Probing Quantum Entanglement from Quantum Correction to Newtonian Potential Energy

Inspired by string theory ideas, we probe quantum entanglement from the gravitational potential energy. Concretely, we reconsider the study of quantum corrections to the Newtonian potential energy by treating a massive two-particle system $m_{1}$ and $m_{2}$ with size dimensions $r_{1}$ ad $$ where the two particles separated by a distance $d$ are under only their mutual classical gravitational interaction V_{r}\left( r_{1}\text{, }% r_{2}\right) . Exploring such a size-dependent gravitational behavior and taking the limit $r_{1}$, $r_{2}\ll d$, we investigate the associated quantum biparticle state and express its evolution after an interaction time $\tau$. Among others, we show that the two masses cannot be separable due to the induced gravitational entanglement in terms of the accumulated quantum phase $\delta \phi =\delta V_{g}\tau /\hbar$. By analogy with the classical gravity, we derive the expression of the resulting extremely weak entanglement force from the corresponding gravitational entanglement energy. Then, we provide certain entanglement diagnostics.