  We use a convolutional neural network (CNN) to study cosmic string detection
in cosmic microwave background (CMB) flat sky maps with Nambu-Goto strings. On
noiseless maps we can measure string tensions down to order $10^{-9}$, however
when noise is included we are unable to measure string tensions below
$10^{-7}$. Motivated by this impasse, we derive an information theoretic bound
on the detection of the cosmic string tension $G\mu$ from CMB maps. In
particular we bound the information entropy of the posterior distribution of
$G\mu$ in terms of the resolution, noise level and total survey area of the CMB
map. We evaluate these bounds for the ACT, SPT-3G, Simons Observatory, Cosmic
Origins Explorer, and CMB-S4 experiments. These bounds cannot be saturated by
any method.

Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology;High Energy Physics - Theory