Zenith Grant Awardee

William Wootters

Williams College

Project Title

Information and the origin of complex amplitudes

Project Summary

Quantum mechanics is a probabilistic theory, but the way we compute probabilities in quantum mechanics is quite different from what one would expect from, say, rolling dice or tossing coins. We do not count the number of possibilities and ask how many of them lead to a particular result. Instead, we first compute a complex number (involving the square root of negative one) associated with each of the possible outcomes of an experiment, and use this number to find the probability of that outcome. Thus the complex numbers play a central role. It has been recognized since 1936 that the logical structure of quantum mechanics is shared by a certain hypothetical theory in which the complex numbers are replaced by real numbers, but we do not see this real-number theory in nature. The proposed research aims to shed light on nature\'s choice. Specifically, we note that the theory based on real numbers has an appealing feature: information is conveyed optimally from one measurement to the next. Standard quantum mechanics does not seem to have this feature, but we ask whether, by viewing the question from a different angle, we can see a form of optimal information propagation in standard quantum mechanics.

Technical Abstract

In logical structure, the real-vector-space version of quantum theory is very similar to standard quantum theory. However, the information-theoretic properties are different. In particular, in the real theory, one finds that the parameters of a pure state are expressed optimally, in a well-defined sense, in the probabilities of the outcomes of a complete orthogonal measurement. One can say that information is conveyed optimally from preparation to measurement. This optimality is lost, however, when one goes over to standard quantum theory with its complex amplitudes. In the proposed research, we ask whether one can recover optimal information transfer in the complex theory. We take two approaches. (i) We reformulate the problem of information transfer to be bidirectional; this modification is intended to capture the invariance of the unitary part of quantum evolution under time reversal. (ii) We explore the possibility that, despite appearances, the real-vector-space theory is in fact the correct theory of nature. Specifically, we investigate the \'ubit model,\' in which the familiar complex phase factor in probability amplitudes is replaced by an actual real-vector-space subsystem.