In Chapter 2, we will review basic facts of quantum measurement that are usually discussed in basic texts on quantum mechanics. These include a motivating experiment – the Stern–Gerlach effect – and discussions of measurement results, statistics, the Born rule, and wavefunction collapse.

In Chapter 10 we discuss feedback and control as an advanced topic. We introduce how to use the measurement results to control the quantum system, via applying conditional unitary operator. A number of experimental systems are discussed, including active qubit phase stabilization, adaptive phase measurements, and continuous quantum error correction.

Chapter 3 takes a step beyond textbook measurements and introduces generalized measurements, beginning with the motivating experiment of an optical polarization measurement with a calcite crystal. In this case, wavefunction collapse is imperfect, and we will discuss how to describe and predict the statistics of outcomes and how to assign postmeasurement states. This topic is closely related to Bayesian probability theory, and we discussed a “Quantum Bayes Rule.”

The book concludes in Chapter 11, where we give in our epilogue a more philosophical reflection on the state of the field. We discuss what it all means, where the field is going, how quantum computers are the ultimate test of quantum mechanics, and speculate on a future post-quantum science.

In Chapter 4, we take a limit where the coupling of the measurement apparatus to the quantum system is very small, and in this limit, discuss weak measurements and weak values. The later involved a sequence of a weak and a strong measurement. Generalizations of these effects are discussed using the concepts in Chapter 3, and we discuss generalized eigenvalues of quantum observables that can exceed the eigenvalue range and reincorporate the concept of observables in generalized measurement theory.

In Chapter 7, we discuss the fundamental limits of quantum amplification. When quantum effects are amplified to classical signals, noise is added to the signal (there are exceptions, but other trade-offs come into play). A detailed discussion of linear response theory is given, which is applicable to many kinds of quantum-limited measurements. This theory is applied to mesoscopic charge detectors and resonant optical cavities. While fundamental bounds quantum mechanics gives to amplification are important, they do not tell you how to invent a quantum-limited amplifier.

Chapter 5 considers the case of diffusive continuous measurements, where the measurement outcomes and quantum state dynamics are analogous to a Brownian noise process. We motivate this type of measurement by considering the example of a double quantum dot system being measured by a quantum point contact. The intrinsic shot noise of the measurement naturally brings about an effective time-continuous measurement. A second example of a superconducting circuit made from Josephson junctions readout with a microwave frequency electromagnetic wave is also discussed in detail. The mathematics of quantum trajectory theory is then pedagogically built up, resulting in the stochastic Schrodinger equation, the stochastic master equation, and the stochastic path integral. We also discuss experimental data and its comparison with this theoretical formalism. These experiments allow us to peer into the inner workings of wavefunction collapse, giving an empirical handle on the many philosophical issues that arise in quantum measurement.

In Chapter 8, we devote a whole chapter to how quantum amplifiers work. We discuss phase-preserving and phase-sensitive amplification and how to realize these with heterodyne and homodyne measurements. Focusing on quantum superconducting circuits, we discuss three-wave and four-wave mixing, the different types of circuits to build in order to realize amplification, and how it can go wrong.

In Chapter 6, we discuss quantum jumps, or leaps, beginning with a historical discussion of quantum jumps, or leaps, and giving the motivating example of blinking atoms in trapped ions and quantum jumps in superconducting circuits. Despite the disruptive name, we discuss how jumps fit into the category of continuous measurement. In some cases, quantum jumps can be predicted, and even reversed. We build up the mathematical formalism by discussing the quantum Zeno effect, Lindblad-type master equations, leading to jump and no-jump dynamics - an inseparable combination of discrete and continuous dynamics. In fact, the discrete nature of the jump is illusory. We discuss the dynamics of a quantum jump and the transition from jumps to diffusion.

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.