This chapter presents the surprising mathematical result that classical systems can indeed have entanglement. However, the degree to which they can be entangled is strictly limited, while quantum systems have no limit to their amount of entanglement.

This chapter begins a short, two-chapter section on calculations that specifically impact the philosophy of quantum mechanics. A quantitative discussion of the famous Einstein–Podalsky–Rosen (EPR) experiment is given, as well as a mathematical discussion of problems with the many-worlds hypothesis, the Bohmian pilot-wave hypothesis, and the “transactional” hypothesis for interpreting quantum mechanics.

This first chapter gives an introduction to the book and guides the reader through much of the history of the field of quantum mechanics, focusing on what we view as the most important episodes in the field. We discuss the essential properties of quantum mechanics and how they have been reimagined in the decades that followed the era of the founders.

Chapter devoted to the basic quantum properties of entanglement and separability. Introduces the Schmidt decomposition for pure states and the positive partial transpose criterion for mixed states as entanglement witnesses. Introduces the famous Einstein–Podolsky–Rosen paradox and its implementation in terms of qubits, then the Bell inequality, quickly reviewing the experimental demonstrations that quantum mechanics violates this inequality. Gives examples of the use of entanglement in a quantum algorithm to accelerate an information task, namely a database search (Grover algorithm) and the possibility of teleportation of a quantum state.

We present a few of the gedanken- and real experiments that demonstrate the spookiness of quantum mechanics. We discuss the Einstein,Podolsky, and Rosengedankenexperiment that invokes hidden variables to create a paradox. We analyze Bell’s analysis of the paradox, which shows that the predictions of quantum mechanics are inconsistent with local hidden variable theories. We discuss the Schrödinger cat paradox, and the Copenhagen interpretation of quantum mechanics.