Continuous Measurements and Instruments

Kurt A. Jacobs

Alberto Barchielli

The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument, positive operator valued measure, etc., by using quantum stochastic differential equations and by using classical stochastic differential equations (SDE's) for vectors in Hilbert spaces or for trace-class operators. In the same times Ozawa made developments in the theory of instruments and introduced the related notions of a posteriori states and of information gain [1]. In this paper we introduce a simple class of SDE's relevant to the theory of continual measurements and we recall how they are related to instruments and a posteriori states and, so, to the general formulation of quantum mechanics. Then we introduce and use the notion of information gain and the other results of the paper [1] inside the theory of continual measurements. [1] M. Ozawa, On information gain by quantum measurements of continuous observables, J. Math. Phys. 27 (1986) 759-763.

Christopher S. Jackson

Measuring instruments, especially ones that observe continually over time, have a reality to them that is independent of the states that stimulate their senses. This is the Principle of Instrument Autonomy. Although the mathematical concept of an instrument implicitly embodies this principle, the conventional analysis of continual observation has become overwhelmingly focused on state evolution rather than on descriptions of instruments themselves. Because of this, it can be hard to appreciate that an instrument that observes for a finite amount of time has an evolution of its own, a stochastic evolution that precedes the Born rule and Schrödinger equation of the measured system. In this article, the two most established of the continually observing instruments, the Srinivas-Davies photodetector and the Goetsch-Graham-Wiseman heterodyne detector, are reviewed with an emphasis on the autonomous instrument evolution they define, made explicit by application of the recently introduced Kraus-operator distribution function. It is then pointed out how the heterodyne instrument evolution is a complete alternative to the original idea of energy quantization, where the usual ideas of \emph{temperature} and \emph{energy} of a \emph{state} are replaced by the \emph{time} and \emph{positivity} of the \emph{instrument}.

Miklós Rédei, A. Zartl, Erwin Schrödinger +9 more

Alexander Semenovich Holevo

Nina Megier, Walter T. Strunz, Kimmo Luoma

Quantum measurements and the associated state changes are properly described in the language of instruments.We investigate the properties of a time continuous family of instruments associated with the recently introduced class of general Gaussian non-Markovian stochastic Schrödinger equations.In this article we find that when the covariance matrix for the Gaussian noise satisfies a particular δ-function constraint, the measurement interpretation is possible for a class of models with self-adjoint coupling operator.This class contains, for example the spin-boson and quantum Brownian motion models with colored bath correlation functions.Remarkably, due to quantum memory effects the reduced state, in general, does not obey a closed form master equation while the unraveling has a time continuous measurement interpretation.