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Journal of Mathematical Physics

Markovian limit for a reduced operation-valued stochastic process

Alberto Barchielli

Published Apr 1, 1987• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.527569

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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.
The physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process (OVSP). In this article, it is shown how the formalism of quantum stochastic calculus of Hudson and Parthasarathy allows, in a simple way, for constructing a large class of OVSP’s that in particular contains the quantum counting processes of Davies and Srinivas and continual ‘‘Gaussian’’ measurements. This result is obtained by means of a stochastic dilation of the OVSP’s: at the level of the enlarged system probabilities turn out to be expressed in terms of projection valued measures associated with certain time-dependent, commuting, self-adjoint operators.
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.
The process of measurement plays an important role in quantum theory. Measurements can be direct, if the physical system S interacts directly with the measurement apparatus M (as we have assumed until now), or indirect. In the latter case, the physical system first interacts with an ancillary system B , which may have a space of states that is very different from that of S , for instance much larger; after this interaction has finished, M is used to perform a measurement on B , without any direct interaction with A . Because S is then “protected” from any direct interaction with the measurement apparatus, the state of S is not necessarily strongly modified, and may even be only weakly affected. In both cases, the process implies entanglement between several physical systems. In this chapter, we study how this entanglement is created and used for measurements as well as the notion of weak and continuous measurements. These questions play an important role in several of the interpretations of quantum mechanics that we discuss in Chapter 10. Direct measurements The Von Neumann model of quantum measurement [4] provides a general frame for describing the process in terms of correlations appearing (or disappearing) in the state vector associated with the whole system S + M . In this model, the two systems S and M are initially described by a product state ∣ψ 0 〈 and interact during the time of measurement, so that they become entangled; they then reach a final state ∣ψ′〈 and do not interact anymore.
While the linearity of the Schr\"odinger equation and the superposition principle are fundamental to quantum mechanics, so are the backaction of measurements and the resulting nonlinearity. It is remarkable, therefore, that the wave equation of systems in continuous interaction with some reservoir, which may be a measuring device, can be cast into a linear form, even after the degrees of freedom of the reservoir have been eliminated. The superposition principle still holds for the stochastic wave function of the observed system and exact analytical solutions are possible in sufficiently simple cases. We discuss here the coupling to Markovian reservoirs appropriate for homodyne, heterodyne, and photon counting measurements. For these we present a derivation of the linear stochastic wave equation from first principles and analyze its physical content.