boundary entropy [@AL]. For the Lipkin-Meshkov-Glick model, a system of mutually interacting spins embedded in a magnetic field for which analytic results can be derived, the global GE of the ground state and the single-copy entanglement behave as the entanglement entropy close to and at criticality [@LMG].

GE thus serves as a useful tool to investigate quantum criticality in quantum many-body lattice systems. Apart from some exceptions [@hl; @gehlw; @thesis; @roman; @z3pott], almost all work to date exploiting GE to study phase transitions has been restricted to quantum systems in 1D. This is mainly due to the difficulty to compute GE, because it involves a formidable optimization over all possible separable states. Indeed, the calculation of various entanglement measures has been shown recently to be NP-complete [@Ionnou; @Huang]. This is further compounded by the inherent computational difficulties posed by two-dimensional (2D) quantum systems. Nevertheless, significant progress has been made to develop efficient numerical algorithms to simulate 2D quantum many-body lattice systems in the context of tensor network representations [@verstraete; @vidal; @gv2d; @jwx; @bvt; @rosgv; @pwe; @sz; @hco; @wks; @orusreview; @tn_latest]. The algorithms have been successfully exploited to compute, for example, the ground-state fidelity per lattice site [@zhou; @zov; @jhz; @whl; 