  Understanding the influence of measurements on the properties of many-body
systems is a fundamental problem in quantum mechanics and for quantum
technologies. This paper explores how a finite density of stochastic local
measurement modifies a given state's entanglement structure. Considering
various measurement protocols, we explore the typical quantum correlations of
their associated projected ensembles arising from the ground state of the
quantum Ising model. Using large-scale numerical simulations, we demonstrate
substantial differences among inequivalent measurement protocols. Surprisingly,
we observe that forced on-site measurements can enhance both bipartite and
multipartite entanglement. We present a phenomenological toy model and
perturbative calculations to analytically support these results. Furthermore,
we extend these considerations to the non-Hermitian Ising model, naturally
arising in optically monitored systems, and we show that its qualitative
entanglement features are not altered by a finite density of projective
measurements. Overall, these results reveal a complex phenomenology where local
quantum measurements do not simply disentangle degrees of freedom, but may
actually strengthen the entanglement in the system.
