpler than it is, leading to less precise advice and potentially inaccurate legal positions. As the use of automated legal guidance by government agencies is likely to grow in the future, a number of policy interventions are needed. This chapter offers multiple detailed policy recommendations for federal agencies that have introduced, or may introduce, chatbots, virtual assistants, and other automated tools to communicate the law to the public. Our recommendations are organized into five general categories: (1) transparency; (2) reliance; (3) disclaimers; (4) process; and (5) accessibility, inclusion, and equity.

Chapter 6 builds upon the foundation of divergences from Chapter 5, advancing into entropies and relative entropies with an axiomatic approach, and the inclusion of the additivity axiom. The chapter delves into the classical and quantum relative entropies, establishing their core properties and revealing the significance of the KL-divergence introduced in Chapter 5, notably characterized by asymptotic continuity. Quantum relative entropies are addressed as generalizations of classical ones, with a focus on the conditions necessary for these measures in the quantum framework. Several variants of relative entropies are discussed, including Renyi relative entropies and their extensions to quantum domain such as the Petz quantum Renyi divergence, minimal quantum Renyi divergence, and the maximal quantum Renyi divergence. This discourse underlines the relevance of continuity and its relation to faithfulness in relative entropies. The concept of entropy is portrayed as a measure with a broad spectrum of interpretations and applications across fields, from thermodynamics and information theory to cosmology and economics.

The Introduction presents an overview of the use of automated legal guidance by government agencies. It offers examples of chatbots, virtual assistants, and other online tools in use across US federal government agencies and shows how the government is committed to expanding their application. The Introduction sets forth some of the critical features of automated legal guidance, including its tendency to make complex aspects of the law seem simple. The Introduction previews how automated legal guidance promises to increase access to complex statutes and regulations. However, the Introduction cautions that there are underappreciated costs of automated legal guidance, including that its simplification of statutes and regulations is more likely to harm members of the public who lack access to legal counsel than high-income and wealthy individuals. The Introduction provides a roadmap for the remainder of the book.

Introduction to strong and weak maps, including issues around realizability and topological representations, is the topic of this chapter.