  We derive a bound on the ability of a linear optical network to estimate a
linear combination of independent phase shifts by using an arbitrary
non-classical but unentangled input state, thereby elucidating the quantum
resources required to obtain the Heisenberg limit with a multi-port
interferometer. Our bound reveals that while linear networks can generate
highly entangled states, they cannot effectively combine quantum resources that
are well distributed across multiple modes for the purposes of metrology: in
this sense linear networks endowed with well-distributed quantum resources
behave classically. Conversely, our bound shows that linear networks can
achieve the Heisenberg limit for distributed metrology when the input photons
are hoarded in a small number of input modes, and we present an explicit scheme
for doing so. Our results also have implications for measures of
non-classicality.
