What this book is about

Quantum mechanics is an extraordinarily successful theory. The quantum mechanical description of the structures and spectra of atoms and molecules is virtually complete, and in principle, this provides the basis for understanding all of chemistry. Quantum mechanics gives detailed insight into many thermal, electrical, magnetic, optical, and elastic properties of condensed materials, including superconductivity, superfluidity, and Bose-Einstein condensation. Quantum mechanics underlies the theory of nuclear structure, nuclear reactions, and radioactive decay. Quantum electrodynamics (QED), an outgrowth of quantum mechanics and special relativity, is a very successful and detailed description of the interaction of charged leptons (i.e., electrons, muons, and tau leptons) with the electromagnetic radiation field. More generally, relativistic quantum field theory, the extension of quantum mechanics to relativistic fields, is the basis for all successful theoretical attempts so far to describe the phenomena of elementary particle physics.

We assume that you, the reader, have some elementary knowledge of quantum mechanics and that you know something about the historical development of the subject and its main principles and methods. We take advantage of this background, after a brief mathematical review in Chapter 2, by stating the rules of quantum mechanics in Chapter 3. An advantage of this approach is that all the rules are set forth in one place so that we can focus on them. In Chapter 3 we also describe application of the rules to several real physical situations, most significantly experiments with photon polarizations. Following some development of wave mechanics (Chapter 4), we illustrate the rules with additional examples (Chapter 5). We then develop the theory further in subsequent chapters, giving as many examples as we can from the physical world.

It is not necessary to understand the full structure of quantum measurement theory to understand quantum measurements in a wide range of experiments. This pedagogical approach was pointed out to me by Alexander Korotkov, and is the one he uses in his work on continuous measurements [343, 524, 638]. When we measure a single observable, and when this observable is not being changed during the measurement by any dynamics other than the measurement process, then only a small addition to Bayesian inference is required to describe quantum measurements. In fact, once we have made this addition, for infinitesimal time-steps we can include a Hamiltonian under which the measured observable changes with time, and obtain a full description of the continuous measurement of any quantum observable.

There is also another situation in which quantum measurement theory simplifies: when the system is linear, and when the observable being measured is a linear combination of the canonical coordinates. In this case, even when the observable is undergoing linear dynamics, a continuous quantum measurement reduces to a classical continuous measurement of a classical linear system, with the addition of a specified amount of white (flat-spectrum) noise. This noise is the “quantum back-action” of the measurement.

We now turn to the subject of thermodynamics, and to describing the process of thermalization of quantum systems. A quantum system that interacts with another system large enough to obey the usual rules of thermodynamics, is referred to as an open system. The thermal system with which it interacts is often referred to as a thermal bath, or simply as the environment. We will use these terms interchangeably. Modeling the behavior of open quantum systems has many applications. One of these is to model the noise that they are subjected to by their surroundings, noise that we will wish to control using measurement and feedback loops in Chapter 5. There are also two important connections between quantum measurements, information theory, and thermodynamics. The first of these is that the thermodynamic entropy of a macroscopic system can be identified with the information-theoretic entropy that characterizes the lack of knowledge of an observer about the microstates of the system. In turn, if an observer is allowed to make microscopic measurements on the system, then this modifies the thermodynamic rules regarding the conversion of heat to work. Such is the subject of Maxwell’s demon. Curiously it took more than a century to understand how Maxwell’s demon, essentially a microscopic measurement and feedback process, fits within the second law of thermodynamics. In another twist, it turns out that the process of erasing information requires an increase in the thermodynamic entropy of the environment, and this increase is directly related to the amount of information erased.

Introduction and overview

Much that is studied in quantum measurement theory, and thus in this book, is very different from that studied in its classical counterpart, Bayesian inference. While quantum measurement theory is in one sense a minimal extension of the latter to quantum states, quantum measurements cause dynamical changes that never need appear in a classical theory. This has a multitude of ramifications, many of which you will find in this book. One such consequence is a limit to the information that can be obtained about the state of a quantum system, and this results in relationships between information gained and dynamics induced that do not exist in classical systems. Because of the limits to information extraction, choosing the right kinds of measurements becomes important in a way that it never was for classical systems. Finding the best measurements for a given purpose, and working out how to realize these measurements, is often nontrivial.

The relationship between information and disturbance impacts the control of quantum systems using measurements and feedback, and we discuss this application in Chapter 5. The ability to extract information about quantum systems is also important in using quantum systems to make precise measurements of classical quantities such as time, acceleration, and the strengths of electric and magnetic fields. This subject is referred to as quantum metrology, and we discuss it in Chapter 6.

I would like to thank here a number of people to whom I am indebted in one way or another. To begin, there are five people from whose insights I especially benefited in my formative years in physics. In order of appearance: Sze M. Tan, for teaching me classical measurement theory, and introducing me to information theory and thermodynamics; Howard M. Wiseman, for teaching me about quantum measurement theory; Salman Habib, for teaching me about open systems and classical chaos; Tanmoy Bhattacharya, for enlightenment on a great variety of topics, and especially for the insight that measurement is driven by diffusion gradients; Gerard Jungman, for mathematical and physical insights, and for introducing me to many beautiful curiosities.

I am very grateful to a number of people who helped directly to make this book what it is: Os Vy, Luciano Silvestri, Benjamin Cruikshank, Alexandre Zagoskin, Gelo Tabia, Justin Finn, Josh Combes, Tauno Palomaki, Andreas Nunnenkamp, and Sai Vinjanampathy who read various chapters and provided valuable suggestions that improved the book. Xiaoting Wang who derived Eqs. (G.43)–(G.48). Jason Ralph who enlightened me on some superconductor facts that were strangely difficult to extract from the literature. Jason also helped me with the brief history of superconductivity and quantum superconducting circuits in Chapter 7. Justin Guttermuth who saved our asses when we had a house to move into, rooms to paint, a new baby, and I had this book to finish. My colleagues in the UMass Boston physics department for their support — especially Maxim Olchanyi, Bala Sundaram, Vanja Dunjco, and Steve Arnason. And last but not least, my wonderful wife Jacqueline, who helped me with the figures and the cover, and put up with the long hours this book required.

Combining two systems: the tensor product

The state of a quantum system is a vector in a complex vector space. (Technically, if the dimension of the vector space is infinite, then it is a separable Hilbert space.) Here we will always assume that our systems are finite-dimensional. We do this because everything we will discuss transfers without change to infinite-dimensional systems. Further, when one actually simulates a system on a computer, one must always truncate an infinite-dimensional space so that it is finite.

Consider two separate quantum systems. Now consider the fact that if we take them together, then we should be able to describe them as one big quantum system. That is, instead of describing them by two separate state vectors, one for each system, we should be able to represent the states of both of them together using one single state vector. (This state vector will naturally need to have more elements in it than the separate state-vectors for each of the systems, and we will work out exactly how many below.) In fact, in order to describe a situation in which the two systems have affected each other – by interacting in some way – and have become correlated with each other as a result of this interaction, we will need to go beyond using a separate state vector for each. If each system is in a pure state, described by a state vector for it alone, then there are no correlations between the two systems. Using a single state vector to describe the joint state of the two systems is the natural way to describe all possible states that the two systems could be in (correlated or uncorrelated). We now show how this is done.