  Local measurements cannot create entanglement, but they can convert
short-range entanglement to long-range entanglement, as in quantum
teleportation. This phenomenon of measurement-induced entanglement (MIE) has
been widely discussed in recent work on measurement-induced entanglement phase
transitions and related phenomena. Here, we situate MIE in a broader context of
the growth of long-range conditional mutual information (CMI) under
decoherence. We upper-bound the rate at which decoherence can generate
long-range CMI, and derive a characterization of states that saturate this
bound. We point out that the structure of states saturating the CMI upper bound
can be very different under different decoherent dynamics and provide explicit
examples. We additionally explore the dynamics of CMI in random quantum
circuits subject to random local decoherence, as a function of circuit depth.
We argue that the universality class of the finite-depth teleportation
transition, as well as its lower critical dimension, are different for erasures
than for measurements.
