  It is by now well-recognised that the na\"ive application of the projection
postulate on composite quantum systems can induce signalling between their
constituent components, indicative of a breakdown of causality in a
relativistic spacetime context. Here we introduce a necessary and sufficient
condition for an ideal measurement of an observable to be non-signalling. As
well as being particularly simple, it generalises previous no-signalling
conditions in that it allows for degeneracies and can be applied to all bounded
self-adjoint operators. The condition is used to establish that arbitrary sums
of local observables will not signal, in accordance with our expectations from
relativistic quantum field theory. On the other hand, it is shown that the
measurement of the tensor product of commuting local observables, for example
bipartite operators of the form $A\otimes B$, can in fact signal, contrary to
the widely-held belief that such measurements are always locally realisable.
The implications for the notion of measurement in relativistic quantum field
theory are addressed; it appears that the most straightforward application of
the standard quantum formalism generically leads to violations of causality. We
conclude that either the class of observables that can be measured should be
restricted and/or that the na\"ive translation of the measurement framework of
quantum theory, in particular the projection postulate, to quantum field theory
should be re-evaluated.
