Dissipative State Engineering of Complex Entanglement with Markovian Dynamics

TL;DR

This work addresses the challenge of deterministically generating multipartite entanglement by engineering dissipative, Markovian dynamics that target a cluster state as the unique steady state. The authors introduce projection-based Lindblad operators $L_m=|C_N\rangle\langle\phi_m|$ acting in a spin chain with Ising interactions, proving that in the strong-dissipation limit the steady state is the pure cluster state $|C_N\rangle\langle C_N|$ with a finite Liouvillian gap that ensures rapid convergence. Numerical results for 1D and an extension to 2D demonstrate high fidelity (F approaching 1) and robust multipartite entanglement, as signaled by entanglement witnesses, with the dissipation strength $\gamma_{\text{sat}}$ scaling roughly linearly with system size. The scheme is analyzed via mean-field theory and exact simulations, and its experimental viability is discussed for trapped-ion platforms, including strategies for scalable implementation. Overall, the paper provides a physically realizable route to steady-state entanglement generation that may be scalable to large quantum networks and universal cluster-state resources for measurement-based quantum computation.

Abstract

Highly multipartite entangled states play an important role in various quantum computing tasks. We investigate the dissipative generation of a complex entanglement structure as in a cluster state through engineered Markovian dynamics in the spin systems coupled via Ising interactions. Using the Lindblad master equation, we design a projection based dissipative channel that drives the system toward a unique pure steady state corresponding to the desired cluster state. This is done by removing the contribution of the orthogonal states. By explicitly constructing the Liouvillian superoperator in the full $2^N$-dimensional Hilbert space, we compute the steady-state density matrix, the Liouvillian spectral gap, entanglement witness and the fidelity with respect to the ideal cluster state. The results demonstrate that the cluster state emerges as the steady state when the engineered Liouvillian dissipation dominates over the local Ising interaction between spins. Moreover, we find that the fidelity and Liouvillian spectral gap is relatively insensitive to the system size once the saturation dissipation has been achieved that scales linearly with the qubit number. This analysis illustrates a physically realizable path towards steady-state entanglement generation in the spin systems using engineered dissipation.

Figures (8)

• Figure 1: Schematic representation of (a) the linear cluster state in one dimension \ref{['eq:N-Cluster state']}; (b) the square lattice cluster state in two dimension \ref{['eq:sqaure_cluster state']}. Each qubit (vertex) is initialized in the superposition state $|+\rangle$ with the controlled phase CZ gate interaction (edges) between the connecting qubits j,k.
• Figure 2: Dissipative model \ref{['eq:masterequation']} for a linear chain of $N$ spin qubits: each k$^\text{th}$ qubit interacts with its nearest neighbors (k-1$^\text{th}$ and k+1$^\text{th}$) through Ising coupling \ref{['eq:ham']} (Coherent dynamics). Each qubit is coupled to a single memoryless reservoir (dissipation dynamics) in an identical way such that the desired cluster state is the steady state of the Liouvillian \ref{['eq:lemma1']}.
• Figure 3: Schematic illustration of the construction of the Lindblad jump operator in Eq. \ref{['eq:Lindbladope']}. The dissipative dynamics projects an arbitrary state $\rho$ from the full Hilbert space onto the subspace spanned by the target cluster state. Equivalently, the action of the jump operator can be interpreted as driving the system toward a specific energy eigenstate $E_0$, as depicted in the adjacent panel, thereby stabilizing the cluster state as the unique steady state of the evolution.
• Figure 4: Dynamics of dissipative model \ref{['eq:masterequation']} with Markovian environment: Variation of the steady state spin expectation values with the controlling parameters (a) no dissipation ($\gamma=0$); (b) with dissipation at a fixed value of $h_g=2.0$ (closed circle) and $h_g=1.0$ (open circle). Distinct phases are marked as $s_j$. Exact time evolution of the density matrix and spin expectation values \ref{['spin_exp_finite']} as a function of time $t$ (c) no dissipation ($\gamma=0$); (d) with dissipation ($\gamma_g=5$) for system size with $N=4$ and two different initial states, $\rho_1(0) = |++++\rangle \langle ++++|$ (solid line) and $\rho_2(0) = |0000\rangle \langle 0000|$ (dashed line).
• Figure 5: Fidelity \ref{['eq:fidelity']} and its scaling with the system size $N$ for the dissipative model \ref{['eq:masterequation']} under Markovian dynamics: (a) variation with the dissipative parameter $\gamma_g$ with varying $N$. A zoomed-in plot is shown for the variation of the saturated dissipation $\gamma_{\text{sat}}$ with varying $N$ in the inset capturing a linear scaling; (b) the maximum of the fidelity $F_{\text{sat}}$ is plotted as a function of the system size $N$ showing a power law scaling ($\propto N^{\beta}$). The system parameters are chosen as specified in each panel. For all cases, we choose $h_g=1$.

Theorems & Definitions (2)

• Lemma 1
• proof