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.

Chapter 13 delves into the complex terrain of mixed-state entanglement, extending the discourse from pure-state entanglement to encompass the broader and more practical scenarios encountered in quantum systems. The chapter systematically explores the detection of entanglement in mixed states, introducing criteria and methods such as the Positive Partial Transpose (PPT) criterion and entanglement witnesses, which serve as diagnostic tools for identifying entanglement in a mixed quantum state. Furthermore, it addresses the quantification of entanglement in mixed states, discussing various measures like entanglement cost and distillable entanglement. These concepts highlight the operational aspects of entanglement, including its creation and extraction, within mixed-state frameworks. The chapter also introduces the notion of entanglement conversion distances, providing a quantitative approach to understanding the transformations between different entangled states.

Chapter 11 delves into the manipulation of quantum resources, the core aspect of quantum resource theories that explore the transformation and conversion of quantum states within a given resource theory framework. The chapter introduces the generalized asymptotic equipartition property and the generalized quantum Stein’s lemma, both foundational to understanding the asymptotic behavior of quantum resources. These concepts pave the way for discussing the uniqueness of the Umegaki relative entropy in quantifying the efficiency of resource conversion processes. Furthermore, the text explores asymptotic interconversions, detailing the conditions and limits for converting one resource into another when multiple copies of quantum states are considered. This analysis is pivotal for establishing the reversible exchange rates between different resources in the asymptotic limit. By providing a comprehensive overview of resource manipulation strategies, the chapter equips readers with the theoretical tools needed for advanced study and research in quantum resource theories, emphasizing both the single-shot and asymptotic domains.

Chapter 10 delves into the quantification of quantum resources, an essential aspect of quantum resource theories that determines the value of quantum states for specific applications. It begins by defining resource measures and investigating their fundamental properties such as monotonicity under free operations and convexity. The chapter discusses distance-based resource measures, which quantify how far a given quantum state is from the set of free states. Such measures often utilize divergences and metrics explored in earlier chapters. Techniques to compute the relative entropy of a resource are also covered.

To refine resource measures, the chapter introduces the concept of smoothing, which considers small deviations from the ideal state to make the measures more robust against perturbations. This approach is crucial in single-shot scenarios where finite resources are available. Furthermore, the chapter examines resource monotones and support functions, offering a comprehensive framework for the theoretical and practical assessment of quantum resources.

Chapter 7 discusses quantum conditional entropy, extending the concept of conditional majorization and introducing the notion of negative quantum conditional entropy. The chapter starts with the basic definition of conditional entropy, exploring its key properties like monotonicity and additivity. It further delves into the concepts of conditional min- and max-entropies, emphasizing their roles in quantifying uncertainty in quantum states and their operational significance in quantum information theory.

The text presents conditional entropy as a measure sensitive to the effects of entanglement, showing that negative conditional entropy is a distinctive feature of quantum systems, contrasting with the classical domain where entropy values are nonnegative. This negativity is particularly pronounced in the context of maximally entangled states and is connected to the fundamental differences between classical and quantum information processing. Moreover, the chapter includes theorems and exercises to solidify understanding, like the invariance of conditional entropy under local isometric channels and its reduction to entropy for product states. It concludes by underscoring the inevitability of negative conditional entropy in quantum systems, a topic of both theoretical and practical importance in the quantum domain.

Chapter 8 explores the asymptotic regime of quantum information processing, beginning with quantum typicality, which illustrates the convergence of quantum states toward a typical form with increasing copies. This leads to the asymptotic equipartition property (AEP), indicating that with a high number of copies, probability vectors become uniformly distributed. The method of types is introduced next, a tool from classical information theory that classifies sequences based on their statistical properties. This is crucial for understanding the behavior of large quantum systems and has implications for quantum data compression. Advancing to quantum hypothesis testing, the chapter outlines efficient strategies for distinguishing between two quantum states through repeated measurements. Central to this is the Quantum Stein’s lemma, which asserts the exponential decline in the error probability of hypothesis testing as the sample size of quantum systems increases. The chapter highlights the deep interplay between typicality, statistical methods, and hypothesis testing, laying the groundwork for asymptotic interconversion of quantum resources.

Chapter 16, centered on the resource theory of nonuniformity, serves as an essential precursor to discussions on thermodynamics as a resource theory. It presents nonuniformity as a fundamental quantum resource, using it as a toy model to prepare for more complex thermodynamic concepts. In this model, free states are considered to be maximally mixed states, analogous to Gibbs states with a trivial Hamiltonian, providing a simplified context for exploring quantum thermodynamics. The chapter carefully outlines how nonuniformity is quantified, offering closed formulas for the conversion distance, nonuniformity cost, and distillable nonuniformity. These measures are explored both in the single-shot and the asymptotic domains. The availability of closed formulas makes this model particularly insightful, demonstrating clear, quantifiable relationships between various measures of nonuniformity.

The study of the quantum–classical correspondence has been focused on the quantum measurement problem. However, most of the discussion in the preceding chapters is motivated by a broader question: Why do we perceive our quantum Universe as classical? Therefore, emergence of the classical phase space and Newtonian dynamics from the quantum Hilbert space must be addressed. Chapter 6 starts by re-deriving decoherence rate for non-local superpositions using the Wigner representation of quantum states. We then discuss the circumstances that, in some situations, make classical points a useful idealization of the quantum states of many-body systems. This classical structure of phase space emerges along with the (at least approximately reversible) Newtonian equations of motion. Approximate reversibility is a non-trivial desideratum given that the quantum evolution of the corresponding open system is typically irreversible. We show when such approximately reversible evolution is possible. We also discuss quantum counterparts of classically chaotic systems and show that, as a consequence of decoherence, their evolution tends to be fundamentally irreversible: They produce entropy at the rate determined by the Lyapunov exponents that characterize classical chaos. Thus, quantum decoherence provides a rigorous rationale for the approximations that led to Boltzmann’s H-theorem.

Chapter 5 explores the consequences of decoherence. We live in a Universe that is fundamentally quantum. Yet, our everyday world appears to be resolutely classical. The aim of Chapter 5 is to discuss how preferred classical states, and, more generally, classical physics, arise, as an excellent approximation, on a macroscopic level of a quantum Universe. We show why quantum theory results in the familiar “classical reality” in open quantum systems, that is, systems interacting with their environments. We shall see how and why, and to what extent, quantum theory accounts for our classical perceptions. We shall not complete this task here—a more detailed analysis of how the information is acquired by observers is needed for that, and this task will be taken up in Part III of the book. Moreover, Chapter 5 shows that not just Newtonian physics but also equilibrium thermodynamics follows from the same symmetries of entanglement that led to Born’s rule (in Chapter 3).

Quantum Darwinism demonstrates not only that preferred states are selected for their stability but also that information about them is broadcast by the same environment that causes decoherence and einselection. That environment acts both as a censor and as an advertising agent that disseminates information about pointer states while suppressing complementary information. Chapter 8 explores the implications and limitations of quantum Darwinism using models inspired by the structure of the Universe we inhabit. We perceive our Universe using light and other means of information transmission. We explore models that have a well-defined relation with our everyday reality, and where one can also selectively relax some of the idealized assumptions and investigate the consequences. Light is the communication channel through which we obtain most of our information. Fortunately, it is an ideal channel in the sense of quantum Darwinism, and simple but realistic cases are exactly solvable. The solution presented herein demonstrates the inevitability of the consensus between observers who rely on scattered photons: The emergence of classical objective reality (classical because pointer states are einselected, and objective because redundancy imposes consensus) is inevitable. This is how the classical world we perceive emerges from within the quantum Universe we inhabit.

The aim in Chapter 7 is to take into account the role of the means of information transmission on the nature of the states that can be perceived. Our point of departure is the recognition that the information we obtain is acquired by observers who monitor fragments of the same environment that decohered the system, einselecting preferred pointer states in the process. Moreover, we only intercept a fraction of the environment. The only information about the system that can be transmitted by its fraction must have been reproduced in many copies in that environment. This process of amplification limits what can be found out to the states einselected by decoherence. Quantum Darwinism provides a simple and natural explanation of this restriction, and, hence, of the objective existence—the essence of classicality—for the einselected states. This chapter introduces and develops information-theoretic tools and concepts (including, e.g., redundancy) that allow one to explore and characterize correlations and information flows between systems, environments, and observers, and illustrates them on an exactly solvable yet non-trivial model.

Chapter 4 begins to discuss decoherence, and, thus, to address the overarching question: How does the classical world—classical states that are responsible for the objective reality of our everyday experience—emerge from within the Universe that is, as we know from compelling experimental evidence, made out of quantum stuff. The short answer to this question is that decoherence selects (from the vast number of superpositions that populate Hilbert space in the process of environment-induced superselection (also known as einselection) the few states that are—in contrast to all the other alternatives—stable in spite of their immersion in the environment. Decoherence is illustrated with a detailed discussion of two models. A spin decohered by an environment of spins as well as quantum Brownian motion have become paradigmatic models of decoherence for good reason: They are exactly solvable and yet they capture (albeit in an idealized manner) the emergence of the preferred classical states in settings that are relevant for quantum measurements and for Newtonian dynamics in effectively classical phase space.