Generalized uncertainty relations and efficient measurements in quantum systems
Abstract
Description
We consider two variants of a quantum-statistical generalization of the Cramer-Rao inequality that establishes an invariant lower bound on the mean square error of a generalized quantum measurement. The proposed complex variant of this inequality leads to a precise formulation of a generalized uncertainty principle for arbitrary states, in contrast to Helstrom's symmetric variant in which these relations are obtained only for pure states. A notion of canonical states is introduced and the lower mean square error bound is found for estimating of the parameters of canonical states, in particular, the canonical parameters of a Lie group. It is shown that these bounds are globally attainable only for canonical states for which there exist efficient measurements or quasimeasurements.
17 pages. Translated from Russian and typeset in LaTeX from Teoreticheskaya i Matematichescheskaya Fizika, Vol. 26, No.3 pp. 316--329, March, 1976
17 pages. Translated from Russian and typeset in LaTeX from Teoreticheskaya i Matematichescheskaya Fizika, Vol. 26, No.3 pp. 316--329, March, 1976