  The complex projective space $\mathbb{P}(\mathbb{C}^n)$ can be interpreted as
the space of all quantum pure states of size $n$. A distance on this space,
interesting from the perspective of quantum physics, can be induced from a
classical distance defined on the $n$-point probability simplex by the `earth
mover problem'. We show that this construction leads to a quantity satisfying
the triangle inequality, which yields a true distance on complex projective
space belonging to the family of quantum $2$-Wasserstein distances.
