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

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... m-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.