  A fundamental distinction between many-body quantum states are those with
short- and long-range entanglement (SRE and LRE). The latter cannot be created
by finite-depth circuits, underscoring the nonlocal nature of Schr\"odinger cat
states, topological order, and quantum criticality. Remarkably, examples are
known where LRE is obtained by performing single-site measurements on SRE, such
as the toric code from measuring a sublattice of a 2D cluster state. However, a
systematic understanding of when and how measurements of SRE give rise to LRE
is still lacking. Here, we establish that LRE appears upon performing
measurements on symmetry-protected topological (SPT) phases -- of which the
cluster state is one example. For instance, we show how to implement the
Kramers-Wannier transformation by adding a cluster SPT to an input state
followed by measurement. This transformation naturally relates states with SRE
and LRE. An application is the realization of double-semion order when the
input state is the $\mathbb Z_2$ Levin-Gu SPT. Similarly, the addition of
fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner
transformation of a general state. More generally, we argue that a large class
of SPT phases protected by $G \times H$ symmetry gives rise to anomalous LRE
upon measuring $G$-charges, and we prove that this persists for generic points
in the SPT phase under certain conditions. Our work introduces a new practical
tool for using SPT phases as resources for creating LRE, and uncovers the
classification result that all states related by sequentially gauging Abelian
groups or by Jordan-Wigner transformation are in the same equivalence class,
once we augment finite-depth circuits with single-site measurements. In
particular, any topological or fracton order with a solvable finite gauge group
can be obtained from a product state in this way.
