  The efficient sensing of weak environmental perturbations via special
degeneracies called exceptional points in non-Hermitian systems has gained
enormous traction in the last few decades. However, in contrast to the
extensive literature on parity-time (PT) symmetric systems, the exotic
hallmarks of anti-PT symmetric systems are only beginning to be realized now.
Very recently, a characteristic resonance of vanishing linewidth in anti-PT
symmetric systems was shown to exhibit tremendous sensitivity to intrinsic
nonlinearities. Given the primacy of sensing in non-Hermitian systems, in
general, and the immense topicality of anti-PT symmetry, we investigate the
statistical bound to the measurement sensitivity for any arbitrary perturbation
in a dissipatively coupled, anti-PT symmetric system. Using the framework of
quantum Fisher information and the long-time solution to the full master
equation, we analytically compute the Cramer-Rao bound for the system
properties like the detunings and the couplings. As an illustrative example of
this formulation, we inspect and reaffirm the role of a long-lived resonance in
dissipatively interacting systems for sensing applications. \end{abstract}
