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arXiv:quant-ph/0401114 · Quantum Information and Computation

Instrumental processes, entropies, information in quantum continual measurements

Alberto Barchielli, G. Lupieri

Published Dec 1, 2004• 16 citations
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Abstract

In this paper we will give a short presentation of the quantum L\'evy-Khinchin formula and of the formulation of quantum continual measurements based on stochastic differential equations, matters which we had the pleasure to work on in collaboration with Prof.\ Holevo. Then we will begin the study of various entropies and relative entropies, which seem to be promising quantities for measuring the information content of the continual measurement under consideration and for analysing its asymptotic behaviour.

DOI
10.26421/qic4.6-7-3

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In this paper we will give a short presentation of the quantum L'evy-Khinchin formula and of the formulation of quantum continual measurements based on stochastic differential equations, matters which we had the pleasure to work on in collaboration with Prof.\ Holevo. Then we will begin the study of various entropies and relative entropies, which seem to be promising quantities for measuring the information content of the continual measurement under consideration and for analysing its asymptotic behaviour.
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.
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure [ 1 , 2 , 3 , 4 , 5 , 6 ], arisen inside the operational approach [ 1 , 7 ] to quantum mechanics, by using functional integrals [ 8 , 9 , 10 ], by using quantum stochastic differential equations [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 ] and by using classical stochastic differential equations (SDE’s) [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 4 , 19 , 20 , 21 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 6 ]. Various types of SDE’s are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both.
The theory of measurements continuous in time in quantum mechanics (quantum continual measurements) has been formulated by using the notions of instrument and positive operator valued measure, functional integrals, quantum stochastic differential equations and classical stochastic differential equations (SDE's). Various types of SDE's are involved, and precisely linear and non linear equations for vectors in Hilbert spaces and for trace-class operators. All such equations contain either a diffusive part, or a jump one, or both. In this paper we introduce a class of linear SDE's for trace-class operators, relevant to the theory of continual measurements, and we recall how such SDE's are related to instruments and master equations and, so, to the general formulation of quantum mechanics. We do not present the Hilbert space formulation of such SDE's and we make some mathematical simplifications: no time dependence is introduced into the coefficients and only bounded operators on the Hilbert space of the quantum system are considered. Then we introduce the notion of a posteriori state and the non linear SDE satisfied by such states and we give conditions from which such equation is assured to preserve pure states and to send any mixed state into a pure one for large times. Finally we review the known results about the existence and uniqueness of invariant measures in the purely diffusive case and we give some concrete examples of physical systems.
In the past few years there has been an increasing interest in a certain class of stochastic differential equations (SDE’s) in Hilbert spaces for applications in quantum mechanics (measurements continuous in time [1-5]) and in quantum optics (photon-detection theory and numerical simulations of master equations [6-10]). Part of the mathematical theory of these equations has been developed in [11], where also “structural properties” of this class of equations have been studied. In the paper [11] it has been shown that such equations are connected with certain semigroups of linear operators and the form of the generator of semigroups related to such SDE’s has been established.
While a positive operator valued measure gives the probabilities in a quantum measurement, an instrument gives both the probabilities and the a posteriori states. By interpreting the instrument as a quantum channel and by using the monotonicity theorem for relative entropies many bounds on the classical information extracted in a quantum measurement are obtained in a unified manner. In particular, it is shown that such bounds can all be stated as inequalities between mutual entropies. This approach based on channels gives rise to a unified picture of known and new bounds on the classical information (Holevo's, Shumacher-Westmoreland-Wootters', Hall's, Scutaru's bounds, a new upper bound and a new lower one). Some examples clarify the mutual relationships among the various bounds.