  It has been shown that any generalized measurement can be decomposed into a
sequence of weak measurements corresponding to a stochastic process. However,
the weak measurements may require almost arbitrary unitaries, which are
unlikely to be realized by any real measurement device. Furthermore, many
measurement processes are destructive, like photon counting procedures that
terminate once all photons are consumed. One cannot expect to have full control
of the evolution of a state under such destructive measurements, and the
possible unitaries allow only a limited set of weak measurements. In this
paper, we consider a qubit model of destructive weak measurements, which is a
toy version of an optical cavity, in which the state of an electromagnetic
field mode inside the cavity leaks out and is measured destructively while the
vacuum state |0> leaks in to the cavity. At long times, the state of the qubit
inevitably evolves to be |0>, and the only available control is the choice of
measurement on the external ancilla system. Surprisingly, this very limited
model can still perform an arbitrary projective measurement on the qubit in any
basis, where the probability of getting an outcome satisfies the usual Born
rule. Combining this method with probabilistic post processing, the result can
be extended to any generalized measurement with commuting POVM elements. This
implies, among other results, that any two-outcome POVM on a qubit can be
decomposed into a sequence of destructive weak measurements by this restricted
measurement device.
