Truncated channel representations for coupled harmonic oscillators

TL;DR

This paper analyzes a simple coupled quantum harmonic oscillator system by truncating its quantum channel representation, deriving explicit error bounds, and demonstrating error correction to mitigate leakage errors.

Contribution

It introduces a method to truncate the quantum channel of coupled oscillators and provides explicit error bounds for the approximation.

Findings

Derived truncated transition amplitudes with explicit error bounds.

Numerical case study in off-resonant, weakly-coupled regime.

Showed quantum error correction reduces leakage errors.

Abstract

Coupled quantum harmonic oscillators, studied by many authors using many different techniques over the decades, are frequently used toy-models to study open quantum systems. In this manuscript, we explicitly study the simplest oscillator model -- a pair of initially decoupled quantum harmonic oscillators interacting with a spring-like coupling, where the bath oscillator is initially in a thermal-like state. In particular, we treat the completely positive and trace preserving map on the system as a quantum channel, and study the truncation of the channel by truncating its Kraus set and its output dimension. We thereby derive the truncated transition amplitudes of the corresponding truncated channel. Finally, we give a computable approximation for these truncated transition amplitudes with explicit error bounds, and perform a case study of the oscillators in the off-resonant and weakly-coupled regime numerically. We demonstrate explicitly that the substantial leakage error can be mitigated via quantum error correction.