Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device. In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments , will be investigated in this chapter. There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process . A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements . Some other authors use the term indirect measurements . Three levels of description of measurements It is convenient to think of three levels of description when considering measurements.
Until now we have treated measurement apparatuses as devices taking quantum systems as their inputs and producing measurement outcomes as outputs. At this level of description, measurement apparatuses are fully described by observables, as discussed in Chapter 3. The measurement outcome distribution is determined by the particular input state and the observable describing the measurement device. In this chapter we explore a mathematical framework suitable for the description of measurement apparatuses for which, in addition to the measurement outcomes, the measured systems are available after the measurement is performed. In such cases we may try to obtain more information on the system by measuring some other observable, or, perhaps by filtering out some preferred output states we may use our measurement apparatus as a preparator. Clearly, in this kind of situation we need a description of measurement apparatuses that is more detailed than that of observables. Therefore, two more refined concepts, measurement models and instruments , will be investigated in this chapter. There is no customary name for a measurement model. As used here, it was first formalized in a general way by Ozawa [106], [107] who referred to the measurement process . A comprehensive general reference on quantum measurements is the monograph by Busch et al. [35], where measurement models are called premeasurements . Some other authors use the term indirect measurements . Three levels of description of measurements It is convenient to think of three levels of description when considering measurements.
In Chapters 2 and 3 we assumed a model according to which an experiment is split into two parts – preparation and measurement. This led us to the associated concepts of states and observables, respectively. On the basis of this picture we can think about two types of apparatus, which can be characterized abstractly as follows. Preparators are devices producing quantum states. No input is required, but a quantum output is produced. Measurements are input–output devices that accept a quantum system as their input and produce a classical output in the form of measurement outcome distributions. This setting is summarized in Figure 2.2. Furthermore, in subsection 3.5.2 we discussed coarse-graining relations; coarse-graining can be depicted as a box with a classical input and a classical output. This line of thought indicates that at least one type of input–output box is so far missing from our discussion – a device taking a quantum input and producing a quantum output. Hence, we say that • quantum channels are input–output devices transforming quantum states into quantum states. Each quantum channel can be placed between arbitrary preparation and measurement devices (Figure 4.1). Channels, and a slightly more general concept, that of operations , are the topics of this chapter. Transforming quantum systems In this section we discuss the mathematical definition of operations and channels. The notion of an operation was introduced by Haag and Kastler [70] within the framework of algebraic quantum field theory. Later Kraus [88], Lindblad [93] and Davies [51] discussed the requirement for operations to be completely positive.
The process of measurement plays an important role in quantum theory. Measurements can be direct, if the physical system S interacts directly with the measurement apparatus M (as we have assumed until now), or indirect. In the latter case, the physical system first interacts with an ancillary system B , which may have a space of states that is very different from that of S , for instance much larger; after this interaction has finished, M is used to perform a measurement on B , without any direct interaction with A . Because S is then “protected” from any direct interaction with the measurement apparatus, the state of S is not necessarily strongly modified, and may even be only weakly affected. In both cases, the process implies entanglement between several physical systems. In this chapter, we study how this entanglement is created and used for measurements as well as the notion of weak and continuous measurements. These questions play an important role in several of the interpretations of quantum mechanics that we discuss in Chapter 10. Direct measurements The Von Neumann model of quantum measurement [4] provides a general frame for describing the process in terms of correlations appearing (or disappearing) in the state vector associated with the whole system S + M . In this model, the two systems S and M are initially described by a product state ∣ψ 0 〈 and interact during the time of measurement, so that they become entangled; they then reach a final state ∣ψ′〈 and do not interact anymore.
A measurement model is a framework that describes a quantum measurement process. In this article we restrict attention to MMs on finite-dimensional Hilbert spaces. Suppose we want to measure an observable A whose outcomes Ax are represented by positive operators (effects) on a Hilbert Space H. We call H the base or object system. We interact H with a probe system on another Hilbert space K by means of a quantum channel. The probe system contains a probe (or meter or pointer) observable F whose outcomes Fx are measured by an apparatus that is frequently (but need not be) classical in practice. The MM protocol gives a method for determining the probability of an outcome Ax for any state of H in terms of the outcome Fx. The interaction channel usually entangles this state with an initial probe state of K that can be quite complicated. However, if the channel is nondisturbing in a sense that we describe, then the entanglement is considerably simplified. In this article, we give formulas for observables and instruments measured by nondisturbing MMs. We begin with a general discussion of nondisturbing operators relative to a quantum context. We present two examples that illustrate this theory in terms of unitary nondisturbing channels.
Measurements play an integral part in any scientific theory. Indeed, the observations revealing causal behaviour and symmetries in our world of experience lie at the heart of determining the truth or the falsity of scientific hypotheses. In this regard, quantum theory provides one of the most stringently tested and comprehensive description of the natural world. This thesis is an examination of the mathematical structures of quantum measurements with an emphasis on the measurement implementations of different operational devices: quantum observables, channels and instruments. The measurability of all the quantum properties a system bears is, in principle, guaranteed by quantum measurement theory. However, as physical processes, measurements are constrained by the laws of physics and it has been long known that certain conservation laws can inhibit the measurements of some of the quantum observables. In this work it is shown that such constraints stem from the fact that the measured observables inherit the symmetries of the measurement devices implementing them. A quantum measurement set-up can implement a multitude of operational devices as a function of the initial state of the apparatus. In other words, one can realize various maps – even incompatible ones – simply by controlling the apparatus’ state. Due its close resemblance with the classical analogue, this operation is often referred as quantum programming. In this thesis the capabilities and restrictions of quantum programming are explored using a measurement-theoretic approach. Moreover, quantitative methods are developed to address the mutual relations between the programming states of an arbitrary pair of observables or channels.
A detailed theory of quantum premeasurement dynamics is presented in which a unitary composite-system operator that contains the relevant object-measuring-instrument interaction brings about the final premeasurement state. It does not include collapse, and it does not consider the environment. It is assumed that a discrete degenerate or non-degenerate observable is measured. Premeasurement is defined by the calibration condition, which requires that every initially statistically sharp value of the measured observable has to be detected with statistical certainty by the measuring instrument. The entire theory is derived as a logical consequence of this definition using the standard quantum formalism. The study has a comprehensive coverage, hence the article is actually a topical review. Connection is made with results of other authors, particularly with basic works on premeasurement. The article is a conceptual review, not a historical one. General exact premeasurement is defined in 7 equivalent ways. Nondemolition premeasurement, defined by requiring preservation of any sharp value of the measured observable, is characterized in 10 equivalent ways. Overmeasurement, i. e., a process in which the observable is measured on account of being a function of a finer observable that is actually measured, is discussed. Disentangled premeasurement, in which, by definition, to each result corresponds only one pointer-observable state in the final composite-system state, is investigated. Ideal premeasurement, a special case of both nondemolition premeasurement and disentangled premeasurement, is defined, and its most important properties are discussed. Finally, disentangled and entangled premeasurements, in conjunction with nondemolition or demolition premeasurements, are used for classification of all premeasurements.
One of the main purposes of a physical theory is to describe and predict events observed in physical experiments. In this chapter we introduce the key physical concepts used in the description of quantum experiments and present their mathematical formalization as Hilbert space operators. Duality of states and effects Fundamental in quantum theory is the concept of a state, which is usually understood as a description of an ensemble of similarly prepared systems. An effect , on the other hand, is a measurement apparatus that produces either ‘yes’ or ‘no’ as an outcome. The concept of an effect was coined by Ludwig [96] and made popular by Kraus [89]. The duality of states and effects means, essentially, that when a state and an effect are specified we can calculate a probability distribution. This can be compared with the outcomes of real experiments, and it gives a physical meaning to the mathematical Hilbert space machinery. Basic statistical framework A basic situation in physics is the following: we have an object system under investigation, and we are trying to learn something about it by doing an experiment. As a result, a measurement outcome is registered. A statistical theory, such as quantum theory, does not in general predict which individual outcomes will occur in any particular measurement; it merely predicts their probabilities of occurrence. Hence, we take the output of an experiment (which consists of many measurement runs) to be a probability distribution on a set Ω of the possible measurement outcomes.