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We present a numerical strategy to efficiently estimate bipartiteentanglement measures, and in particular the Entanglement of Formation, formany-body quantum systems on a lattice. Our approach exploits the Tree TensorOperator tensor network ansatz, a positive loopless representation for densitymatrices which, as we demonstrate, efficiently encodes information on bipartiteentanglement, enabling the up-scaling of entanglement estimation. Employingthis technique, we observe a finite-size scaling law for the entanglement offormation in 1D critical lattice models at finite temperature for up to 128spins, extending to mixed states the scaling law for the entanglement entropy.

Entanglement of formation of mixed many-body quantum states via Tree Tensor Operators