  Unlike unitary dynamics, measurements of a subsystem can induce long-range
entanglement via quantum teleportation. The amount of measurement-induced
entanglement or mutual information depends jointly on the measurement basis and
the entanglement structure of the state (before measurement), and has
operational significance for whether the state is a resource for
measurement-based quantum computing, as well as for the computational
complexity of simulating the state using quantum or classical computers. In
this work, we examine entropic measures of measurement-induced entanglement
(MIE) and information (MII) for the ground-states of quantum many-body systems
in one- and two- spatial dimensions. From numerical and analytic analysis of a
variety of models encompassing critical points, quantum Hall states, string-net
topological orders, and Fermi liquids, we identify universal features of the
long-distance structure of MIE and MII that depend only on the underlying phase
or critical universality class of the state. We argue that, whereas in $1d$ the
leading contributions to long-range MIE and MII are universal, in $2d$, the
existence of a teleportation transition for finite-depth circuits implies that
trivial $2d$ states can exhibit long-range MIE, and the universal features lie
in sub-leading corrections. We introduce modified MIE measures that directly
extract these universal contributions. As a corollary, we show that the leading
contributions to strange-correlators, used to numerically identify topological
phases, are in fact non-universal in two or more dimensions, and explain how
our modified constructions enable one to isolate universal components. We
discuss the implications of these results for classical- and quantum-
computational simulation of quantum materials.
