Paul Busch

Heisenberg's uncertainty principle

Heisenbergs uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accepted. The uncertainty principle is shown to appear in three manifestations, in the form of uncertainty relations: for the widths of the position and momentum distributions in any quantum state; for the inaccuracies of any joint measurement of these quantities; and for the inaccuracy of a measurement of one of the quantities and the ensuing disturbance in the distribution of the other quantity. Whilst conceptually distinct, these three kinds of uncertainty relations are shown to be closely related formally. Finally, we survey models and experimental implementations of joint measurements of position and momentum and comment briefly on the status of experimental tests of the uncertainty principle.

Quantum States and Generalized Observables: A Simple Proof of Gleason’s Theorem

A quantum state can be understood in a loose sense as a map that assigns a value to every observable. Formalizing this characterization of states in terms of generalized probability distributions on the set of effects, we obtain a simple proof of the result, analogous to Gleasons theorem, that any quantum state is given by a density operator. As a corollary we obtain a von Neumann-type argument against noncontextual hidden variables. It follows that on an individual interpretation of quantum mechanics the values of effects are appropriately understood as propensities.

Indeterminacy Relations and Simultaneous Measurements in Quantum Theory

This paper concerns derivations and interpretations of the uncertainty relations. The exclusive validity of the statistical interpretation is called into question. An individualistic interpretation, formulated by means of the concept of unsharp observables, is justified through a model of a joint measurement of position and momentum.

The determination of the past and the future of a physical system in quantum mechanics

The determination of the past and the future of a physical system are complementary aims of measurements. An optimal determination of the past of a system can be achieved by an informationally complete set of physical quantities. Such a set is always strongly noncommutative. An optimal determination of the future of a physical system can be obtained by a Boolean complete set of quantities. The two aims can be reconciled to a reasonable degree with using unsharp measurements.

Informationally complete sets of physical quantities

The notion of informational completeness is formulated within the convex state (or operational) approach to statistical physical theories and employed to introduce a type of statistical metrics. Further, a criterion for a set of physical quantities to be informationally complete is proven. Some applications of this result are given within the algebraic and Hilbert space formulations of quantum theory.

LUDERS THEOREM FOR UNSHARP QUANTUM MEASUREMENTS

Abstract A theorem of Luders states that an ideal measurement of a sharp discrete observable does not alter the statistics of another sharp observable if, and only if, the two observables commute. It will be shown that this statement extends to certain pairs of unsharp observables. Implications for local relativistic quantum theory will be discussed.

Time observables in quantum theory

Abstract “Time” as an observable of a physical system is to be understood with reference to the evolution of some nonstationary quantity. Thus, any observable “time” is the time of occurrence of an event of a certain type, defined by the appearance of some specified value of the dynamical quantity in question. This interpretation of time observables is illustrated by means of some examples.

Some realizable joint measurements of complementary observables

Noncommuting quantum observables, if considered asunsharp observables, are simultaneously measurable. This fact is exemplified for complementary observables in two-dimensional state spaces. Two proposals of experimentally feasible joint measurements are presented for pairs of photon or neutron polarization observables and for path and interference observables in a photon split-beam experiment. A recent experiment proposed and performed by Mittelstaedt, Prieur, and Schieder in Cologne is interpreted as a partial version of the latter example.

Colloquium: Quantum root-mean-square error and measurement uncertainty relations

Recent years have witnessed a controversy over Heisenbergs famous error-disturbance relation. Here we resolve the conflict by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. We discuss two approaches to adapting the classic notion of root-mean-square error to quantum measurements. One is based on the concept of noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for {\em state-dependent} errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for {\em state-independent} errors have been proven.

Insolubility of the quantum measurement problem for unsharp observables

Abstract The quantum mechanical measurement problem is the difficulty of dealing with the indefiniteness of the pointer observable at the conclusion of a measurement process governed by unitary quantum dynamics. There has been hope to solve this problem by eliminating idealizations from the characterization of measurement. We state and prove two ‘insolubility theorems’ that disappoint this hope. In both the initial state of the apparatus is taken to be mixed rather than pure, and the correlation of the object observable and the pointer observable is allowed to be imperfect. In the insolubility theorem for sharp observables , which is only a modest extension of previous results, the object observable is taken to be an arbitrary projection valued measure. In the insolubility theorem for unsharp observables , which is essentially new, the object observable is taken to be a positive operator valued measure. Both theorems show that the measurement problem is not the consequence of neglecting the ever-present imperfections of actual measurements.