This chapter reviews the main points in the theory of instantaneous measurements. A description of continuous measurements as limits of series of instantaneous measurements turns out to be very cumbersome and in fact inadequate. A characteristic feature of quantum systems is that measurements in them unavoidably affect their dynamics. The well known manifestation of this back reaction is the uncertainty principle, discussion of which can be found in any textbook on quantum mechanics. The operator plays the role of a propagator for a quantum system under continuous measurement. The quantum Zeno effect is an example of a continuous measurement giving a trivial result. However, this is a consequence of the very special character of the measurement procedure described by a discrete set of orthogonal projectors each of which is a bounded operator. The description of approximate measurements was derived for the case when precise measurement is a discrete-spectrum measurement.
This chapter reviews the main points in the theory of instantaneous measurements. A description of continuous measurements as limits of series of instantaneous measurements turns out to be very cumbersome and in fact inadequate. A characteristic feature of quantum systems is that measurements in them unavoidably affect their dynamics. The well known manifestation of this back reaction is the uncertainty principle, discussion of which can be found in any textbook on quantum mechanics. The operator plays the role of a propagator for a quantum system under continuous measurement. The quantum Zeno effect is an example of a continuous measurement giving a trivial result. However, this is a consequence of the very special character of the measurement procedure described by a discrete set of orthogonal projectors each of which is a bounded operator. The description of approximate measurements was derived for the case when precise measurement is a discrete-spectrum 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.
Description: This work investigates a fundamental but often overlooked limitation of physical measurement: the finite duration required to perform any measurement. We demonstrate that the impossibility of instantaneous measurement imposes intrinsic limits on the assignment of exact numerical values to physical quantities, even within purely classical systems. By developing a unified temporal framework, the paper shows that classical dynamical uncertainty arising from system evolution during the measurement interval shares a common origin with quantum measurement back-action. A generalized uncertainty relation is derived that incorporates both classical dynamical effects and quantum measurement noise, revealing a temperature-dependent crossover between classical and quantum regimes. The theory predicts a measurable classical–quantum transition in back-action noise and proposes a concrete experimental test using cavity optomechanical systems. Practical implications for precision metrology, quantum-enhanced sensing, and the operational interpretation of physical quantities are discussed. These results suggest that measurement precision is inherently temporal rather than instantaneous, providing new insight into the foundations of measurement in both classical and quantum physics. Keywords:measurement theory; quantum measurement; classical dynamics; uncertainty relations; back-action; optomechanics; metrology; temporal averaging
Discrete and continuous measurements All the examples of measurements so far discussed have in common the fact that the experimental output is either a single number or a finite set of numbers. However, in real measurements the output is often a record of some continuous function of time, from which one can get an understanding of the behavior of the measured quantity during some time interval. This type of measurement is called continuous. Correspondingly, measurements that give a discrete set of numbers can be called discrete. Let us note that measurements are sometimes called continuous if they are made not instantaneously, but over a finite duration of time. Evidently, this is not a reasonable viewpoint; all real measurements last for a finite time. The dividing line between continuous and discrete measurements is in fact the character of the output signal: does it give information about the measured quantity only at some chosen moment of time, or does it permit a continuous monitoring of the quantity's time evolution. The necessity to develop the quantum theory of continuous measurements arose in the 1960s, when devices for making continuous measurements were developed with sensitivities close to the quantum level (masers, parametric amplifiers, optical heterodyne receivers, …).
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 chapter presents the measurement problem from the point of view of the thermal interpretation of quantum physics. Everything physicists measure is measured in a thermal environment, for which statistical thermodynamics is relevant. The thermal interpretation agrees with how one interprets measurements in thermodynamics, the macroscopic part of quantum physics, derived via statistical mechanics. By its very construction, the thermal interpretation naturally matches the classical properties of our quantum world: The thermal interpretation assigns states-and a realistic interpretation for them-to individual quantum systems, in a way that large quantum systems are naturally described by classical observables. Since measurement devices are large quantum systems, this is the key to solving the measurement problem. We discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. Since quantum physics makes many deterministic predictions, for example regarding observed spectra, but also assertions about probabilities, we distinguish deterministic and statistical measurements. To clarify the meaning of the concept of measurement, we postulate (in Section 11.1) a measurement principle that defines what it means in the thermal interpretation to measure a quantity with a specified uncertainty. This turns the measurement problem from a philosophical riddle into a scientific problem in the domain of quantum statistical mechanics, namely how the quantum dynamics correlates macroscopic readings from an instrument with properties of the state of a measured microscopic system. The essence of scientific practice is the reproducibility of measurements, discussed in Section 11.2. Unlike in traditional interpretations, single, nonreproducible observations do not count as measurements, since this would violate the reproducibility of measurements-the essence of scientific practice. As a consequence, the measurement of a Hermitian quantity A is regarded as giving an uncertain value approximating the q-expectation {A} rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of A. This difference is most conspicuous in the interpretation of single discrete microscopic events. Except in very special circumstances, these are not reproducible. Thus, they have no scientific value in themselves and do not constitute measurement results. Scientific value is, however, in ensembles of such observations, which result in approximate measurements of q-probabilities and q-expectations. Sections 11.3 and 11.4 distinguish deterministic and statistical measurements, depending on whether a single observation is reproducible, and discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. How to measure the q-probabilities of events, described in terms of POVMs, is discussed in Section 11.5. Section 11.6 poses the measurement problem as a problem of the statistical mechanics of measurement devices, and discuss recent work by Allahverdyan, Balian, and Nieuwenhuizen on its resolution. Section 11.7 discusses the problem how, in the thermal interpretation, discrete events can arise, although qexpectations are by their very nature continuous. This is resolved by pointing to work by Breuer and Petruccione on piecewise deterministic processes (PDP). Section 11.8 sheds additional light on the origin of discrete events by an analysis of the role dissipative bistability plays in obtaining discreteness in classical physics.
This chapter focuses on the notions of classical instruments and of classical measurements. It reviews the theories, based on the concept of macroscopic properties, that is, on the essential complexity of the measuring apparatus. One of the most distinctive features of quantum mechanics is the fact that it often assigns precise structures—or discrete "choices" among possible different structures—to the physical systems it describes. Hisenberg's uncertainty relations bear on the dispersions of ensembles as a whole, whereas the limitations that serve to define "classical measurements" essentially concern our possibilities of actually performing measurements in different elements of the ensemble. Thus from the uncertainty relations alone one can certainly not derive the conclusion that nonclassical measurements are impossible. The chapter considers the restricted set of observables. It is convenient to use the word "macrostate" to designate the whole set of microstates on which all the classical observables forming a definite complete set have definite values.