The structure of reversible computation determines the self-duality of quantum theory

TL;DR

This paper demonstrates that the self-duality of quantum theory arises from a fundamental computational primitive called bit symmetry, linking reversible transformations to the structure of quantum states and measurements.

Contribution

It proves that in probabilistic theories, bit symmetry implies self-duality, providing a dynamical explanation for quantum theory's self-dual structure.

Findings

Bit symmetry enforces self-duality in probabilistic theories.

Reversible transformations can map any logical bit to any other.

Bit symmetry imposes stronger constraints on bipartite states than no-signalling.

Abstract

Predictions for measurement outcomes in physical theories are usually computed by combining two distinct notions: a state, describing the physical system, and an observable, describing the measurement which is performed. In quantum theory, however, both notions are in some sense identical: outcome probabilities are given by the overlap between two state vectors - quantum theory is self-dual. In this paper, we show that this notion of self-duality can be understood from a dynamical point of view. We prove that self-duality follows from a computational primitive called bit symmetry: every logical bit can be mapped to any other logical bit by a reversible transformation. Specifically, we consider probabilistic theories more general than quantum theory, and prove that every bit-symmetric theory must necessarily be self-dual. We also show that bit symmetry yields stronger restrictions on the set of allowed bipartite states than the no-signalling principle alone, suggesting reversible time evolution as a possible reason for limitations of non-locality.