Reversibility and the structure of the local state space

Jonathan Richens,Sabri W Al-Safi

doi:10.1088/1367-2630/17/12/123001

Jonathan Richens, Sabri W Al-Safi

Open Access

https://doi.org/10.1088/1367-2630/17/12/123001

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Abstract

The richness of quantum theory’s reversible dynamics is one of its unique operational characteristics, with recent results suggesting deep links between the theory’s reversible dynamics, its local state space and the degree of non-locality it permits. We explore the delicate interplay between these features, demonstrating that reversibility places strong constraints on both the local and global state space. Firstly, we show that all reversible dynamics are trivial (composed of local transformations and permutations of subsytems) in maximally non-local theories whose local state spaces satisfy a dichotomy criterion; this applies to a range of operational models that have previously been studied, such as d-dimensional ‘hyperballs’ and almost all regular polytope systems. By separately deriving a similar result for odd-sided polygons, we show that classical systems are the only regular polytope state spaces whose maximally non-local composites allow for non-trivial reversible dynamics. Secondly, we show that non-trivial reversible dynamics do exist in maximally non-local theories whose state spaces are reducible into two or more smaller spaces. We conjecture that this is a necessary condition for the existence of such dynamics, but that reversible entanglement generation remains impossible even in this scenario.

Highlights

In the quest to understand why quantum mechanics accurately predicts natural phenomena, it is prudent to investigate the properties that distinguish it from classical mechanics and from other conceivable theories of nature.Exploring these properties leads to the development of algorithms for information-based tasks [1], and provides insight into counter-intuitive quantum phenomena such as the prediction of non-local correlations [2, 3], teleportation [4], and the impossibility of cloning [5]
III we show how transformations are defined and give a useful necessary and sufficient condition for a reversible transformation to be trivial; in Section IV we show that all reversible dynamics are trivial in maximally non-local theories whose local systems are dichotomic; in Section V we show that non-trivial transformations exist if one or more local systems are reducible, and conjecture that this is a necessary condition for reversible interactions in maximally non-local theories
We have investigated reversible dynamics in maximally non-local general probabilistic theories, demonstrating in Section IV that reversible dynamics are trivial in the case where all subsystems are non-classical and dichotomic

Summary

Reversibility and the structure of the local state space

We show that all reversible dynamics are trivial (composed of local transformations and permutations of subsytems) in maximally non-local theories whose local state spaces satisfy a dichotomy criterion; this applies to a range of operational models that have previously been studied, such as d-dimensional “hyperballs” and almost all regular polytope systems. We show that non-trivial reversible dynamics do exist in maximally non-local theories whose state spaces are reducible into two or more smaller spaces. We conjecture that this is a necessary condition for the existence of such dynamics, but that reversible entanglement generation remains impossible even in this scenario

INTRODUCTION

DICHOTOMIC SYSTEMS

DISCUSSION