Chapter 5 considers the case of diffusive continuous measurements, where the measurement outcomes and quantum state dynamics are analogous to a Brownian noise process. We motivate this type of measurement by considering the example of a double quantum dot system being measured by a quantum point contact. The intrinsic shot noise of the measurement naturally brings about an effective time-continuous measurement. A second example of a superconducting circuit made from Josephson junctions readout with a microwave frequency electromagnetic wave is also discussed in detail. The mathematics of quantum trajectory theory is then pedagogically built up, resulting in the stochastic Schrodinger equation, the stochastic master equation, and the stochastic path integral. We also discuss experimental data and its comparison with this theoretical formalism. These experiments allow us to peer into the inner workings of wavefunction collapse, giving an empirical handle on the many philosophical issues that arise in quantum measurement.
Chapter 5 considers the case of diffusive continuous measurements, where the measurement outcomes and quantum state dynamics are analogous to a Brownian noise process. We motivate this type of measurement by considering the example of a double quantum dot system being measured by a quantum point contact. The intrinsic shot noise of the measurement naturally brings about an effective time-continuous measurement. A second example of a superconducting circuit made from Josephson junctions readout with a microwave frequency electromagnetic wave is also discussed in detail. The mathematics of quantum trajectory theory is then pedagogically built up, resulting in the stochastic Schrodinger equation, the stochastic master equation, and the stochastic path integral. We also discuss experimental data and its comparison with this theoretical formalism. These experiments allow us to peer into the inner workings of wavefunction collapse, giving an empirical handle on the many philosophical issues that arise in quantum measurement.
In the last chapter, we dealt with the wide class of physical operations that preserve the Gaussian character of the initial state, treating them as discrete transformations. Here, we describe what could loosely be considered the continuous-time version of such transformations. This chapter is intended to provide the reader with the tools to describe a wide range of notable quantum dynamics, with a very broad applicative scope. After a note on the action of linear and quadratic Hamiltonians, whose description follows directly from the theory of Gaussian states presented in Chapter 3 but was not explicitly clarified so far, we shall address systems linearly coupled to a reservoir of modes that are delta-correlated in time (a situation commonly referred to as ‘white noise’), and thus give rise to Markovian dynamics which we shall refer to as ‘diffusive’ for reasons that will be made clear in due time. Our story will commence with a derivation of the Gaussian dynamics resulting from such an environmental coupling, in the form of dynamical equations for the first and second statistical moments. Besides their interest per se , the latter will be instrumental in obtaining an unconventional derivation of the master equations that govern general, not necessarily Gaussian, quantum states under the same conditions. The corresponding quantum Langevin equations for the canonical operators in the Heisenberg picture will also be derived. Next, we will turn our attention to the study of the system's dynamics that result from the measurement of the environment. This is a topic with a longstanding tradition in quantum mechanics, grounded in the formalism of quantum stochastic calculus that emerged within the mathematical physics community as early as the seventies, and then re-introduced as the quantum trajectories approach in quantum optics almost twenty years later. The ‘weak’ POVMs resulting from observing a weakly coupled environment (as opposed to ‘strong’, projective measurements), carried out continuously in time (a prescription customarily subsumed by the term ‘monitoring’), leads to a stochastic dynamics for the system, where the randomness of the measurement outcomes is compactly and most elegantly represented by infinitesimal stochastic increments, giving rise to continuous but not differentiable quantum dynamics. Existing approaches to this subject tend to be quite impervious to the uninitiated. In the quantum optics literature, this is largely due to the technicalities involved in incorporating the discontinuous update of the quantum state prescribed by the von Neumann postulate. More mathematically oriented treatments have instead typically preferred an algebraic approach where the usual difficulty of describing measurements in the Heisenberg picture is compounded by the adoption of a rather abstract hierarchy of filters, in part inspired by classical control theory, to describe different possible updates of the quantum state. This state of affairs is further complicated by the added difficulties that come with the necessary introduction of some form of quantum stochastic calculus. Our approach to monitoring will therefore take a different route. Like in our derivation of diffusion and master equations, we shall keep restricting to linear couplings with a white noise environment, forfeiting full generality This situation is well modelled by the so called input-output formalism, whose physical origin will be discussed in detail in Chapter 9 , which approximates reliably, among other instances, the interaction between internal resonating modes and electromagnetic radiation leaking in and out of an optical cavity. First, we shall introduce stochastic Wiener increments and the Itô rule, which will arise naturally from the identification of the continuum field's modes subject to monitoring. We shall deliberately keep our arsenal of stochastic methods to these bare essentials, sacrificing to some extent both depth and generality, but still providing the reader with a powerful and seminal demonstration of the formalism. We shall then restrict to Gaussian general-dyne detections and apply the exact formalism developed in the previous chapter to the continuous filtering of the system's state conditioned by such measurements' outcomes. Next, as in our derivation of the unconditional master equations, we shall use the Gaussian dynamics as a key to unlock the full stochastic master equations governing general states. Hence, we will build on the quantum theory of filtering in the Gaussian regime to derive dynamical equations for linear feedback control and also apply our theoretical tools to the enhanced production of squeezing. Finally, we will turn to the closely related paradigm of coherent feedback, where environmental modes do not undergo measurements but are fed back to interact coherently with the system, see in detail how it modifies Gaussian diffusive dynamics and assess its potential for control by reconsidering the production of squeezing assisted by interferometric coherent feedback loops.
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.
Abstract Quantum measurement has always been thought of as a formal, sudden projection that changes the system state instantly. However, in real life, the process of measurement involves a physical interaction that takes time to gather the information. The paper discusses the continuous quantum measurement that challenges the traditional instant measurement idea with the detector’s accumulated data continuously uninterruptedly. The paper starts by laying down the basic mathematics, which gradually leads to the modern Stochastic master equations and quantum trajectories through generalized measurements positive operator-valued measures and quantum instruments. The crucial experimental setups like circuit QED, trapped ions, and optomechanics are delineated where these notions have been put into practice. The significant ramifications of ongoing observation are then discussed, starting with the fundamental Zeno effect, measurement backaction, and continuing through the active steering of quantum states and the emergence of measurement-induced phase transitions in many-body systems. One of the main ideas of the paper is the attribute of measurement as a powerful control method which enables real-time feedback, state stabilization, and continuous quantum error correction. Finally, the paper points out the open problems that remain in the areas of many-body systems’ scalability and the coupling of machine learning with real-time quantum control.
A concise and self-contained derivation of hybrid quantum-classical dynamics is given in terms of Markovian master equations. Many previously known results are rederived and revised and some of them are completed or corrected. Using a method as simple as possible, our goal is a brief introduction to the state of the art of hybrid dynamics, with a limited discussion of the implications for foundations and without discussion of further relevance in the measurement problem, quantum gravity, chemistry, numeric methods, etc. Hybrid dynamics is defined as a special case of composite quantum dynamics where the observables of one of the two subsystems are restricted to the commuting set of diagonal operators in a fixed basis. With this restriction, the derivation of hybrid dynamical equations is clear conceptually and simple technically. Jump and diffusive dynamics follow in the form of hybrid master equations. Their stochastic interpretation (called unravelings) is derived. We discuss gauge-type ambiguities, problems of uniqueness, and covariance of the diffusive master equation. Also conditions of minimum noise and of monitoring the quantum trajectory are derived. We conclude that the hybrid formalism is equivalent to the standard Markovian theory of time-continuous quantum measurement (monitoring) on the one hand and is a motivating alternative formalism on the other.
This book adopts a novel, physics-first approach to quantum measurement, using physical experiments as the basis to describe the underlying mathematical formalism. Topics covered include weak measurements, quantum measurement reversal, quantum trajectories and the stochastic path integral formalism. The theory of quantum measurement is also covered in detail, including discussion of how it can be tested and demonstrated in a laboratory: how to build quantum-limited amplifiers, fundamental noise limits imposed on measurement by quantum mechanics, and the design of superconducting circuits. This text is an excellent introduction for students with a basic understanding of quantum mechanics wanting to learn more about measurement theory, and the inclusion of a wide selection of end-of-chapter exercises make this book ideal for emerging courses on the topic. Key chapters introducing the foundations of quantum computing and the history of measurement theory are equally accessible to a broader, less specialised audience.
This tutorial note develops a self-contained engineering-oriented model that connects thecontinuous dynamics of a two-level quantum system to a classical binary readout. The purpose ispedagogical rather than foundational: no new interpretation of quantum mechanics, measurementpostulate, or quantum analog-to-digital conversion protocol is proposed. Instead, establishedelements of open-system dynamics, generalized measurement, stochastic master equations, pointer-state selection, and threshold detection are organized into a single chain that can be simulatedwith conventional state-space tools.A qubit driven by a Hamiltonian H = ℏ(Ωσx + ∆σz )/2 is first described by a Lindbladequation and by the corresponding Bloch equations. Repeated nonselective partial measurementsare represented by a completely positive trace-preserving dephasing channel, from which the exactdiscrete-time coherence decay rate Γϕ = −fs ln (1 − ϵ) is obtained. The distinction between thisensemble-averaged channel and a single experimental realization is then made explicit througha diffusive stochastic master equation. In that conditional description, the detector producesa continuous noisy record; integration over a finite acquisition time and comparison with athreshold yields the final Boolean symbol. The resulting engineering chain is therefore not “wavefunction to bit,” but rather qubit state → detector statistics → analog record → integration →binary decision.Three time scales are emphasized: coherent dynamics, measurement-induced dephasing, andreadout integration. A Nyquist-like sampling inequality is retained only as a temporal-resolutionguideline for ensemble-reconstructed expectation values, not as a sufficient theorem for single-shotquantum-state reconstruction. Finally, the unconditional Bloch equations are mapped ontoan analog state-space macromodel implementable in LTspice. The note includes a correctedreference netlist and a reproducible validation protocol comparing the discrete channel, thecontinuous master equation, averaged stochastic trajectories, and the SPICE realization.