The utmost distance for quantum entanglement

Abstract: A common viewpoint is that a particle could be quantum entangled with another particle arbitrarily far away. But in this paper we suggest that there is an utmost distance for the existence of quantum entanglement between two particles, beyond which the initial quantum entanglement would be broken by some quantum gravitational effect. The utmost distance is proposed to be $L_{QE}=\lambda^\alpha l_{p}^{1-\alpha}$, where $\lambda$ is the quantum wavelength of the particles and $l_{p}= 1.616 \times 10^{-35} m$ is the Planck length. The most probable value of the parameter $\alpha$ is $2$ or $3$. As other quantum-gravitational effects, this effect is very weak and hard to be detected in foreseeable experiment.