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 $|0\ensuremath{\rangle}+\ensuremath{\alpha}|1\ensuremath{\rangle}$ with $\ensuremath{\alpha}=|\ensuremath{\alpha}|{e}^{i\ensuremath{\phi}}$, we propose a scheme to prepare a class of three-qubit pure states. These states take the form of ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}={c}_{0}|000\ensuremath{\rangle}+{c}_{1}|100\ensuremath{\rangle}+{c}_{2}|010\ensuremath{\rangle}+{c}_{3}|001\ensuremath{\rangle}$. The coefficients $({c}_{0},$ ${c}_{1},$ ${c}_{2}, \text{and} {c}_{3})$ can be manipulated through interaction parameters $(|\ensuremath{\alpha}|,$ $\ensuremath{\phi}, \text{and} T)$. In line with Xie and Eberly's work [Phys. Rev. Lett. 127, 040403 (2021)], we investigate the genuine tripartite entanglement for ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}$ by using the measure of concurrence fill. Drawing on Hao et al.'s research [Phys. Rev. Lett. 128, 120402 (2022)], we examine tripartite steering for ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}$ 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 ${|\ensuremath{\psi}\ensuremath{\rangle}}_{123}$ exhibits entanglement, steering remains unattainable in a substantial portion of the parameter space.