Numerical approaches to entangling dynamics from variational principles

TL;DR

This paper develops and compares numerical methods based on variational principles to detect and analyze entanglement dynamics in quantum systems, highlighting the stability and applicability of different discretization strategies.

Contribution

It introduces and evaluates variational discretization schemes for entanglement detection, revealing the instability of certain approaches and providing broadly applicable numerical tools.

Findings

Linear splitting methods confirm analytical solutions for small time steps.

Discretization before restriction leads to numerical instability.

The methods are applicable to various Hamiltonians in quantum information.

Abstract

In this work, we address the numerical identification of entanglement in dynamical scenarios. To this end, we consider different programs based on the restriction of the evolution to the set of separable (i.e., non-entangled) states, together with the discretization of the space of variables for numerical computations. As a first approach, we apply linear splitting methods to the restricted, continuous equations of motion derived from variational principles. We utilize an exchange interaction Hamiltonian to confirm that the numerical and analytical solutions coincide in the limit of small time steps. The application to different Hamiltonians shows the wide applicability of the method to detect dynamical entanglement. To avoid the derivation of analytical solutions for complex dynamics, we consider variational, numerical integration schemes, introducing a variational discretization for Lagrangians linear in velocities. Here, we examine and compare two approaches: one in which the system is discretized before the restriction is applied, and another in which the restriction precedes the discretization. We find that the "first-discretize-then-restrict" method becomes numerically unstable, already for the example of an exchange-interaction Hamiltonian, which can be an important consideration for the numerical analysis of constrained quantum dynamics. Thereby, broadly applicable numerical tools, including their limitations, for studying entanglement over time are established for assessing the entangling power of processes that are used in quantum information theory.