The Acquisition of Information from Quantum Measurements.

The following thought-experiment is considered. A finite ensemble of identically prepared quantum systems (e.g., photons prepared by a polarizing filter) is analyzed with a fixed analyzing device (e.g., a Nicol prism). An experimenter tries to determine the preparation parameters of the ensemble (e.g., the orientation of the polarizing filter) by counting how many times each outcome occurs. These measurements are fundamentally probabilistic, and therefore the experimenter cannot avoid an error due to statistical fluctuations. The analysis of these statistical fluctuations in terms of Shannon's information forms the basis of this dissertation. Two main results emerge. (1) In certain situations, the law of nature which determines the probabilities of the various outcomes is such as to provide the best conceivable discrimination among the different values of the preparation parameters. (2) The angle in Hilbert space between two rays (i.e., two sets of values of the preparation parameters) is proportional to the number of intermediate distinguishable rays, where "distinguishability" depends on the size of the statistical fluctuations. These results would typically not be true if the relevant laws of physics were different from what they are. Finally, the question is raised whether the idea of defining "distance" in terms of distinguishability can be applied to other instances of geometry in physics (i.e., other than Hilbert space).