This chapter explores accurate formulation of the approach, including a description of the evolution of the measured system and the generalized unitarity condition for this evolution. It argues that evolution under continuous measurement has both quantum and classical features. The chapter deals with the evolution of the system subject to continuous measurement. It demonstrates in general form and considers the specific example of a free particle scattered because its position is measured. The quantum theory of measurements has been discussed not only because of its conceptual interest but also for practical purposes. The chapter discusses the application of the measurement amplitude for estimation of probabilities of different measurement outputs. However, the same amplitude can be used to describe the evolution of the system subject to the measurement. The chapter shows the continuous measurement changes the evolution of the quantum system undergoing the measurement.
This chapter explores accurate formulation of the approach, including a description of the evolution of the measured system and the generalized unitarity condition for this evolution. It argues that evolution under continuous measurement has both quantum and classical features. The chapter deals with the evolution of the system subject to continuous measurement. It demonstrates in general form and considers the specific example of a free particle scattered because its position is measured. The quantum theory of measurements has been discussed not only because of its conceptual interest but also for practical purposes. The chapter discusses the application of the measurement amplitude for estimation of probabilities of different measurement outputs. However, the same amplitude can be used to describe the evolution of the system subject to the measurement. The chapter shows the continuous measurement changes the evolution of the quantum system undergoing the measurement.
This chapter explores the path-integral theory of continuous quantum measurements. It explains the principal ideas of this theory on a qualitative level with a minimum of mathematical apparatus. Quantum mechanics appeared as a theory of microscopic bodies when it had been proved that the motion of microscopic systems cannot be described in the framework of classical physics. However, quantum effects may be important even for macroscopic bodies. From a certain point of view the main object in quantum mechanics is probability amplitude because it expresses the principal difference between quantum and classical theory. Quantum mechanics differs in that not probabilities but probability amplitudes should be summed up for a quantum system. The uncertainty principle is good for expressing features of instantaneous quantum measurements but it is inconvenient for continuous measurements. The variance of outputs of a precise measurement may be considered to be quantum measurement noise.
Introduction A very general concept of a quantum trajectory would be the path taken by the state of a quantum system over time. This state could be conditioned upon measurement results, as we considered in Chapter 1. This is the sort of quantum trajectory we are most interested in, and it is generally stochastic in nature. In ordinary use, the word trajectory usually implies a path that is also continuous in time. This idea is not always applicable to quantum systems, but we can maintain its essence by defining a quantum trajectory as the path taken by the conditional state of a quantum system for which the unconditioned system state evolves continuously. As explained in Chapter 1, the unconditioned state is that obtained by averaging over the random measurement results which condition the system. With this motivation, we begin in Section 4.2 by deriving the simplest sort of quantum trajectory, which involves jumps (that is, discontinuous conditioned evolution). In the process we will reproduce Lindblad's general form for continuous Markovian quantum evolution as presented in Section 3.6. In Section 4.3 we relate these quantum jumps to photon-counting measurements on the bath for the model introduced in Section 3.11, and also derive correlation functions for these measurement records. In Section 4.4 we consider the addition of a coherent field (the ‘local oscillator’) to the output before detection. In the limit of a strong local oscillator this is called homodyne detection, and is described by a continuous (diffusive) quantum trajectory.
In this chapter we shall deal with the quantum dynamics of an open system. By open quantum system we mean here a system which interacts with an environment (see Sec. 9.4): since we are not interested in the dynamics of the environment, we shall have to describe the evolution of the system in some “effective” way. In particular, if we consider only the evolution of the system, it will be non-unitary, and this will represent the subject of this chapter. As we know, Hamiltonian quantum dynamics is unitary and changes pure states into pure states. On the other hand, non-unitary dynamics changes an initially pure state into a mixture, which must be described by a density matrix (see Ch. 5). In the case of macroscopic systems, the coupling with the environment may be arbitrarily reduced and therefore its influence can be made correspondingly small (see Sec. 1.1). Microscopic systems, however, always couple to the environment and this coupling cannot be considered negligible. This is the reason why the quantum theory of open systems is one of the most important and fundamental chapters of quantum mechanics that, though born in quantum optics, has many implications in almost all fields of physics. The present chapter can be seen as a further development of the measurement theory (see Ch. 9), as open systems manifest a decoherent dynamics (see in particular Sec. 9.4).
This chapter investigates quantum continuous measurements from the point of view of their group-theoretical structure. There are two sources of a group-theoretical structure. One is that evolution develops continuously in time. Another source of group-theoretical structure is the equivalence of different measurement outputs. This equivalence can be described by the transverse group transforming alternative outputs into each other. To treat the group structure connected with time evolution, the notation has to be made more detailed, with explicit marking of the initial and final instants. A quantum system under continuous measurement exhibits evolution of a quite new type, having classical and quantum features simultaneously. The Galilean type semigroups are in fact very similar in the two cases, but the representations arising in describing the evolution must be more complicated in the presence of a measurement.
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.
We study the dynamics of a quantum system Γ with an environment Ξ made of N elementary quantum components. We aim at answering the following questions: can the evolution of Γ be characterized by some general features when N becomes very large, regardless of the specific form of its interaction with each and every component of Ξ ? In other terms: should we expect all quantum systems with a macroscopic environment to undergo a somehow similar evolution? And if yes, of what type? In order to answer these questions we use well established results from large- N quantum field theories, particularly referring to the conditions ensuring a large- N quantum model to be effectively described by a classical theory. We demonstrate that the fulfillment of these conditions, when properly imported into the framework of the open quantum systems dynamics, guarantees that the evolution of Γ is always of the same type of that expected if Ξ were a measuring apparatus, no matter the details of the actual interaction. On the other hand, such details are found to determine the specific basis w.r.t. which Γ undergoes the decoherence dictated by the dynamical description of the quantum measurement process. This result wears two hats: on the one hand it clarifies the physical origin of the formal statement that, under certain conditions, any channel from ρΓ to ρΞ takes the form of a measure-and-prepare map, as recently shown in Ref. \cite{BrandaoPH15}; on the other hand, it formalizes the qualitative argument that the reason why we do not observe state superpositions is the continual measurement performed by the environment.