Entanglement’s Fleeting Dance Now Trackable with New Computational Techniques

Scientists are increasingly focused on understanding and quantifying entanglement in dynamic quantum systems. Christian Offen (University of Birmingham), Boris Wembe and Laura Ares (Paderborn University) et al. present novel numerical approaches to identify entanglement arising from dynamical processes, utilising variational principles and restrictions to separable states.

This research is significant because it establishes broadly applicable numerical tools, alongside a critical analysis of their limitations, for assessing the entangling power of quantum processes. By comparing different variational and numerical integration schemes, including a detailed examination of discretization order, the authors demonstrate how to effectively, and avoid pitfalls in, numerically modelling entanglement over time. To this end, they consider different programmes based on the restriction of the evolution to the set of separable (ie, non-entangled) states, together with the discretisation of the space of variables for numerical computations. As a first approach, they apply linear splitting Methods to the restricted, continuous equations of motion derived from variational principles. They utilise 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. Discretisation order impacts stability in constrained quantum dynamics simulations Scientists are investigating analytical solutions for complex dynamics, considering 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. Determining whether a quantum state at a fixed time is separable or inseparable is known to be an NP-hard problem. In the special case of two and three-dimensional bipartite systems, necessary and sufficient criteria exist, allowing for efficient identification of entangled states. In contrast, high-dimensional and multipartite systems introduce additional intricacies, for example, because of the existence of multiple partitions, making criteria for entanglement certification significantly more challenging to derive. To address these challenges in an experimentally feasible manner, analytical criteria in the form of entanglement witnesses have been developed. One well-established framework to derive optimal witnesses is through the separability eigenvalue equations. However, the sophistication of the separability problem reduces the availability of known exact solutions to limited scenarios. For this reason, numerical approaches for solving the separability eigenvalue equations have been investigated to help with the detection of multipartite entanglement in complex systems. A limitation of the aforementioned approaches is that they apply to static scenarios only, probing entanglement at fixed times. Also, one ought to expect a speedup in terms of physical time, not only complexity, when a quantum processor has access to entangling dynamics, compared to a non-entangling evolution. Therefore, to certify entanglement in dynamical scenarios, several approaches have been developed over the last years. It is not uncommon to find relations analyzing the difference between the entanglement of initial and final states, resulting in an input-output-based quantum channel characterization with respect to its entangling power. However, any entanglement at intermediate times is thereby not detected, especially when the initial and final states are separable. For this reason, variational principles were utilized to witness time-dependent entanglement, which recently have been generalized to open-system dynamics. Already in the closed-system approach, however, it becomes increasingly challenging to find exact solutions. Thus, there is a need for numerical Methods, applicable to multipartite and high-dimensional systems, to benchmark the entangling part of the evolution of a composite quantum system. Further, numerical investigations, as carried out in the following, can even avoid the derivation of sophisticated equations of motion, dubbed separability Schrödinger equations (SSE), for the entangling evolution. In this work, we develop numerical Methods for the identification of time-dependent entanglement by restricting the dynamics to be in a separable state for all times. First, we apply linear splitting Methods to the restricted, continuous equations of motion derived from variational principles. We gauge the accuracy of the method by comparing the numerical and analytical solutions for a proof-of-concept Hamiltonian and demonstrate the general applicability by detecting entanglement for different Hamiltonians and various system sizes. While numerical splitting Methods are a natural choice for integrating the SSE, these require analytically deriving the continuous equations first. To bypass this step, we propose a variational discretization approach that directly yields integration schemes for the restricted dynamics. We consider two scenarios in this context: restrict, discretize and discretize, restrict. In the first approach, the restriction is applied before the discretization whereas, in the second one, the order is reversed. We compare the two Methods and show that the discretize, restrict approach becomes numerically unstable when applied to the exchange-interaction Hamiltonian, highlighting a critical issue for the numerical treatment of constrained systems in numerical analysis. In this section, we very briefly review the method of SSE for characterizing the entangling capabilities of a Hamiltonian process. We consider N complex Hilbert spaces H1, ., HN of dimension d, where d is a positive integer, and H = H1 ⊗ H2 ⊗.⊗ HN. In the special case where N = 2, the tensor product H1 ⊗ H2 is a Hilbert space of dimension d2. Determining whether a state |ψ⟩ ∈ H is separable or entangled is a challenging task. In the special case where N = 2, the Peres-Horodecki criterion provides a sufficient condition for entanglement. In contrast, high-dimensional and multipartite systems introduce additional intricacies, for example, because of the existence of multiple partitions, making criteria for entanglement certification significantly more challenging to derive. To address these challenges in an experimentally feasible manner, analytical criteria in the form of entanglement witnesses have been developed. One well-established framework to derive optimal witnesses is through the separability eigenvalue equations. However, the sophistication of the separability problem reduces the availability of known exact solutions to limited scenarios. For this reason, numerical approaches for solving the separability eigenvalue equations have been investigated to help with the detection of multipartite entanglement in complex systems. A limitation of the aforementioned approaches is that they apply to static scenarios only, probing entanglement at fixed times. Also, one ought to expect a speedup in terms of physical time, not only complexity, when a quantum processor has access to entangling dynamics, compared to a non-entangling evolution. Therefore, to certify entanglement in dynamical scenarios, several approaches have been developed over the last years. It is not uncommon to find relations analyzing the difference between the entanglement of initial and final states, resulting in an input-output-based quantum channel characterization with respect to its entangling power. However, any entanglement at intermediate times is thereby not detected, especially when the initial and final states are separable. For this reason, variational principles were utilized to witness time-dependent entanglement, which recently have been generalized to open-system dynamics. Already in the closed-system approach, however, it becomes increasingly challenging to find exact solutions. Thus, there is a need for numerical Methods, applicable to multipartite and high-dimensional systems, to benchmark the entangling part of the evolution of a composite quantum system. Further, numerical investigations, as carried out in the following, can even avoid the derivation of sophisticated equations of motion, dubbed separability Schrödinger equations (SSE), for the entangling evolution. Next, we introduce an approximated flow map ΦSSE ∆t, with ∆t 0 being a discretization parameter, of the exact time-∆t-flow ΦSSE ∆t . To compute this numerical ΦSSE ∆t, we consider the components of a given state (|a 1 ⟩.,|a N ⟩) ∈H ×, where upper indices in brackets [·] are used to describe the effects of the update operations. Then, the updated components (|a[N] 1 ⟩.,|a[N] N ⟩) = ΦSSE ∆t (|a 1 ⟩.,|a N ⟩) are computed by performing N update operations, |a[l] 1 ⟩= |a[l−1] 1 ⟩.,|a[l] l−1⟩= |a[l−1] l−1 ⟩ |a[l] l ⟩= ΦSSE a[l−1] 1.,a[l−1] l−1,a[l−1] l+1.,a[l−1] N,∆t|a[l−1] l ⟩ |a[l] l+1⟩= |a[l−1] l+1 ⟩.,|a[l] N ⟩= |a[l−1] N ⟩, for l ∈{1.,N}. In other words, the components are updated sequentially by the flow of the corresponding reduced operator; that is, in Eq. (4), the first component is updated while all other components remain unchanged, then the second component is updated while all other components remain unchanged, etc. To an initial value (|a(0) 1 ⟩.,|a(0) N ⟩) ∈H ×, where we denote iterates with upper indices in parenthesis (·), the iteration (|a(j+1) 1 ⟩.,|a(j+1) N ⟩) = ΦSSE ∆t (|a(j) 1 ⟩.,|a(j) N ⟩), for j ∈{0,1.}, constitutes a numerical approximation of first order O(∆t) of the exact solution of Eq. (4). Both, the exact flow at time ∆t, ΦSSE ∆t, and the numerical flow map ΦSSE ∆t are nonlinear, yet inner-product-preserving (thus, norm-preserving) transformations, generalizing unitary maps from the SE to the flow of the nonlinear SSE. In this sense, the numerical method is structure-preserving. The above numerical method is known as a splitting method, wherein each summand in the split is obtained by setting all but one right-hand side in Eq. (4) to zero. Indeed, ΦSSE ∆t amounts to computing the Lie-Trotter splitting, a first-order numerical method. Higher-order splittings require more complicated compositions of the flow maps of each summand: for instance, we can perform a second-order numerical method Φ SSE ∆t for approximating ΦSSE ∆t by the Strang-Splitting method (|a′′′ 1 ⟩,|a′′′ 2 ⟩) = Φ SSE ∆t (|a1⟩,|a2⟩), for the example N = 2 and with |a′ 1⟩= |a1⟩ |a′ 2⟩= ΦSSE a1,∆t/2|a2⟩ |a′′ 1⟩= ΦSSE a2,∆t|a1⟩ |a′′ 2⟩= |a′ 2⟩ |a′′′ 1 ⟩= |a′′ 1⟩ |a′′′ 2 ⟩= ΦSSE a1,∆t/2|a′′ 2⟩. By construction, Φ SSE ∆t in Eq. (11) is a unitary-like, nonlinear transformation on the Cartesian product space H1 × H2. For higher-order splitting Methods, as well as the non-bipartite case, see the literature on numerical splitting Methods. For splitting Methods where the operators are unbounded, we additionally refer to Ref0.0.For any splitting method, when the numerical iterates (|a(j) 1 ⟩.,|a(j) N ⟩) are mapped to |ψ(j)⟩= |a(j) 1 ⟩⊗.⊗|a(j) N ⟩∈H, the norm p ⟨ψ(j)|ψ(j)⟩remains constant. Moreover, for a Hamiltonian that decomposes into local parts, H = H1 ⊗1H2 ⊗.⊗1HN + 1H1 ⊗H2 ⊗1H3 ⊗.⊗1HN +.+ 1H1 ⊗.⊗1HN−1 ⊗HN, the SSE (4) decouple. Thus, a numerical solution based on a splitting method coincides with the exact solution because each individual equation in (4) is solved exactly by the numerical scheme when the equations decouple. In the remainder of this section, we apply the method in Eqs. (9) and (10) to different Hamiltonians, comparing the solutions for the SSE and the SE. The source code for the numerical experiments may be found on GitHub. For convenience, we set h = We show how, for small enough intervals, the numerical method renders the exact solution, in cases they are known, up to second order. Bloch sphere trajectories differentiate pure and mixed state dynamics under exchange interaction Trajectories on the Bloch sphere demonstrate distinct dynamics for restricted and unrestricted evolution, particularly for a time step of 0.001 and the initial state |ψ(0)⟩= |0⟩⊗(|0⟩+|1⟩)/ √ 2 with an exchange interaction. The restricted dynamics maintain the subsystem state as pure, confining the trajectory to the surface of the sphere, while unrestricted evolution yields a mixed state with trajectories travelling through the sphere’s interior. Numerical convergence to analytical solutions was confirmed for sufficiently small time steps, and backward error analysis can be applied to further refine the Trotter and Strang-Splitting schemes. Applying the method to a randomly sampled Hermitian operator with five qubits, the study revealed significant differences in the trajectories of each subsystem between separable state evolution and strongly separable state evolution. The partial trace analysis showed that unrestricted trajectories converge towards a maximally entangled state, contrasting sharply with the restricted evolution which cannot produce entanglement. Overlaps between separable and inseparable solutions rapidly decayed, and the speed of entangling versus non-entangling evolution yielded differing rates of state change as measured by the nuclear norm. For a system of three qudits, ladder operators were used to define a Hamiltonian, and dynamics were visualized on the Poincaré sphere, a higher-dimensional analogue of the Bloch sphere. Trajectories for the reduced systems disagreed, clearly indicating quantum correlations and entanglement resulting from the interaction. A comparison of two-party and three-party correlators revealed revivals in the three-party case, absent in the two-party correlator, at least within the observed time scale. The research established that linear and unrestricted separable state evolution solutions can be numerically obtained and compared with nonlinear and restricted strongly separable state evolution solutions, characterizing interesting entangling dynamics. Using a randomly generated five-qubit Hamiltonian, the application extended to arbitrary multi-qubit interactions, and multi-party correlators were used to study distinct types of interactions and their impact on the entangling power of processes coupling angular momenta of three particles. Variational Discretisation and Stability in Separable State Dynamics Scientists have developed numerical methods for identifying entanglement during dynamic processes by restricting quantum evolution to only separable states. These techniques utilize variational principles and discretization of variables to enable computational analysis of entanglement over time. The approach was initially validated using a simple exchange interaction Hamiltonian, where numerical solutions aligned with analytical predictions for small time steps, and subsequently demonstrated applicability to more complex Hamiltonians. Further refinement involved a variational discretization framework, exploring two implementation orders: first discretizing then restricting, and first restricting then discretizing. Numerical experiments revealed that the “discretize-then-restrict” method exhibits numerical instability, even with the exchange interaction Hamiltonian, indicating the “restrict-discretize” order is preferable for stable and accurate results. This work establishes a basis for numerically investigating time-dependent entanglement within variational and geometric frameworks, offering computational tools for simulating complex quantum systems. The established framework is also adaptable for studying other quantum phenomena beyond entanglement, potentially aiding the assessment of nonclassical properties in quantum technologies. A limitation acknowledged by the researchers is the potential for numerical instability when the discretization step precedes the restriction to separable states. Future research may focus on applying these methods to quantify dynamic entanglement using the Schmidt number and exploring their use in assessing the performance of quantum-technological applications. 👉 More information 🗞 Numerical approaches to entangling dynamics from variational principles 🧠 ArXiv: https://arxiv.org/abs/2602.05726 Tags: