  Cosmic birefringence -- the rotation of the polarization of the cosmic
microwave background (CMB) photons as they travel to the Earth -- is a smoking
gun for axion-like particles (ALPs) that interact with the photon. It has
recently been suggested that topological defects in the ALP field called cosmic
strings can cause polarization rotation in quantized amounts that are
proportional to the electromagnetic chiral anomaly coefficient $\mathcal A$,
which holds direct information about physics at very high energies. In this
work, we study the detectability of a random network of cosmic strings through
estimating rotation using quadratic estimators (QEs). We show that the QE
derived from the maximum likelihood estimator is equivalent to the recently
proposed global-minimum-variance QE; the classic Hu-Okamoto QE equals the
global-minimum-variance QE under special conditions, but is otherwise still
nearly globally optimal. We calculate the sensitivity of QEs to cosmic
birefringence from string networks, for the Planck satellite mission, as well
as for third- and fourth-generation ground-based CMB experiments. Using
published binned rotation power spectrum derived from the Planck 2015
polarization data, we obtain a constraint $\mathcal A^2\,\xi_0 < 0.93$ at the
95\% confidence level, where $\xi_0$ is the total length of strings in units of
the Hubble scale per Hubble volume, for a phenomenological but reasonable
string network model describing a continuous distribution of string sizes.
Looking forward, experiments such as the Simons Observatory and CMB-S4 will
either discover or falsify the existence of an ALP string network for the
theoretically plausible parameter space $\mathcal A^2\,\xi_0 \gtrsim 0.01$.
