Algebraic properties of quantum reference frames: Does time fluctuate?

TL;DR

This paper explores how quantum reference frames, especially time, are inherently restricted by their quantum nature, revealing new limitations on their fluctuations, correlations, and interactions with systems.

Contribution

It introduces the concept of almost-positive states to mathematically analyze quantum reference frames and uncovers previously unrecognized restrictions on frame-system interactions.

Findings

Quantum reference frames have inherent restrictions on fluctuations and correlations.

Almost-positive states provide a new mathematical framework for analyzing quantum reference frames.

Relaxing assumptions in clock models introduces strong new restrictions.

Abstract

Quantum reference frames are expected to differ from classical reference frames because they have to implement typical quantum features such as fluctuations and correlations. Here, we show that fluctuations and correlations of reference variables, in particular of time, are restricted by their very nature of being used for reference. Mathematically, this property is implemented by imposing constraints on the system to make sure that reference variables are not physical degrees of freedom. These constraints not only relate physical degrees of freedom to reference variables in order to describe their behavior, they also restrict quantum fluctuations of reference variables and their correlations with system degrees of freedom. We introduce the notion of "almost-positive" states as a suitable mathematical method. An explicit application of their properties to examples of recent interest in quantum reference frames reveals previously unrecognized restrictions on possible frame-system interactions. While currently discussed clock models rely on assumptions that, as shown here, make them consistent as quantum reference frames, relaxing these assumptions will expose the models to new restrictions that appear to be rather strong. Almost-positive states also shed some light on a recent debate about the consistency of relational quantum mechanics.