Abstract: Randomized measurements provide a practical scheme to probe complex many-body quantum systems. While they are very powerful to extract local information, global properties such as bipartite entanglement remain hard to probe, requiring a number of measurements or classical post-processing resources growing exponentially in the system size. In this talk, I will address the problem of estimating pure and mixed-state entanglement via partial-transposed (PT) moments, and show that efficient estimation strategies exist under assumptions which are very natural from the point of view of quantum simulation, namely that all the spatial correlation lengths are finite. Focusing on 1D systems, I will introduce a set of approximate factorization conditions on the system density matrix and show that the latter allow us to reconstruct entanglement entropies and PT moments from information on local subsystems. Combined with the RM toolbox, this yields a simple strategy for entanglement estimation which is provably accurate and which only requires polynomially-many measurements and post-processing operations. I will discuss the generality of the approximate factorization conditions, showing their validity in large classes of many-body states, including matrix-product density operators and thermal states of local Hamiltonians. I will argue that the proposed method could be practically useful to detect bipartite mixed-state entanglement for large numbers of qubits available in today’s quantum platforms.