Information In Direct And Indirect Quantum Measurements

Miguel Ballesteros, Martin Fraas, Jürg Martin Fröhlich +1 more

Long sequences of successive direct (projective) measurements or observations of a few "uninteresting" physical quantities of a quantum system may reveal indirect, but precise and unambiguous information on the values of some very "interesting" observables of the system. In this paper, the mathematics underlying this claim is developed; i.e., we attempt to contribute to a mathematical theory of indirect and, in particular, non-demolition measurements in quantum mechanics. Our attempt leads us to make novel uses of classical notions and results of probability theory, such as the "algebra of functions measurable at infinity", the Central Limit Theorem, results concerning relative entropy and its role in the theory of large deviations, etc.

Eduard Prugovečki

Dan Cristian Marinescu, Gabriela M. Marinescu

Miguel Ballesteros, Martin Fraas, Jürg Martin Fröhlich +1 more

The problem of reconstructing information on a physical system from data acquired in long sequences of direct (projective) measurements of some simple physical quantities - histories - is analyzed within quantum mechanics; that is, the quantum theory of indirect measurements, and, in particular, of non-demolition measurements is studied. It is shown that indirect measurements of time-independent features of physical systems can be described in terms of quantum-mechanical operators belonging to an algebra of asymptotic observables. Our proof involves associating a natural measure space with certain sets of histories of a system and showing that quantum-mechanical states of the system determine probability measures on this space. Our main result then says that functions on that space of histories measurable at infinity (i.e., functions that only depend on the tails of histories) correspond to operators in the algebra of asymptotic observables.

K. -E. Hellwig

Michel Bauer, Denis Bernard

Motivated by recent experiments on quantum trapped fields, we give a rigorous proof that repeated indirect quantum nondemolition (QND) measurements converge to the collapse of the wave function as predicted by the postulates of quantum mechanics for direct measurements. We also relate the rate of convergence toward the collapsed wave function to the relative entropy of each indirect measurement, a result which makes contact with information theory.