  Wigner's celebrated theorem, which is particularly important in the
mathematical foundations of quantum mechanics, states that every bijective
transformation on the set of all rank-one projections of a complex Hilbert
space which preserves the transition probability is induced by a unitary or an
antiunitary operator. This vital theorem has been generalised in various ways
by several scientists. In 2001, Moln\'ar provided a natural generalisation,
namely, he provided a characterisation of (not necessarily bijective) maps
which act on the Grassmann space of all rank-$n$ projections and leave the
system of Jordan principal angles invariant (see [20] and [17]). In this paper
we give a very natural joint generalisation of Wigner's and Moln\'ar's
theorems, namely, we prove a characterisation of all (not necessarily
bijective) transformations on the Grassmann space which fix the quantity
$\mathrm{tr} PQ$ (i.e.~the sum of the squares of cosines of principal angles)
for every pair of rank-$n$ projections $P$ and $Q$.
