Error bounds for Lie Group representations in quantum mechanics

TL;DR

This paper establishes state-dependent error bounds for Lie group representations in quantum mechanics, linking the difference in unitaries to energy and group metrics, applicable to various representations and channels.

Contribution

It introduces a general method to bound errors in Lie group representations in quantum mechanics, independent of specific representations and applicable to channels.

Findings

Provides bounds for unitaries based on energy and group metrics

Applicable to any connected Lie group and projective representations

Extends to energy-constrained channel distances

Abstract

We provide state-dependent error bounds for strongly continuous unitary representations of connected Lie groups. That is, we bound the difference of two unitaries applied to a state in terms of the energy with respect to a reference Hamiltonian associated to the representation and a left-invariant metric distance on the group. Our method works for any connected Lie group and the metric is independent of the chosen representation. The approach also applies to projective representations and allows us to provide bounds on the energy constrained diamond norm distance of any suitably continuous channel representation of the group.