Robust Tripartite Entanglement Generation via Correlated Noise in Spin Qubits

Abstract

We investigate the generation of genuine tripartite entanglement in a triangular spin-qubit system due to spatially correlated noise. In particular, we demonstrate how the formation of a highly entangled dark state -- a W state -- enables robust, long-lived tripartite entanglement. Surprisingly, we find that environmentally induced coherent coupling does not play a crucial role in sustaining this entanglement. This contrasts sharply with the two-qubit case, where the induced coupling significantly influences the entanglement dynamics. Furthermore, we explore two promising approaches to enhance the tripartite entanglement by steering the system towards the dark state: post-selection and coherent driving. Our findings offer a robust method for generating high-fidelity tripartite entangled states with potential applications in quantum computation.

Figures (3)

• Figure 1: (a) A system of three spin qubits is arranged in an equilateral triangle and coupled to a noisy environment. (b) The states of the three-qubit system [spheres in (b)] can be labeled by the eigenvalues $\chi \in \{0, \pm 1\}$ of the chirality operator $\hat{\chi} =\vec{\sigma}_1 \cdot \vec{\sigma}_2 \times \vec{\sigma}_3/2\sqrt{3}$ and the eigenvalues $S^z \in \{\pm 3/2, \pm 1/2\}$ of the total spin-$z$ operator $\hat{S}^z = \sum_i \hat{\sigma}_i^z/2$. Noise induces transitions (arrows) between the eigenstates, while correlated spatial noise introduces an imbalance in the transition amplitudes. When the non-local dissipation is strong, one type of transition can be suppressed (orange dashed arrows), and this allows the formation of a dark state (dashed box).
• Figure 2: Entanglement dynamics demonstrating long-lived entanglement for parameters such that one of the jump amplitudes $\gamma_k$ in Eq. \ref{['eq:gamma_k']} vanishes. (a) Evolution of the tripartite negativity $\mathcal{N}_{123}$ for different values of $\phi$ when the system is initialized in the state $\ket{\downarrow\uparrow\uparrow}$. (b) Entanglement dynamics for varying correlated noise. (c) Entanglement dynamics for varying coherent interaction amplitude $\mathcal{J}$. (d) Upon initialization in a spin product state in the total spin-$z$ sectors $S^z=-1/2$ (red) and $S^z=-3/2$ (green), the system exhibits revival and bursts of entanglement. Parameters used throughout unless stated otherwise: $\mathcal{J} = 0$, $|A|/a=0.5$, $\phi = \pi$, and $\Delta/a= 100$.
• Figure 3: Enhancement of entanglement generation: (a)-(b) post selection and (c)-(d) driving. (a) By periodically measuring the total spin-$z$ and post-selecting states with $S^z=1/2$, the decay from the spin-sector $S^z=1/2$ to the spin sector $S^z=3/2$ is inhibited. Since the decay rates are imbalanced, this effectively leads to transfer of probability to the state with the slowest decay rate. (b) Entanglement dynamics determined by Eq. \ref{['eq:postSelectionEoM']} for $|A|/a = 0.5$, and various $\alpha$. Post-selection increases the tripartite entanglement. (c) Energy level diagram in the presence of coherent coupling with $\psi = 0$. The states within a given total spin-$z$ sector are split. The system is driven with frequency corresponding to the splitting between the ground state and a single $W$ state. (d) Entanglement dynamics in the presence of driving for $\mathcal{J}/a =10$ and various values of the correlated noise. Parameters used throughout: $\phi= \pi$, $\mathcal{J}/a = 0$, and $\Delta/a = 100$ unless specified otherwise.