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Modeling Generalized Measurements with Stochastic Processes

Martin Varbanov, Todd A. Brun

Published Jan 17, 2007• 0 citations
OpenAlex

Abstract

One of the abstract concepts of measurement in quantum information theory is the \\textit{generalized measurement}, a special subclass of which is the \\textit{positive-operator value measurements}. A different paradigm of measurement comes from Quantum Optics, when one considers measurements continuous in time. Their natural description involves the concept of \\textit{a stochastic process}--a time-dependent random variable. In this paper we construct a \\textit{continuous} stochastic process to simulate any generalized measurement--in the long time limit it gives the same outcomes with the right probabilities as the measurement being simulated. This stochastic evolution takes the form of a random walk in a space of measurement parameters, and the continuous measurement process must in general take into account the outcomes of earlier measurements.

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Decomposing generalized measurements into continuous stochastic processes

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•Jan 25, 2007•0 citations
One of the abstract concepts of measurement in quantum information theory is the \textit{generalized measurement}, a special subclass of which is the \textit{positive-operator value measurements}. A different paradigm of measurement comes from Quantum Optics, when one considers measurements continuous in time. Their natural description involves the concept of \textit{a stochastic process}--a time-dependent random variable. In this paper we construct a \textit{continuous} stochastic process to simulate any generalized measurement--in the long time limit it gives the same outcomes with the right probabilities as the measurement being simulated. This stochastic evolution takes the form of a random walk in a space of measurement parameters, and the continuous measurement process must in general take into account the outcomes of earlier measurements.
One of the broadest concepts of measurement in quantum theory is the generalized measurement. Another paradigm of measurement---arising naturally in quantum optics, among other fields---is that of continuous-time measurements, which can be seen as the limit of a consecutive sequence of weak measurements. They are naturally described in terms of stochastic processes, or time-dependent random variables. We show that any generalized measurement can be decomposed as a sequence of weak measurements with a mathematical limit as a continuous stochastic process. We give an explicit construction for any generalized measurement, and prove that the resulting continuous evolution, in the long-time limit, collapses the state of the quantum system to one of the final states generated by the generalized measurement, being decomposed, with the correct probabilities. A prominent feature of the construction is the presence of a feedback mechanism---the instantaneous choice weak measurement at a given time depends on the outcomes of earlier measurements. For a generalized measurement with nnn outcomes, this information is captured by a real nnn-vector on an nnn-simplex, which obeys a simple classical stochastic evolution.
In the present paper we consider the problem of the description of an arbitrary generalized quantum measurement with outcomes in a measurable space. Analysing the unitary invariants of a separable statistical realization of a quantum instrument, we present the most general form of a possible integral representation of an instrument. We introduce the notion of a stochastic realization of an instrument and establish a one-to-one correspondence between the class of unitarily and phase equivalent separable statistical realizations and the equivalence class of stochastic realizations of an instrument. We further single out the invariant class of unitarily and phase equivalent separable statistical realizations for which the integral representation of an instrument is the same for all statistical realizations from this class and is wholly determined by the invariants of this class. We call the special form of this integral representation the quantum stochastic representation of an instrument. We show that the description of a generalized direct quantum measurement can be considered in the frame of a new general approach based on the notion of a family of quantum stochastic evolution operators satisfying the orthonormality relation. This approach gives not only the complete statistical description of any generalized direct quantum measurement but the complete description in a Hilbert space of the stochastic behaviour of a quantum system under a generalized direct measurement in the sense of specifying the probabilistic transition law governing the change from the initial state to a final one under a single measurement. Under this approach a unitary evolution of an isolated quantum system is included as a special case. In the frame of the proposed approach, which we call quantum stochastic approach, all possible schemes of measurements upon a quantum system can be considered. In the case of repeated or continuous in time measurements the quantum stochastic approach allows to define, in the most general case, the notion of the family of posterior pure state trajectories (quantum trajectories in discrete or continuous time) in the Hilbert space of a quantum system and to give their probabilistic treatment.
The time-dependent measurement probabilities for the simple two-state quantum oscillator seem to invite description as a classical two-state stochastic process. It has been shown that such a description cannot be achieved using a Markov process. Constructing a more general non-Markov process is a challenging task, requiring as it does the proper generalizations of the Markovian Chapman-Kolmogorov and master equations. Here we describe those non-Markovian generalizations in some detail, and we then apply them to the two-state quantum oscillator. We devise two non-Markovian processes that correctly model the measurement statistics of the oscillator, we clarify a third modeling process that was proposed earlier by others, and we exhibit numerical simulations of all three processes. Our results illuminate some interesting though widely unappreciated points in the theory of non-Markovian stochastic processes. But since quantum theory does not tell us which one of these quite different modeling processes "really" describes the behavior of the oscillator, and also since none of these processes says anything about the dynamics of other (noncommuting) oscillator observables, we can see no justification for regarding any of these processes as being fundamentally descriptive of quantum dynamics. (c) 2001 American Institute of Physics.
The topics presented in this thesis have continuous quantum measurements and quantum walks at their core. The first topic being discussed centers around simulating a generalize measurement with a finite number of outcomes using a continuous measurement process with a continuous measurement history. We provide conditions under which it is possible to prove that such a process exists and that at long times it simulates faithfully the generalized measurement. We give the stochastic equations governing the feedback between the measurement history and the instantaneous weak measurements. The second topic examines a definition of hitting time for continuous-time quantum walks. A crucial component for such a definition is the use of weak measurements. Several methods using alternative but equivalent definitions of weak, continuous measurements are employed to derive a formula for the hitting time. The behavior of the thus defined hitting time is studied subsequently, in general and for specific graphs. The last topic explores continuous-time quantum walks on graphs with infinite tails. The equations for propagating and bound states are derived and the S-matrix is defined. Their properties, such as orthogonality of the propagating and bound states, unitarity of the S-matrix, are discussed. Formulas for the S-matrix under operations of cutting, adding or connecting tails are derived.
Operation-valued stochastic processes give a formalization of the concept of continuous (in time) measurements in quantum mechanics. In this article, a first stage M of a measuring apparatus coupled to the system S is explicitly introduced, and continuous measurement of some observables of M is considered (one can speak of an indirect continuous measurement on S). When the degrees of freedom of the measuring apparatus M are eliminated and the weak coupling limit is taken, it is shown that an operation-valued stochastic process describing a direct continuous observation of the system S is obtained.
Abstract This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned on the measurement record, is given by a piecewise deterministic process in Hilbert space involving continuous time-evolution periods which are broken by randomly occurring, sudden quantum jumps. This so-called unravelling of the quantum master equation in the form of a stochastic process is based on a close relation between quantum dynamical semigroups and piecewise deterministic processes. The general theory is illustrated by means of a number of examples, such as direct, homodyne, and heterodyne photodetection. The chapter also treats the corresponding description of continuous measurements through stochastic Schr ö dinger equations and stochastic differential equations for the density matrix, as well as the stochastic representation of multitime correlations functions.