Generating large-scale Greenberger-Horne-Zeilinger-like states in lattice spin systems

Abstract

Greenberger-Horne-Zeilinger (GHZ) state is a typical maximally entangled state which is pursued in both fundamental research and emerging quantum technologies. Preparing large-scale GHZ states in lattice spin systems is particularly appealing for quantum advantages, but conventional schemes face great challenges in scalability. Here we propose a universal and scalable scheme to generate large-scale GHZ-like states, which share similar entanglement and metrological properties with standard GHZ states, in lattice spin systems through global Floquet engineering. Our scheme requires only global operations and shows great advantage for large particle number. It is applicable to systems with arbitrary interaction ranges, offering a practical pathway for large-scale implementation of many-body entangled states in various systems.

Figures (6)

• Figure 1: Schematic diagram of the scheme to generate GHZ-like states in lattice spin systems governed by the power-law Ising model. (a) An illustration of the pulse sequences. The red and blue pulses denote $\pm\pi/2$ pulses along $x$ and $y$ axis, respectively. The spins effectively interact through $s_j^y s_k^y$ (yellow shaded), $s_j^x s_k^x$ (purple shaded) and $s_j^z s_k^z$ (green shaded) in the three successive parts of one period, each taking the time of $\tau$. (b) An illustration of the model. The spins located in a square lattice interact through power-law Ising interaction, with interaction strength decaying with distance $r$ as $1/r^\alpha$. The noncommutativity of the three segments in one period gives rise to the effective three-body interaction.
• Figure 2: Characterization of the GHZ-like state generated in our scheme, calculated for (a) $20\times1$ spins, $\alpha=1$, $K\tau=0.1$ and (b) $4\times4$ spins, $\alpha=3$, $K\tau=0.06$. (a1), (b1) The probability distributions $P(m)$ of the obtained states, where $m$ denotes the eigenvalue of $S_x$. (a2), (b2) Parity oscillations of the obtained states. Black dashed curves show the results of perfect GHZ states $\expval{\Pi(\theta)}_{\mathrm{GHZ}}=\cos N\theta$. (a3), (b3) Time evolution of the quantum Fisher information $F_\mathrm{Q}$, obtained using exact diagonalization (ED, red solid) and discrete truncated Wigner approximation (DTWA, blue dashed).
• Figure 3: The influence of the pulse separation $\tau$ on the evolution for 2D lattice of $20\times 20$ spins with decaying factor $\alpha =2$. (a) The time evolution of the QFI $F_{\mathrm{Q}}$ for different pulse separations, compared with the result of the zero-momentum Hamiltonian $H_{\mathrm{ZM}}$ (red dashed curve) and the Heisenberg limit $F_{\mathrm{Q}}=N^{2}$ (black dotted line). The time evolution of finite-momentum spin-wave excitation $N_{\mathrm{FM}}$ is shown in the inset. Here the evolution time has been rescaled with a factor $\chi _{\mathrm{eff}}(\tau )=\lambda NK^{2}\tau /6$. (b) The maximal QFI (red square) and the corresponding total evolution time $t_{\mathrm{tot}}$ (blue circle) versus the pulse separation $\tau$. Open squares (circles) denote the optimal QFI over all generator directions $F_\mathrm{Q}^\mathrm{opt}$ (and the corresponding evolution time), whereas solid squares (circles) correspond to the QFI in regard to the generator $S_x$, $F_\mathrm{Q}^{S_x}$ (and the corresponding evolution time). The red dashed line indicates the maximal $F_{\mathrm{Q}}^{S_x}$ of the zero-momentum Hamiltonian. The QFI is obtained using the DTWA (averaged over 1000 trajectories).
• Figure 4: The performance of the scheme versus different lattice length $L$, pulse separation $\tau$ and decaying factor $\alpha$ for (a) 1D and (b) 2D lattices. (a1, b1) The ratio between the maximal quantum Fisher information of the scheme and the effective one ($H_{\mathrm{ZM}}$) $F_{\mathrm{Q}}/F_{\mathrm{Q}}^{\mathrm{eff}}$ as a function of $L$ and $\tau$, calculated at $\alpha =1.5$. The black dashed line shows the fitting result of $F_{\mathrm{Q}}/F_{\mathrm{Q}}^{\mathrm{eff}}=0.8$. (a2, b2) The power-law exponent $\mu$ of the suitable pulse separation $\tau _{\mathrm{s}}$ which approximately obeys $\tau _{\mathrm{s}}\sim L^{-\mu }$, changing with $\alpha$. Both the numerical fitting results (blue circle) and the value predicted by finite-momentum spin-wave excitations as Eq. \ref{['mu']} (red solid line) are presented. (a3, b3) The total evolution time as a function of $L$ for different $\alpha$, compared with the fitted scaling law $t_{\mathrm{tot}}\sim L^{-\nu}\ln L$ with $\nu$ given by Eq. \ref{['nu']} (solid lines with corresponding colors). Numerical results are obtained using the DTWA (averaged over 1000 trajectories).
• Figure 5: The maximal QFI as a function of the decoherence rate for both local ($\gamma$) and global dephasing ($\Gamma$), evaluated under the proposed experimental parameters ($N=12$ spins in 1D lattice, $K=560\,\mathrm{Hz}$, and $\alpha =1.0$). The pulse separation is fixed at $\tau=0.18\,\text{ms}$. The black dashed line represents the ideal case without decoherence. Inset: expectation value of the parity $\Pi =\prod_{j=1}^{N} \sigma _{j}^{z}$ after applying $e^{i\theta S_{x}}$, calculated at a decoherence rate of $10\text{Hz}$ (gray line). Calculations are based on the exact diagonalization.