  When quantum systems interact with the environment they lose their quantum
properties, such as coherence. Quantum erasure makes it possible to restore
coherence in a system by measuring its environment, but accessing the whole of
it may be prohibitive: realistically one might have to concentrate only on an
accessible subspace and neglect the rest. If that is the case, how good is
quantum erasure? In this work we compute the largest coherence $\langle
\mathcal C\rangle$ that we can expect to recover in a qubit, as a function of
the dimension of the accessible and of the inaccessible subspaces of its
environment. We then imagine the following game: we are given a uniformly
random pure state of $n+1$ qubits and we are asked to compute the largest
coherence that we can retrieve on one of them by optimally measuring a certain
number $0\leq a\leq n$ of the others. We find a surprising effect around the
value $a\approx n/2$: the recoverable coherence sharply transitions between 0
and 1, indicating that in order to restore full coherence on a qubit we need
access to only half of its physical environment (or in terms of degrees of
freedom to just the square root of them). Moreover, we find that the
recoverable coherence becomes a typical property of the whole ensemble as $n$
grows.
