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.
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.
In this thesis we study the problem of continuous measurement on an open quantum system.The starting point of our analysis is the time-evolution of an open quantum system where the evolution of the environment is not explicitly described, i.e. traced out.More explicitly we restrict to time-evolutions having the structure of a semigroup or evolution system.That is, for time-points 0 ≤ r ≤ s ≤ t there exists a two-parameter family of completely-positive maps E * (t, s), such that E * (t, s)E * (s, r) = E * (t, r) and the systems state evolved according to ρ(tPhysically this means that in our Ansatz the interaction between system and environment is already fixed and can only be weak.The main goal of this thesis is now to describe, mathematically rigorous, all continuous measurements compatible with the given evolution.A measurement is compatible with the given time-evolution, if the evolution does not change when measurement results are ignored.In other words the measurement is indirect and does not further disturb the systems evolution.Our results can be interpreted as a rigorous construction of continuous matrix product states, and are related to the Hudson-Parthasarathy quantum stochastic calculus.The analysis in this thesis separates into two parts.In the first we study the class of quantum systems having an evolution system structure as described above.Precisely we study the theory of minimal solutions to a Cauchy equation with a generator of Gorini-Kossakowski-Sudarshan-Lindblad type.We do this for the case of time-dependent and unbounded generators.This analysis culminates in the description of a counterexample due to Holevo of a semigroup which is not the minimal solution to such a standard Lindblad equation.In the second part we construct a description of all measurements compatible with a given evolution and thus achieve our main goal.The analysis starts with the discrete-time version of the problem, i.e.where the time-points r, s, t in the above description of the evolution are chosen from a finite set.This discrete-time problem is completely solved by Stinespring dilation theory.We then construct our solution of the continuous-time case as a refinement limit over arbitrary discretizations of the evolution.Hence, our construction gives a continuous-time analogue of the Stinespring dilation for semigroups and evolution systems.It can be interpreted as a complete quantum description of the information the system radiates into the environment, i.e. it yields a quantum state of the environment.We complement the description of the state with a description of the observables for continuous measurements and the calculation expectation values.
Preface INTRODUCTION TO CONTINUOUS QUANTUM MEASUREMENTS Quantum and Classical Systems Amplitudes and Alternatives Paths and Continuous Measurements The Action Uncertainty Principle INSTANTANEOUS AND SEQUENTIAL MEASUREMENTS Measurement of a Quantum System Quantum Zeno Paradox Approximate and Sequential Measurements TECHNIQUE OF PATH INTEGRALS Propagators and Path Integrals Definition of a Path Integral The Path Integral for an Oscillator Gaussian Path Integrals CONTINUOUS MEASUREMENT AND EVOLUTION OF THE MEASURED SYSTEM The Measurement Amplitude Effective Lagrangian Evolution of a Quantum System Subject to Continuous Measurement Scattering by the Measuring Medium CONTINUOUS MEASUREMENTS OF OSCILLATORS Position Monitoring (Path measurement) Estimation of Force Acting on an Oscillator Spectral Measurements of an Oscillator Evolution of a Harmonic Oscillator Subject to Spectral Measurement Measurement of a Path of a Nonlinear Oscillator CONTINUOUS QUANTUM NONDEMOLITION MEASUREMENTS Quantum Nondemolition Measurements Monitoring of Linear Momentum Velocity versus Momentum QND Measurements of Oscillators MEASUREMENT OF AN ELECTROMAGNETIC FIELD Exposition of the Method Measurement Amplitude Analysis of the Results Another Definition of the Field Strength Explicit Model of a Meter TIME IN QUANTUM COSMOLOGY Statement of the Problem Measurements in Quantum Cosmology Minisuperspace without Measurement Minisuperspace under Measurement Analysis of the results Concluding Remarks THE ACTION UNCERTAINTY PRINCIPLE Uncertainties in Continuous Measurements Classical Regime of Measurements Quantum Regime and AUP AUP and Equations of Motion Examples from Quantum Mechanics Quantum Nondemolition Measurements AUP in Quantum Gravity Concluding Remarks GROUP-THEORETICAL STRUCTURE OF QUANTUM CONTINUOUS MEASUREMENTS Evolution under Measurement Transverse Group for Alternatives Longitudinal Semigroup for Evolution Unification of Transverse and Longitudinal Structures Concluding Remarks PATHS AND MEASUREMENTS: FURTHER DEVELOPMENT Quantum and Classical Features in Continuous Measurement The Two-Slit Experiment Wave-Particle Dualism Conceptual Problems References Index
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.
The Unitary Dynamical Group in Schrödinger's Picture Up to now we have been concerned with the description of a physical system at a fixed instant of time. The time evolution of quantum systems constitutes a many-faced, wide issue which, however, is somewhat tangential to the main aim of this volume. We shall touch upon it only briefly, restricting ourselves to the more familiar aspects of the problem. We assume, to begin with, that the physical system has no superselection rules, so that states and physical quantities correspond one-to-one to density operators and self-adjoint operators, respectively, on the Hilbert space. To describe the dynamical evolution of the system one has to specify the way the representatives of states and physical quantities evolve in, which is fixed in time, since the physical system is supposed to preserve its identity. There is an old tenet according to which there are two kinds of time evolution: the discontinuous, nondeterministic evolution undergone by the quantum system when a measurement is made on it, and the continuous, deterministic evolution caused by the interaction with external forces or with other quantum systems. The first kind of evolution will be dealt with in Chapter 8, where we shall find it less alarming than is often claimed.
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, …).
This chapter is concerned with the measure of quantum states . This requires one to introduce the subtle notion of quantum measurement , an operation that has no counterpart in the classical domain. To this effect, we first need to develop some new tools, starting with Dirac notation , a formalism that is not only very elegant but is relatively simple to handle. The introduction of Dirac notation makes it possible to become familiar with the inner product for quantum states, as well as different properties for operators and states concerning projection , change of basis , unitary transformations , matrix elements , similarity transformations , eigenvalues and eigenstates , spectral decomposition and diagonal representation , matrix trace and density operator or matrix . The concept of density matrix makes it possible to provide a very first and brief hint of the analog of Shannon's entropy in the quantum world, referred to as von Neumann's entropy , to be further developed in Chapter 21. Once we have all the required tools, we can focus on quantum measurement and analyze three different types referred to as basis-state measurements , projection or von Neumann measurements , and POVM measurements. In particular, POVM measurements are shown to possess a remarkable property of unambiguous quantum state discrimination (UQSD), after which it is possible to derive “absolutely certain” information from unknown system states. The more complex case of quantum measurements in composite systems described by joint or tensor states is then considered.