Quantum Dynamics: A Dilation-Based Approach

TL;DR

This thesis explores a dilation-based approach to modeling open quantum system dynamics, representing reduced dynamics via unitary evolution on an extended system, offering new mathematical insights and approximation techniques.

Contribution

It introduces a dilation-based framework for finite-dimensional quantum dynamics, including exact and approximate dilation results and analysis of singular behaviors.

Findings

Exact dilation results for analytic dynamical curves.

Approximate dilations for Lipschitz-continuous curves.

Analysis of singular behavior at t=0.

Abstract

In the study of open quantum systems, one commonly describes the evolution of a system of interest through reduced dynamics, obtained by treating the environment indirectly rather than as a part of the full model. This thesis presents an expository account of an alternative, dilation-based viewpoint in the finite-dimensional setting, where a family of reduced dynamics is represented through unitary evolution on a larger system consisting of the original system together with an ancillary environment. After reviewing the reduced-dynamics perspective and the language of quantum channels, we formulate finite-dimensional quantum dynamics as channel-valued dynamical curves and use this framework to discuss Stinespring dilations of such curves. We then present exact dilation results for analytic dynamical curves, explain the singular behavior that can arise at t=0, and describe approximation results showing that Lipschitz-continuous dynamical curves admit approximate finite-dimensional Stinespring dilations. The thesis therefore provides a mathematically focused introduction to dilation-based modeling of quantum dynamics and argues that a change of perspective can lead to new ways of formulating problems in the theory of open quantum systems.