Restricted Monte Carlo wave function method and Lindblad equation for identifying entangling open-quantum-system dynamics

Abstract

We develop an extension of the Monte Carlo wave function approach that unambiguously identifies dynamical entanglement in general composite, open systems. Our algorithm performs tangential projections onto the set of separable states, leading to classically correlated quantum trajectories. By comparing this restricted evolution with the unrestricted one, we can characterize the entangling capabilities of quantum channels without making use of input-output relations. Moreover, applying this method is equivalent to solving the nonlinear master equation in Lindblad form introduced in \cite{PAH24} for two-qubit systems. We here extend these equations to multipartite systems of qudits, describing non-entangling dynamics in terms of a stochastic differential equation. We identify the impact of dynamical entanglement in open systems by applying our approach to several correlated decay processes. Therefore, our methodology provides a complete and ready-to-use framework to characterize dynamical quantum correlations caused by arbitrary open-system processes.

Figures (4)

• Figure 1: A Kraus operator $K^m$ takes the product state $\ket{\psi_A(t)\psi_B(t)}$ outside the manifold of separable states, cf. curve that is lifted from the manifold for improved visibility. A projection onto the tangent space, resulting in $\ket{\psi^\prime}=\ket{\psi^m(t+\tau)}$, of the manifold allows one to determine the updated product state $\ket{\psi_A(t+\tau)\psi_B(t+\tau)}$.
• Figure 2: (a) Scheme of the decay process given in Eq. \ref{['eq:exampleGenerators']}. Comparison between the separable (dotted, purple line) and conventional (solid, turquoise line) Monte Carlo wave function methods: (b) Entanglement, characterized by the negativity of the partially transposed state. (c) Population of the ground state $\ket{00}$. (d) Probability to occupy one of the intermediate levels $|\Phi_+\rangle$ and $|\Phi_-\rangle$. One standard deviation uncertainty in lighter-colored bands. Note that the evolution of the probability to occupy the excited state can be computed by subtracting to $1$ the sum of probabilities displayed in (c) and (d). Parameters for the numerical simulation: $\gamma_{11\to\Phi_+}=9=\gamma_{\Phi_-\to00}, \gamma_{11\to\Phi_-}=1=\gamma_{\Phi_+\to00}$, step size $\varepsilon=0.2$, and a sample size of $600$Zenodo.
• Figure 3: Comparison between the separable (dashed, purple line) and conventional (solid, turquoise line) Monte Carlo wave function method for the decay process given in Eq. \ref{['eq:exampleGeneratorsSep']}, with one standard deviation uncertainty (lighter colored bands). Contrasting the process in Fig. \ref{['fig:exampleLoss']}, the here described decay mechanism does not lead to entanglement, panel (b). Also, decay into the ground state, panel (c), and the occupation of the intermediate states, panel (d), show no significant deviation from another. This confirms the expected consistency of the separable Monte Carlo technique when no entanglement is involved in the process. Selected parameters for the numerical simulation are the same as in Fig. \ref{['fig:exampleLoss']}.
• Figure 4: Comparison between the separable (dashed, purple line) and conventional (solid, turquoise line) Monte Carlo wave function method for the process in Eq. \ref{['eq:exampleCNOTGenerator']} with one standard deviation uncertainty (lighter colored bands): (a) For the initial state $(|0\rangle+|1\rangle)/\sqrt2\otimes\ket{0}$, entanglement is quantified via negativity under partial transposition. (b) For the same initial state, overlap between $\rho(t)$ and the Bell state $\ket{\Phi_+}$. The trajectories do not converge due to the entanglement present of state for $t\to\infty$ for the unrestricted evolution. Panels (c) and (d) depict the results for the initial state $\ket{10}$. The expectation that both the separable and inseparable technique ought to lead to the same classically correlated outcome for this input is confirmed. The parameters for the numerical simulations are $\varepsilon=0.2$ and a sample size of $400$Zenodo.