Quantum entanglement requires a minimum of two quantum systems to exist, and each quantum system has to have a minimum of two levels. This is exactly what a two-qubit system is, which in this chapter is explored on various levels: state description, entanglement measures, useful theorems, quantum gates, hidden variable theory, quantum teleportation.

In this chapter, we (a) present the notion of a polyhedral representation and illustrate its importance, (b) demonstrate via Fourier--Motzkin eliminaton that every polyhedrally representable set is polyhedral, and (c) outline the calculus of polyhedral representations. As an immediate application, we demonstrate that a bounded and feasible LP problem is solvable.

In this chapter, we (a) present epigraph characterization of cone-convexity, (b) introduce cone-monotonicity, and describe differential criteria of cone-convexity and cone-monotonicity, (c) present instructive examples of cone-convex and cone-monotone functions, (d) outline basic operations preserving cone-convexity and cone-monotonicity. Taken together, (b)--(d) provide simple and powerful tools allowing one to detect and utilize cone-convexity and cone-monotonicity.

In this chapter, we (a) present an algebraic characterization of extreme points of polyhedral sets and extreme rays of polyhedral cones, (b) describe extreme points of several important polyhedral sets, including the Birkhoff--von-Neumann Theorem on extreme points of the polytope of doubly stochastic matrices, (c) establish the theorem on the structure of polyhedral sets stating that nonempty polyhedral sets are exactly the sets representable as sums of convex hulls of nonempty finite sets and conic hulls of finite sets, and vice versa, (d) extract from the latter theorem basic descriptive results of linear programming theory, and (e) present and justify the Majorization Principle.

In this chapter, we present preliminaries on convex functions -- definitions via convexity inequality and via the convexity of the epigraph, basic examples, Jensen’s inequality, convexity of sublevel sets; we introduce the notion of the domain of a convex function and its representation as a function taking values in the extended real axis and introduce the concept of a proper convex function.

Deducing the quantum state of your device is essential for diagnosing and perfecting it, and the methods needed for this are introduced in this chapter. We also extend the discussion to methods used to validate noisy, intermediate-scale quantum computers when they grow too large for tomography to be used.

The generic properties of physical qubits are discussed in detail: in particular the need for an energy gap to ensure cooling and its implications for the size of devices. The basic notions of controlling qubits by external forces shows us how single-qubit gates are implemented.

In this chapter, we present and illustrate Caratheodory’s Theorem (in plain and conic forms), Radon’s Theorem, and Helly’s Theorem (for finite and for infinite families of convex sets).

In this chapter, we (a) present the definition and game theory interpretation of saddle points, (b) describe primal and dual optimization problems induced by an antagonistic game, (c) provide a characterization of saddle points in terms of optimal solutions and optimal values of primal and dual problems induced by the game, and (d) formulate and prove he Minimax Lemma and the Sion--Kakutani Theorem on existence of saddle points in convex--concave antagonistic games.

Several other technologies under development to exploit quantum power are discussed in this chapter. You will learn about quantum key distribution; improving measurements of phase shifts is used as an example to demonstrate the power of entanglement in beating the standard quantum limit. How the latter is used to improve detection of objects is also discussed. Finally, modelling complicated quantum systems by designing simpler and easier to control systems, represented by quantum circuits, simplifies the studying of such systems, allowing us to gain better insight into their physics and to make better predictions about them.