Teiko Heinonen

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.

Optimal measurements in quantum mechanics

Abstract Four common optimality criteria for measurements are formulated using relations in the set of observables, and their connections are clarified. As case studies, 1–0 observables, localization observables, and photon counting observables are considered.

Position and momentum observables on R and on R3

We characterize all position and momentum observables on R and on R3. We study some of their operational properties and discuss their covariant joint observables.

Noise and disturbance in quantum measurement

The operational meaning of some measures of noise and disturbance in measurements is analyzed and their limitations are pointed out. The cases of minimal noise and least disturbance are characterized.

The norm-1-property of a quantum observable

A normalized positive operator measure

On the coexistence of position and momentum observables

X\mapsto E(X)

Covariant fuzzy observables and coarse-graining

has the norm-1-property if

Intrinsic unsharpness and approximate repeatability of quantum measurements

\no{E(X)}=1

Covariant phase difference observables in quantum mechanics

whenever

Fuzzy position and momentum observables

E(X)\ne O