Quantum Theory Prior to 1925

Clayton A. Gearhart

Irina Basieva, Emmanuel M. Pothos, Jennifer S. Trueblood +2 more

Cromwell's rule (also known as the zero priors paradox) refers to the constraint of classical probability theory that if one assigns a prior probability of 0 or 1 to a hypothesis, then the posterior has to be 0 or 1 as well (this is a straightforward implication of how Bayes's rule works). Relatedly, hypotheses with a very low prior cannot be updated to have a very high posterior without a tremendous amount of new evidence to support them (or to make other possibilities highly improbable). Cromwell's rule appears at odds with our intuition of how humans update probabilities. In this work, we report two simple decision making experiments, which seem to be inconsistent with Cromwell's rule. Quantum probability theory, the rules for how to assign probabilities from the mathematical formalism of quantum mechanics, provides an alternative framework for probabilistic inference. An advantage of quantum probability theory is that it is not subject to Cromwell's rule and it can accommodate changes from zero or very small priors to significant posteriors. We outline a model of decision making, based on quantum theory, which can accommodate the changes from priors to posteriors, observed in our experiments.

E.-K. Tan, John Jeffers, Stephen Mark Barnett

Injected two-level atoms, prepared and measured in various states, make an effective measurement of the cavity field of a micromaser prior to the injection of the atoms. In contrast to other existing literature on the micromaser, we have used the unconventional retrodictive quantum theory which allows us to calculate, in particular, the phase properties of the field state at an earlier time prior to the interaction with the injected atoms.

Daniel M. Greenberger, Klaus Hentschel, Friedel Weinert

Yang Yu, Philip Goyal

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.

Klaus Hentschel