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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 $ρ_0$ and its reduced substate $ρ_A = \mbox{Tr}_B ρ_0$, we compute two entanglement measures of $ρ_0$, i.e. ${\cal E}_{EoF} (ρ_0) = S_{von} (ρ_A)$ and ${\cal E}_γ (ρ_0) = S_γ (ρ_A)$, where $S_{von}$ and $S_γ$ are the von Neumann and Rényi entropies, up to the first order of the GUP parameter $α$. It is shown that ${\cal E}_γ (ρ_0)$ increases with increasing $α$ when $γ= 2, 3, \cdots$. The remarkable fact is that ${\cal E}_{EoF} (ρ_0)$ does not have first-order of $α$. Based on there results we conjecture that ${\cal E}_γ (ρ_0)$ increases or decreases with increasing $α$ when $γ> 1$ or $γ< 1$ respectively for nonnegative real $γ$.