Operational formulation of time reversal in quantum theory

TL;DR

This paper provides an operational framework for understanding time reversal in quantum theory, resolving longstanding controversies by defining a generalized, time-symmetric formulation that includes broader symmetry transformations.

Contribution

It introduces a rigorous operational probabilistic approach to time reversal, defining a generalized quantum theory with new symmetry transformations and clarifying the distinction between states and effects.

Findings

Established a precise definition of time-reversal symmetry in quantum theory.

Proved an analogue of Wigner's theorem for the generalized framework.

Discovered larger classes of symmetry transformations than previously assumed.

Abstract

The symmetry of quantum theory under time reversal has long been a subject of controversy because the transition probabilities given by Born's rule do not apply backward in time. Here, we resolve this problem within a rigorous operational probabilistic framework. We argue that reconciling time reversal with the probabilistic rules of the theory requires a notion of operation that permits realizations via both pre- and post-selection. We develop the generalized formulation of quantum theory that stems from this approach and give a precise definition of time-reversal symmetry, emphasizing a previously overlooked distinction between states and effects. We prove an analogue of Wigner's theorem, which characterizes all allowed symmetry transformations in this operationally time-symmetric quantum theory. Remarkably, we find larger classes of symmetry transformations than those assumed before. This suggests a possible direction for search of extensions of known physics.