Measurements Conspire Nonlocally to Restructure Critical Quantum States

We study theoretically how local measurements performed on critical quantum ground states affect longdistance 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 ground states, a continuous family of critical states in one dimension whose structure is characterized 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 translation-invariant set of measurement outcomes, we show that there is a transition in the character of the postmeasurement quantum state for K < 1, and highlight a formal analogy with the effect of a static defect on transport through a Tomonaga-Luttinger liquid. 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 postselect 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.