Relation of the entanglement entropy and uncertainty product in ground states of coupled anharmonic oscillators

We explore quantitatively the fundamental connection between the entropy and local uncertainty product as an entanglement measure within the lowest energy eigenstate of a class of two-coupled anharmonic oscillator systems. We find that these two quantum aspects approach a unique relation as the entanglement between the oscillators becomes large if the potential of the first collective mode displays a generic single-well potential structure. This implies that the uncertainty product can be a good choice for the characterization of entanglement when the entropy is sufficiently large. Interestingly, although this relation is robust against the anharmonic perturbation, we find that it can be altered effectively by a quantum catastrophe.