Quantum Entanglement with Generalized Uncertainty Principle

TL;DR

This paper investigates how quantum entanglement measures are affected by the generalized uncertainty principle (GUP) in a coupled harmonic oscillator system, revealing that certain entanglement measures increase with GUP parameter, while others do not at first order.

Contribution

It introduces GUP corrections to quantum entanglement measures in a coupled harmonic oscillator, analyzing their dependence on the GUP parameter and proposing a conjecture on their behavior.

Findings

Entanglement measure ${ m EoF}$ does not have first-order GUP correction.

Entanglement measure ${ m E}_ ext{ extgamma}$ increases with GUP parameter for $ extgamma=2,3, extellipsis$.

${ m E}_ ext{ extgamma}$'s behavior depends on $ extgamma$ being greater or less than 1.

Abstract

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 $\rho_A = \mbox{Tr}_B \rho_0$, we compute two entanglement measures of $\rho_0$, i.e. ${\cal E}_{EoF} (\rho_0) = S_{von} (\rho_A)$ and ${\cal E}_{\gamma} (\rho_0) = S_{\gamma} (\rho_A)$, 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 ${\cal E}_{\gamma} (\rho_0)$ increases with increasing $\alpha$ when $\gamma = 2, 3, \cdots$. The remarkable fact is that ${\cal E}_{EoF} (\rho_0)$ does not have first-order of $\alpha$. Based on there results we conjecture that ${\cal E}_{\gamma} (\rho_0)$ increases or decreases with increasing $\alpha$ when $\gamma > 1$ or $\gamma < 1$ respectively for nonnegative real $\gamma$.