 We developed a scheme to reduce the computational complexity of extracting low-lying entanglement spectra from quantum Monte Carlo simulations. This scheme relies on the path integral formulation of the reduced density matrix, which allows us to overcome the exponential growth of computational complexity associated with these calculations. The scheme was tested on the Heisenberg spin ladder with long entangled boundaries between two chains, and our results supported the Lie-Haldane conjecture regarding the entanglement spectrum of topological phases. Furthermore, we explained this conjecture using the wormhole effect in the path integral, and showed that it could be extended to systems beyond gapped topological phases. Simulations of the bilayer-antiferomagnetic Heisenberg model with two de-entangled boundaries across the 2-plus-1-do-3-quantum phase transition confirmed the correctness of the wormhole picture. Finally, we noted that the relative strength of the wormhole effect compared to the edge energy gap determines the behavior of the low-lying entanglement spectrum of the system. This article was authored by Jingyang and Ziyang Meng. We are article.tv, links in the description below.