  Quantum critical systems constitute appealing platforms for the exploration
of novel measurement-induced phenomena due to their innate sensitivity to
perturbations. We study the impact of measurement on paradigmatic Ising quantum
critical chains using an explicit protocol, whereby correlated ancilla are
entangled with the critical chain and then projectively measured. Using a
perturbative analytic framework supported by extensive numerical simulations,
we demonstrate that measurements can qualitatively alter long-distance
correlations in a manner dependent on the choice of entangling gate, ancilla
measurement basis, measurement outcome, and nature of ancilla correlations. We
derive numerous quantitative predictions for the behavior of correlations in
select measurement outcomes, and also identify two strategies for detecting
measurement-altered Ising criticality in measurement-averaged quantities.
First, averaging the square of the order-parameter expectation value over
measurement outcomes retains memory of order parameter condensation germinated
in fixed measurement outcomes -- even though on average the order parameter
itself vanishes. Second, we show that, in certain cases, observables can be
averaged separately over measurement outcomes residing in distinct symmetry
sectors, and that these `symmetry-resolved averages' reveal measurement effects
even when considering standard linearly averaged observables. We identify
complementary regimes in which symmetry-resolved averages and post-selection
can be pursued reasonably efficiently in experiment, with the former
generically outperforming the latter in the limit of sufficiently weak
ancilla-critical chain entanglement. Our framework naturally adapts to more
exotic quantum critical points and highlights opportunities for potential
experimental realization in NISQ hardware and in Rydberg arrays.
