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Error bounds for Lie Group representations in quantum mechanics

Lauritz van Luijk, Niklas Galke, Alexander Hahn, Daniel Burgarth

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.

Error bounds for Lie Group representations in quantum mechanics

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.
Paper Structure (22 sections, 10 theorems, 116 equations, 2 figures)

This paper contains 22 sections, 10 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The metric
  4. Lie algebras with $\operatorname{Ad}$-invariant inner product
  5. Main results
  6. Proper unitary representations
  7. Projective representations and estimates on unitary channels
  8. Examples and applications
  9. Spin representations of $\mathrm{SU}(2)$
  10. Special orthogonal group and FLO representation
  11. Displacement operators
  12. Quasi-free bosonic transformations
  13. Two-mode bosonic representation of $\operatorname{SU}(1,1)$
  14. Representation of the Lorentz group
  15. Proofs
  16. ...and 7 more sections

Key Result

Lemma 2

$d$ is a left-invariant metric on $G$ and the induced metric topology agrees with the topology of $G$.

Figures (2)

  • Figure 1: Comparison of our bound for $\mathrm{SO}(2m)$ in \ref{['eq:SO_channel_compare_log']} (red) with the bound in Oszmaniec2022 (blue) for two particular channels. In this numerical simulation, we consider $m=2$, $g=\exp(B_{21})$ and $h=\exp(B_{21}+aB_{31})$ with $B_{jk}$ defined in \ref{['eq:so_basis']}. We find that our bound is tighter (red vs. blue) in this case.
  • Figure 2: Comparison of our bound for the symplectic group in \ref{['eq:symplectic_channel_compare']} with the bound in Becker2021 for a particular Trotter problem on a $\log$-$\log$ axis. That is, $g=\mathrm{e}^{-t(X+Y)}$ and $h=(\mathrm{e}^{-tX/L} \mathrm{e}^{-Yt/L})^L$ for $t\in\mathbb R$ and $X,Y\in\mathfrak{sp}(2m,\mathbb R)$. In this particular simulation, we chose $m=1$, $t=1$, $X=\Omega$ (harmonic oscillator) and $Y=\Omega\sigma_x$ (generator of the squeezing transformation), where $\sigma_x$ denotes the first Pauli matrix. We compute the bounds for the energy-constrained diamond norm of the associated unitary channels up to the second Fock state, i.e., $\langle \psi,{N} \psi\rangle\leq 2$. The energy is taken with respect to the number operator ${N}$. Our bound correctly covers the $\mathcal{O}(1/L)$ scaling of the Trotter product formula.

Theorems & Definitions (23)

  • Definition 1: Left-invariant metric
  • Lemma 2
  • Proposition 3
  • Lemma 4
  • Theorem 5
  • Proposition 6
  • Definition 7
  • Remark 8
  • Lemma 9
  • Remark 10
  • ...and 13 more