Principle of Information Increase: An Operational Perspective on Information Gain in the Foundations of Quantum Theory

Abstract

A measurement performed on a quantum system is an act of gaining information about its state. However, in the foundations of quantum theory, the concept of information is multiply defined, particularly in the area of quantum reconstruction, and its conceptual foundations remain surprisingly under-explored. In this paper, we investigate the gain of information in quantum measurements from an operational viewpoint in the special case of a two-outcome probabilistic source. We show that the continuous extension of the Shannon entropy naturally admits two distinct measures of information gain, differential information gain and relative information gain, and that these have radically different characteristics. In particular, while differential information gain can increase or decrease as additional data are acquired, relative information gain consistently grows and, moreover, exhibits asymptotic indifference to the data or choice of Bayesian prior. In order to make a principled choice between these measures, we articulate a Principle of Information Increase, which incorporates a proposal due to Summhammer that more data from measurements leads to more knowledge about the system, and also takes into consideration black swan events. This principle favours differential information gain as the more relevant metric and guides the selection of priors for these information measures. Finally, we show that, of the symmetric beta distribution priors, the Jeffreys binomial prior is the prior that ensures maximal robustness of information gain for the particular data sequence obtained in a run of experiments.