Conceptual inadequacy of the Shannon information in quantum measurements

In a classical measurement the Shannon information is a natural measure of our ignorance about properties of a system. There, observation removes that ignorance in revealing properties of the system which can be considered to preexist prior to and independent of observation. Because of the completely different root of a quantum measurement as compared to a classical measurement, conceptual difficulties arise when we try to define the information gain in a quantum measurement using the notion of Shannon information. The reason is that, in contrast to classical measurements, quantum measurements, with very few exceptions, cannot be claimed to reveal a property of the individual quantum system existing before the measurement is performed.