Using linear algebra, the mathematical techniques needed for describing and manipulating qubits are laid out in detail, including quantum circuits. Moreover, the chapter also explains the state evolution of an isolated quantum system, as is predicted by the Schrödinger equation, as well as non-unitary irreversible operations such as measurement. More details of classical and quantum randomness and their mathematical representation is discussed, leading to the density matrix. representation of a quantum state.

In this chapter, we (a) outline operations preserving convexity of functions, (b) present differential criteria for convexity, (c) establish convexity of several important multivariate functioins, (d) present the gradient inequality, and (e) establish local boundedness and Lipschitz continuity of convex functions.

This is the chapter that gets down to applying concepts from the previous chapters about qubits to construct a quantum computer. It teaches how numbers can be stored in quantum computers and how their functions can be evaluated. It also demonstrates the computational speed-up that quantum computers offer over their classical counterparts through the study of Deutsch, Deutsch-Jozsa, and Bernstein-Vazirani algorithms. Finally, it gives a practical demonstration of speed-up in search algorithms provided by Grover’s search algorithm.

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.