Tripartite entanglement and tripartite steering in three-qubit pure states induced by vacuum--one-photon superpositions

Abstract

Utilizing a tritter with variable parameter $T$ and induced by vacuum--one-photon superpositions $\left\vert 0\right\rangle +\alpha\left\vert 1\right\rangle $ with $\alpha =\left\vert \alpha \right\vert e^{i\phi }$, we propose a scheme to prepare a class of three-qubit pure states. These states take the form of $\left\vert \psi \right\rangle _{123}=c_{0}\left\vert 000\right\rangle +c_{1}\left\vert 100\right\rangle +c_{2}\left\vert 010\right\rangle +c_{3}\left\vert 001\right\rangle $. The coefficients ($c_{0}$, $c_{1}$, $c_{2}$, and $c_{3}$) can be manipulated through interaction parameters ($\left\vert \alpha \right\vert $, $\phi $, and $T$). In line with Xie and Eberly's work[Phys. Rev. Lett. 127, 040403 (2021)], we investigate the genuine tripartite entanglement for $\left\vert \psi \right\rangle _{123}$ by using the measure of concurrence fill. Drawing on Hao \textit{et al.}'s research [Phys. Rev. Lett. 128, 120402 (2021)], we examine tripartite steering for $\left\vert \psi \right\rangle _{123}$ under certain measurements based on the uncertainty relations criterion. We identify nine potential configurations exhibiting varying steerability across different parameter spaces. It is important to highlight that, while the state $\left\vert \psi \right\rangle _{123}$ exhibits entanglement, steering remains unattainable in a substantial portion of the parameter space.