Local measurements and the entanglement transition in quantum spin chains
Sven Bachmann, Mahsa Rahnama, Gabrielle Tournaire
TL;DR
The paper analyzes how local measurements on symmetry-protected topological states in infinite 1D spin chains induce an entanglement transition from short-range to long-range order. By formulating SRE via split automorphisms and exploiting on-site Abelian G×H symmetries, it proves that post-measurement states cannot remain uniformly short-range entangled, as string-order parameters embedded in the pre-measurement SPT state generate long-range correlations after projection. When the entangler is a quantum cellular automaton, blocking measurements yields an infinite-volume post-measurement state that is long-range entangled, and, under mild commutation conditions, the long-range order emerges even without blocking. The work connects SPT cohomology, projective symmetry representations in the GNS framework, and quasi-local automorphisms to illuminate how measurements reshape entanglement structure, offering a rigorous route to entanglement transitions in 1D quantum systems.
Abstract
We consider the transition between short-range entangled (SRE) and long-range ordered (and therefore long-range entangled) states of infinite quantum spin chains, which is induced by local measurements. Specifically, we assume that the initial state is in a non-trivial symmetry-protected topological phase with local symmetry group $\mathcal{G} = G \times H$, where $G$ is an Abelian subgroup. We show that the on-site measurements of the local $G$-charge on intervals of increasing lengths transform the initial SRE state into a family of states with increasingly long-range correlations. In particular, the post-measurement states cannot be uniformly short-range entangled. In the case where the initial state is obtained from a product state using a quantum cellular automaton, we construct the infinite-volume post-measurement state and exhibit almost local observables that are maximally correlated.