Numerical approaches to entangling dynamics from variational principles

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.

Figures (8)

• Figure 1: Trajectories on the Bloch sphere of restricted (dashed) and non-restricted dynamics (solid) for $\Delta t=0.001$ and the initial state $|\psi(0)\rangle=|0\rangle\otimes(|0\rangle+|1\rangle)/\sqrt{2}$ for an exchange interaction (left, first qubit; right, second qubit).
• Figure 2: Top: Bloch sphere with the evolution of the qubits (Qb1--Qb5) generated by a random Hamiltonian (left: SE \ref{['eq:SE']}, middle: SSE \ref{['eq:SSE']}, right: comparison between trajectories for the second qubit). In the bottom, the left panel depicts the overlap $\langle \psi_{\mathrm{SE}}| \psi_{\mathrm{SSE}}\rangle$. The right panel shows the rate of change of the SSE, $| \frac{d}{d t} | \psi_{\mathrm{SSE}}(t)\rangle \langle \psi_{\mathrm{SSE}}(t)| |_{\mathrm{nucl}}$ (dotted line), and the corresponding quantity for the SE (solid line), using the nuclear/trace norm.
• Figure 3: Top: Dynamics based on a Hamiltonian given in terms of two-party correlators of angular-momentum ladder operators. The evolution of qudits is shown on the Poincaré sphere (left: SE, middle: SSE, right: comparison for second qudit). Bottom: the overlap $\langle \psi_{\mathrm{SE}}| \psi_{\mathrm{SSE}}\rangle$ (left), as well as the rate of change $| \frac{d}{d t} | \psi_{\mathrm{SSE}}(t)\rangle \langle \psi_{\mathrm{SSE}}(t)| |_{\mathrm{nucl}}$ in nuclear norm for the SSE (dotted line, right plot) and for the SE (solid line, right plot) as a function of time.
• Figure 4: Top: Dynamics via angular-momentum ladder operator and $3$-party correlator on the Poincaré sphere (left: SE, middle: SSE, right: comparison second qudit). Bottom: Overlap of the restricted (SSE) and unrestricted (SE) dynamics (left panel), and the individual rates of change (i.e., speed; right panel). Compared to the two-party correlator in \ref{['fig:LadderOperator1']}, the here-discussed three-party correlator yields relatively strong revivals (increases of the overlap) in the similarity of the SE and SEE solutions.
• Figure 5: Visualization of the two distinct approaches: "first-restrict-then-discretize" (left path) and "first-discretize-then-restrict" (right path). For compactness of notation, we show the bipartite case $N=2$ with variables $a,b$ and neglect the complex conjugate input arguments of the Lagrangians. "$\xrightarrow{\Delta}$" denotes discretization by a variational integrator. The diagram does not commute as the left branch yields $S^{\mathrm{sep1}}$ and the right branch $S^{\mathrm{sep2}}$.