  In the lore of quantum metrology, one often hears (or reads) the following
no-go theorem: If you put vacuum into one input port of a balanced Mach-Zehnder
Interferometer, then no matter what you put into the other input port, and no
matter what your detection scheme, the sensitivity can never be better than the
shot noise limit (SNL). Often the proof of this theorem is cited to be in Ref.
[C. Caves, Phys. Rev. D 23, 1693 (1981)], but upon further inspection, no such
claim is made there. A quantum-Fisher-information-based argument suggestive of
this no-go theorem appears in Ref. [M. Lang and C. Caves, Phys. Rev. Lett. 111,
173601 (2013)], but is not stated in its full generality. Here we thoroughly
explore this no-go theorem and give the rigorous statement: the no-go theorem
holds whenever the unknown phase shift is split between both arms of the
interferometer, but remarkably does not hold when only one arm has the unknown
phase shift. In the latter scenario, we provide an explicit measurement
strategy that beats the SNL. We also point out that these two scenarios are
physically different and correspond to different types of sensing applications.
