Signal-to-noise-ratio and SNR-max detection statistics in template bank
  searches for exotic physics transients with networks of quantum sensors
  Signal-to-noise ratio (SNR) detection statistic has wide-spread applications.
A potential event is recorded when the SNR from a specific template exceeds a
threshold set by a desired false positive rate. In template bank searches, the
generalization of the SNR statistic is the SNR-max statistic, defined as the
maximum of the absolute value of SNRs from individual template matching. While
individual SNR realizations are Gaussian distributed, SNR-max probability
distribution is non-Gaussian. Moreover, as the individual template-bank SNRs
are computed using the same network data streams, SNRs become correlated
between templates. Cross-template correlations have sizable effect on the
SNR-max probability distribution, and the threshold SNR-max values. Computing
threshold SNR-max values for large banks is computationally prohibitive and we
develop analytic approaches to computing properties of SNR-max statistic. This
is done for nearly orthogonal template banks and for banks with cross-template
correlation coefficients "squeezed" about the most probable cross-template
correlation value. Since cross-template correlation coefficients quantify
similarity of templates, increasing correlations decrease SNR-max thresholds
for specific values of false positive rates. Increasing the number of templates
in the bank increases the SNR-max thresholds. Our derivations are carried out
for networks that may exhibit colored noise and cross-node correlations.
Specific applications are illustrated with a dark matter search with atomic
clocks and a ''toy'' planar network with cyclic rotational symmetry.
