  To prepare quantum states and extract information, it is often assumed that
one can perform a perfectly projective measurement. Such measurements can
achieve an uncorrelated system and environment state. However, perfectly
projective measurements can be difficult or impossible to perform in practice.
We investigate the limitations of repeated non-projective measurements in
preparing a quantum system. For an $n$-qubit system initially entangled with
its environment and subsequently prepared with measurements, using a sequence
of weak measurements, we show that some entanglement remains unless one of the
measurement operators becomes perfectly projective through an extreme limiting
process. Removing initial (unentangled) correlations between a system and its
environment and the scenario where measurement outcomes are not tracked are
also discussed. We present results for $n$-qubit and $n$-dimensional input
states.
