  Estimating global properties of many-body quantum systems such as entropy or
bipartite entanglement is a notoriously difficult task, typically requiring a
number of measurements or classical post-processing resources growing
exponentially in the system size. In this work, we address the problem of
estimating global entropies and mixed-state entanglement via partial-transposed
(PT) moments, and show that efficient estimation strategies exist under the
assumption that all the spatial correlation lengths are finite. Focusing on
one-dimensional systems, we identify a set of approximate factorization
conditions (AFCs) on the system density matrix which allow us to reconstruct
entropies and PT moments from information on local subsystems. This yields a
simple and efficient strategy for entropy and entanglement estimation. Our
method could be implemented in different ways, depending on how information on
local subsystems is extracted. Focusing on randomized measurements (RMs),
providing a practical and common measurement scheme, we prove that our protocol
only requires polynomially-many measurements and post-processing operations,
assuming that the state to be measured satisfies the AFCs. We prove that the
AFCs hold for finite-depth quantum-circuit states and translation-invariant
matrix-product density operators, and provide numerical evidence that they are
satisfied in more general, physically-interesting cases, including thermal
states of local Hamiltonians. We argue that our method could be practically
useful to detect bipartite mixed-state entanglement for large numbers of qubits
available in today's quantum platforms.
