Zenith Grant Awardee

Alexander Wilce

Susquehanna University

Project Title

Conjugates, Correlation and Quantum Mechanics

Project Summary

Quantum mechanics makes probabilistic predictions about outcomes of experiments. To this extent, it\'s about information. In fact, finite-dimensional quantum mechanics (QM) has been derived (in several ways) from purely information-theoretic principles. One of these is that the state of a pair of systems is determined by an assignment of joint probabilities to outcomes of measurements made on the two systems separately. However, this fails in two well-known, and arguably reasonable, variants of standard QM, called real and quaternionic QM. What happens if we drop this assumption? In some ways, things become simpler. Using much weaker information-theoretic assumptions, we obtain a theory that unifies standard QM with real and quaternionic QM, while leaving just a bit of additional room beyond these. The basic idea is that each system can be paired with a \'conjugate\' copy of itself, in a joint state in which every measurement on either system looks completely random, but is nevertheless exactly correlated with its counterpart on the other. In standard QM, this conjugate system is a time-reversed version of the original. This project aims to clarify the meaning of conjugate systems in general, and to study the information processing power of the theory based on them.

Technical Abstract

Recently, standard finite-dimensional quantum mechanics (QM) has been derived from various packages of information- theoretic assumptions. This strengthens the case that QM is about information. However, some of the assumptions on which these derivations rest are very strong. In particular, all assume that the state of a bipartite system is determined by the joint probabilities it assigns to measurements on the two component systems. This assumption fails in real and quaternionic quantum mechanics. What happens if we drop it? In some ways, things become simpler. A weaker set of postulates, focussing on correlations between finite-dimensional probabilistic systems and isomorphic \'conjugate\' systems (in standard QM, the time-reversed systems associated with conjugate Hilbert spaces) leads to a representation of such systems in terms of euclidean Jordan algebras: direct sums of real, complex and quaternionic systems, the exceptional Jordan algebra, and spin factors (“bits\' represented by n-dimensional balls). Excluding the exceptional Jordan algebra, there is a natural non-signaling tensor product for such systems. The resulting theory unifies real, complex and quaternionic QM, while leaving just a bit of extra room. This project aims to clarify the axiomatic basis of this theory and to study the information processing power of the resulting slightly-more-than quantum theory.