Quantum state entanglement is a crucial ingredient in many quantum algorithms. This chapter introduces the description of multi-qubit quantum systems and then explains what it means to say the state of that system is entangled. If the quantum state of the system is an entangled state, none of the constituents of the system are described by their own individual states. For a two-qubit entangled state, measurements of one of the qubits apparently affect the measurement results of the other even though the two qubits may be far apart. The change in basis states, introduced in Chapter 8, is applied to a two-qubit system, which will be used to illustrate Bell’s theorem. That theorem tells us that the quantum description of the world profoundly violates almost every concept we have about material properties and the measurement of those properties.

Measurements of quantum states are a key feature of QIS and QC. Polarized light, optical polarization devices, and light detectors illustrate how we describe quantum measurements and how the results of measurements are related to the properties of the quantum states describing the photon qubits. The key distinction between quantum physics and classical physics is that the results of individual quantum measurements cannot be predicted; quantum mechanics gives us only the probabilities of observing each of the possible measurement outcomes. The famous Born rule expresses the probability interpretation of quantum states. Those states carry all the information about the probabilities of all possible measurements on the corresponding qubit. In particular, the probabilities are proportional to the squares of the amplitudes of the basis states appearing in the overall quantum state. Bob points out that state preparation and state measurement are two sides of the same coin.

Alice and Bob describe currently available physical systems to be used as quantum qubits. While the quantitative physics is beyond the scope of this book, they give a brief qualitative explanation of the physics of those qubits. The difference between QC and QIS is briefly delineated, including the possibility of a quantum internet. They also explain a variety of other QC and QIS issues including programming quantum computers, future qubits, new QC architectures, machine learning, and AI. They also raise several ethical issues facing QC and QIS and point to information about careers in QC and QIS.

This chapter explains why complex numbers are important for more general quantum mechanical situations. The complex exponential function, aka Euler’s formula, plays a critical role in expressing phase differences between components of a quantum state vector. These phase differences affect the outcome of quantum operations. The Bloch sphere and Bloch vector provide a visual representation of these more complex quantum states with the real and imaginary parts on different axes. The complex exponential is important in the quantum Fourier transform, which is then used to prove an important result for finding the repetition period in the Shor algorithm.

Alice and Bob introduce and explain in detail some of the algorithms that convinced physicists and computer scientists that there might be something to this crazy idea of quantum computing. Each algorithm provides a case study of how the properties of quantum states can be used to provide an advantage for using a quantum computer compared to a classical computer. Alice and Bob describe the Deutsch algorithm, designed to answer a simple question about the properties of functions of binary digits (0 and 1), and the Simon algorithm, which finds a “secret code.” The Simon algorithm, like many other quantum algorithms, is a mix of quantum state algorithms and classical computational methods. Alice and Bob also introduce Hadamard gates that act on multi-qubit systems. Those gates play an important role in many quantum algorithms.