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.
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.
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.
If H1 and H2 are finite-dimensional Hilbert spaces, a channel from H1 to H2 is a completely positive, linear map I that takes the set of states S(H1) for H1 to the set of states S(H2) for H2. Corresponding to I there is a unique dual map I∗ from the set of effects E(H2) for H2 to the set of effects E(H1) for H1. We call I∗(b) the effect b conditioned by I and the set Ic=I∗(E(H2)) the conditioned set of I. We point out that Ic is a convex subeffect algebra of the effect algebra E(H1). We extend this definition to the conditioning I∗(B) for an observable B on H2 and say that an observable A is in Ic if A=I∗(B) for some observable B. We show that Ic is closed under post-processing and taking parts. We also define the conditioning of instruments by channels. These concepts are illustrated using examples of Holevo instruments and channels. We next discuss measurement models and their corresponding observables and instruments. We show that calculations can be simplified by employing Kraus and Holevo separable channels. Such channels allow one to separate the components of a tensor product.
It is not surprising that our language should be incapable of describing processes occurring within atoms, for, as has been remarked, it was invented to describe the experiences of daily life, … Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme–the quantum theory–which seems entirely adequate for the treatment of atomic processes. Werner Heisenberg, The Physical Principles of the Quantum Theory (1930) In this chapter we introduce the mathematical structure of quantum theory, for a finite number of systems and finite dimension, starting from three simple Hilbert-space postulates, and based on the intuitive notions of state, transformation, and effect, within a causal context. Such notions will be formalized in a general non-causal context in Chapter 3, which is devoted to the informational framework. We will assume the reader has a basic knowledge of matrix analysis and of operators over Hilbert spaces. Comments about the extension to infinite dimensions, continuous outcome-spaces, and infinitely many systems can be found in the notes at the end of the chapter. From the three simple Hilbert-space postulates, the present chapter extracts six principles, which will become the postulates for quantum theory in the axiomatic derivation presented in Part III of the book. In addition, we will take the opportunity to present and explore many relevant features of the theory – including its convex and causal structure, the Choi–Jamiołkowski isomorphism, and relevant notions, such as those of POVM, complete positivity of maps, quantum operation and channels, entangled states, state discrimination, tomography of states, and transformations. The purification principle will open the stage to the dilation theorems of the theory of quantum open systems, along with other consequences, such as the existence of faithful and steering states, the theorem of no information without disturbance, quantum teleportation, and more. We will review the quantum no-cloning and no-programming theorems, and finally provide a thorough derivation of the theory of quantum channel inversion, which is the backbone of the general theory of quantum error correction in quantum information. Additional interesting topics will be classical information theory as a restriction of quantum information theory, and a critical reconsideration of the von Neumann postulate and the notion of observable.
Quantum systems range from the traditional microscopic ones like electron and atoms to macroscopic ones like superconductors. There are also quantum fields. This chapter sets out one general properties of these systems. For a classical particle, such a sequence does converge to give a value for the instantaneous velocity with arbitrary accuracy. For a quantum particle, the situation is entirely different. It is accepted that a quantum system, say an electron, has an objective existence. At the outset we assume that a measurement operation exists to produce a value of any given observable of the system. This assumption is built into the concept of observables which are physical quantities which can be measured to produce a value. Since a quantum particle in a general state may not have an objective value of a given observable independent of measurement it is useful to distinguish two kinds of values: possessed values and measured values.
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.
A new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it. – Max Planck, 1936 The only way quantum theory allows us to interact with quantum systems is via quantum processes. Thus, the only way we can extract information about the state of a quantum system is to apply some non-deterministic process and observe which of the branches happened. For that reason, many people refer to the act of applying certain quantum processes as measurement and sometimes even as observation . A consequence of this unfortunate terminology is that one of the most touted ‘strange’ features of quantum theory is often misleadingly described as follows: ‘The mere act of observing a quantum state changes it.’ ‘Misleading’, since the concept of measurement described in the previous paragraph is far from the passive concept of observation familiar from our macroscopic world, but is rather a non-trivial process that will almost always drastically affect the quantum state. So, the mysterious aspect of quantum theory is not that ‘observation’ alters the quantum state but, rather, that it is impossible, even in principle, to ‘observe’ a quantum system in the classical sense. Given this fundamental restriction on how we can extract information from a quantum system, what can we learn about that system by means of a quantum process? The answer is, in fact, very little! First, performing a particular measurement could, for the vast majority of quantum states, yield any outcome i with a non-zero probability. In that case, obtaining an outcome i doesn't tell us much about what the state of the system was. Second, we say ‘was’, because the measurement will moreover irreversibly change the state according to the outcome. So rather than revealing the state of a system, a quantum measurement typically erases that state from history! Nonetheless, these quantum measurements are crucial to quantum theory since they constitute the only interface between us, in our classical world, and the quantum world. The great insight of the quantum computing community is that the non-deterministic changes induced by these quantum measurements are not a nuisance, but rather an extremely useful resource.