  We study theoretically how local measurements perfomed on critical quantum
ground states affect long-distance correlations. These states are highly
entangled and feature algebraic correlations between local observables. As a
consequence, local measurements can have highly nonlocal effects. Our focus is
on Tomonaga-Luttinger liquid (TLL) ground states, a continuous family of
critical states in one dimension whose structure is parameterized by a
Luttinger parameter $K$. We show that arbitrarily weak local measurements,
performed over extended regions of space, can conspire to drive transitions in
long-distance correlations. Conditioning first on a particular measurement
outcome we show that there is a transition in the character of the
post-measurement quantum state for $K<1$, and highlight a formal analogy with
the effect of a static impurity on transport through a TLL. To investigate the
full ensemble of measurement outcomes we consider averages of physical
quantities which are necessarily nonlinear in the system density matrix. We
show how their behavior can be understood within a replica field theory, and
for the measurements that we consider we find that the symmetry of the theory
under exchange of replicas is broken for $K<1/2$. A well-known barrier to
experimentally observing the collective effects of multiple measurements has
been the need to post-select on random outcomes. Here we resolve this problem
by introducing cross-correlations between experimental measurement results and
classical simulations, which act as resource-efficient probes of the
transition. The phenomena we discuss are, moreover, robust to local
decoherence.
