Quantum Entanglement with Generalized Uncertainty Principle

We explore how the quantum entanglement is modified in the generalized uncertainty principle (GUP)-corrected quantum mechanics by introducing the coupled harmonic oscillator system. Constructing the ground state ${\rho }_0$ and its reduced substate ${\rho }_A=T{r}_B{\rho }_0$, we compute two entanglement measures of ${\rho }_0$, i.e. ${\mathcal{E}}_{EoF}({\rho }_0)=S_{von}({\rho }_A)$ and ${\mathcal{E}}_{\gamma }({\rho }_0)=S_{\gamma }({\rho }_A)$, where $S_{von}$ and $S_{\gamma }$ are the von Neumann and R'{e}nyi entropies, up to the first order of the GUP parameter $\alpha$. It is shown that ${\mathcal{E}}_{\gamma }({\rho }_0)$ increases with increasing $\alpha$ when $\gamma =2,3,\cdots$. The remarkable fact is that ${\mathcal{E}}_{EoF}({\rho }_0)$ does not have first-order of $\alpha$. Based on there results we conjecture that ${\mathcal{E}}_{\gamma }({\rho }_0)$ increases or decreases with increasing $\alpha$ when $\gamma >1$ or $\gamma < 1$respectivelyfornonnegativereal$\gamma$.