  We analyze the ultimate quantum limit of resolving two identical sources in a
noisy environment. We prove that in the presence of noise causing false
excitation, such as thermal noise, the quantum Fisher information of arbitrary
quantum states for the separation of the objects, which quantifies the
resolution, always converges to zero as the separation goes to zero. Noisy
cases contrast with a noiseless case where it has been shown to be nonzero for
a small distance in various circumstances, revealing the superresolution. In
addition, we show that false excitation on an arbitrary measurement, such as
dark counts, also makes the classical Fisher information of the measurement
approach to zero as the separation goes to zero. Finally, a practically
relevant situation resolving two identical thermal sources, is quantitatively
investigated by using the quantum and classical Fisher information of finite
spatial mode multiplexing, showing that the amount of noise poses a limit on
the resolution in a noisy system.
