Numerical approaches to entangling dynamics from variational principles

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.

💡 Research Summary

This paper tackles the problem of detecting entanglement that develops during the time evolution of a composite quantum system. The authors adopt a variational‑principles framework in which the state is constrained to remain a product (separable) state at all times. From the stationary‑action principle they derive a set of nonlinear coupled equations – the separable Schrödinger equations (SSE) – which are the constrained analogue of the ordinary Schrödinger equation. Because analytical solutions of the SSE are rarely available, the authors develop two complementary numerical strategies.

The first strategy applies linear splitting methods directly to the continuous SSE. By decomposing the full flow into a sequence of partial flows, each acting on a single subsystem while keeping the others fixed, they obtain a Lie‑Trotter (first‑order) scheme and a Strang‑splitting (second‑order) scheme. The method is structure‑preserving: each partial flow is unitary‑like and norm‑preserving, and for Hamiltonians that decompose into local terms the splitting reproduces the exact solution. The authors validate the approach on a simple exchange‑interaction Hamiltonian (a two‑qubit swap operation). With a sufficiently small time step (Δt = 0.001) the numerical trajectories of the restricted dynamics coincide with the analytical SSE solution up to O(Δt²). Bloch‑sphere plots clearly show that the restricted evolution stays on the surface (pure separable states) whereas the unrestricted evolution penetrates the interior (mixed, entangled reduced states).

The second strategy bypasses the need to derive the continuous SSE by discretizing the variational action itself. This variational discretization yields an integration scheme that can be applied directly to the original Lagrangian. The authors compare two orderings: (i) “restrict‑then‑discretize” (apply the separability constraint first, then discretize) and (ii) “discretize‑then‑restrict” (discretize first, then impose the constraint). Using the same exchange‑interaction Hamiltonian they demonstrate that the second ordering becomes numerically unstable: the iterates diverge or violate norm conservation, indicating that the nonlinear structure of the constrained dynamics is destroyed when the constraint is imposed after discretization.

To illustrate the generality of the methods, the paper also presents results for (a) a randomly generated five‑qubit Hamiltonian and (b) a three‑qutrit system with ladder‑operator interactions that generate r‑party correlations. In the random‑Hamiltonian case the unrestricted dynamics quickly drives the system toward a maximally mixed reduced state, signalling strong entanglement generation, while the restricted dynamics remains on the pure‑state manifold. In the ladder‑operator example the authors show how the constrained and unconstrained trajectories differ in speed and in the decay of overlap between the two solutions.

Overall, the work provides a practical toolbox for benchmarking the entangling power of quantum processes. It shows that (1) splitting methods give accurate, structure‑preserving approximations of the SSE, (2) variational discretization is feasible but the order of applying constraints matters critically, and (3) the approach scales to multipartite, higher‑dimensional systems. The findings have immediate relevance for quantum simulation, quantum control, and the design of quantum information protocols where one wishes to quantify or limit entanglement generation over time.