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.

In 1973 Roger Penrose proposed an inequality which states that, in an isolated gravitational system with nonnegative local energy density, the total mass of the system must be at least as much as that contributed by any black holes contained within. The original motivation for proposing the inequality is the fact that a counterexample of this inequality would produce a counterexample for the cosmic censorship conjecture. While the Penrose inequality in full generality remains an open problem, we present in this chapter the main ideas of three proofs of the Riemannian Penrose inequality, a special case that arises in the time-symmetric setting. In this setting, the Penrose inequality is formulated on an asymptotically flat manifold with nonnegative scalar curvature that contains closed minimal surfaces, and asserts a bound of the total mass in terms of the area of the outermost minimal surface. We begin with Lam's proof for the graphical case, then move to the inverse mean curvature flow approach of Huisken and Ilmanen, and finish with a sketch of Bray's conformal flow technique.

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.

The theory of special relativity incorporates a modification of Newtonian mechanics together with electromagnetism. A natural question to consider is how gravitation fits into the framework of relativity. In this chapter we focus our analysis of this question along two main ideas, that of the equivalence between uniform acceleration and a uniform gravitational field, and that of the gravitational redshift. These will lead us to the Einstein equation, which we then show can be given a variational formulation. We present some solutions of the Einstein equation, with particular attention given to the Schwarzschild spacetime and its Kruskal extension.

This chapter develops the geometry of and analysis on initial data sets that arise in models of isolated gravitational systems. We begin with some detailed discussion and analysis involving the Laplace operator on asymptotically flat manifolds, which we use to develop density and deformation results on scalar curvature, leading to a proof of the Riemannian positive mass theorem. In the last section of the chapter we develop a technique for localized scalar curvature deformation, and we apply it to glue an asymptotically flat end with vanishing scalar curvature to an end of a Riemannian Schwarzschild metric, maintaining zero scalar curvature throughout.

Many physical models admit an initial value formulation. In this chapter we discuss an initial value formulation for the vacuum Einstein equation. A vacuum initial data set will be given geometrically as a manifold endowed with Riemannian metric and a symmetric two-tensor. That these give the first and second fundamental forms of an embedding into a Lorentzian manifold satisfying the vacuum Einstein equation imposes, via the Gauss and Codazzi equations, constraints on the initial data. These conditions, which govern the space of allowable initial data sets for the vacuum Einstein equation, comprise the Einstein constraint equations, the study of solutions to which form an interesting and rich subject for geometric analysis.

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.

The tangent space at any point on a Lorentzian manifold can be partitioned into three classes, timelike, null and spacelike vectors, from which the causal structure derives. In this chapter we introduce some basic concepts of Lorentzian causality needed in the discussion of the Penrose singularity theorem in the next 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.