  Standard projective measurements represent a subset of all possible
measurements in quantum physics, defined by positive-operator-valued measures.
We study what quantum measurements are projective simulable, that is, can be
simulated by using projective measurements and classical randomness. We first
prove that every measurement on a given quantum system can be realised by
classical processing of projective measurements on the system plus an ancilla
of the same dimension. Then, given a general measurement in dimension two or
three, we show that deciding whether it is projective-simulable can be solved
by means of semi-definite programming. We also establish conditions for the
simulation of measurements using projective ones valid for any dimension. As an
application of our formalism, we improve the range of visibilities for which
two-qubit Werner states do not violate any Bell inequality for all
measurements. From an implementation point of view, our work provides bounds on
the amount of noise a measurement tolerates before losing any advantage over
projective ones.
