  Entanglement, measurement, and classical communication together enable
teleportation of quantum states between distant parties, in principle with
perfect fidelity. To what extent do correlations and entanglement of a
many-body wavefunction transfer under imperfect teleportation protocols? We
address this question for the case of an imperfectly teleported quantum
critical wavefunction, focusing on the ground state of a critical Ising chain.
We demonstrate that imperfections, e.g., in the entangling gate adopted for a
given protocol, effectively manifest as weak measurements acting on the
otherwise pristinely teleported critical state. Armed with this perspective, we
leverage and further develop the theory of measurement-altered quantum
criticality to quantify the resilience of critical-state teleportation. We
identify classes of teleportation protocols for which imperfection $(i)$
preserves both the universal long-range entanglement and correlations of the
original quantum critical state, $(ii)$ weakly modifies these quantities away
from their universal values, and $(iii)$ obliterates long-range entanglement
altogether while preserving power-law correlations, albeit with a new set of
exponents. We also show that mixed states describing the average over a series
of sequential imperfect teleportation events retain pristine power-law
correlations due to a `built-in' decoding algorithm, though their entanglement
structure measured by the negativity depends on errors similarly to individual
protocol runs. These results may allow one to design teleportation protocols
that optimize against errors -- highlighting a potential practical application
of measurement-altered criticality.
