ar X iv : q ua nt - p h / 98 08 00 9 v 2 9 F eb 1 99 9 An Example of Non - Attainability of Expected Quantum Information

Braunstein and Caves showed how quantum information can be used to define a metric for quantum states, relying on attainability of a quantum information bound. We show that the bound is not generally attainable, but that a two-stage procedure of repeated measurements achieves the bound in the limit. We connect to the question of 'non-locality without entanglement': can a joint measurement on n independent copies of a quantum system can yield more information than separate measurements. Though for small n generalized measurements are more informative,the gain is asymptotically negligible in the examples studied. 1 Introduction Braunstein and Caves [1] have clarified the relation between classical expected information i(θ), in the sense of Fisher, and the analogous concept of expected quantum information I(θ), by showing that I(θ) is an upper bound of i(θ; M) with respect to all (dominated) generalized measurements M of the state ρ = ρ(θ) where θ is an unknown parameter and i(θ; M) is the Fisher expected information for θ in the distribution of the outcome of the measurement of M. They indicate moreover that a measurement exists achieving the bound. In the present paper we show by an example, for an elementary spin