# Lossless and Lossy Characterization of the State of Perturbed Anharmonic Diatomics: An Information-Theoretic Compaction of Quantum Dynamics

## TL;DR

This paper presents a method to compactly represent the quantum state of a perturbed anharmonic molecule using symmetries, both exactly and approximately.

## Contribution

A novel matrix-based approach is introduced for reversible and irreversible compaction of quantum states in anharmonic systems.

## Key findings

- The density matrix of an anharmonic molecule can be compacted exactly using dynamical symmetries.
- Approximate compaction with fewer symmetries is shown to be irreversible and its fidelity is quantified.
- The method is demonstrated using a forced Morse oscillator across various dynamic limits.

## Abstract

A lossless, exact compaction of the time-evolved state
of the quantum
dynamical system of a perturbed anharmonic molecule is demonstrated
using dynamical symmetries. The density matrix of the anharmonic molecule
is a linear combination of these symmetries, and it remains so as
a time-dependent perturbation is applied. Accurate, unitary-but-approximate,
and thereby irreversible compaction is further shown using fewer symmetries,
and the fidelity of this lossy compaction is quantified. Perturbations
are typically linear in the operators of a Lie algebra. For a Hamiltonian
that is also linear, one knows well how to reversibly compact the
state of a dynamical system. However, anharmonic vibrations have a
finite number of unequally spaced energy levels, and a good description
of their spectra typically requires an algebraic-type Hamiltonian
that is bilinear in the operators of a Lie algebra. For a bilinear
Hamiltonian we show how a matrix-based approach allows us to compact
both the populations and the coherences, either exactly reversibly
or inexactly irreversibly, with fewer symmetries. A forced Morse oscillator
is used as an explicit analytical and numerical example covering the
entire range of dynamics from the sudden to the adiabatic limits.