  We consider the problem of optimally identifying the state of a probe qudit,
prepared with given prior probability in a pure state belonging to a finite set
of possible states which together span a D-dimensional subspace of the
d-dimensional Hilbert space the qudit is defined in.It is assumed that we do
not know some or all of the states in the set, but for each unknown state we
are given a reference qudit into which this state is encoded. We show that from
the measurement for optimal state identification with d=D one can readily
determine the optimal figure of merit for qudits with d>D, without solving a
new optimization problem. This result is applied to the minimum-error
identification and to the optimal unambiguous identification of two qudit
states with d > 2, where either one or both of the states are unknown, and also
to the optimal unambiguous identification of N equiprobable linearly
independent unknown pure qudit states with d > N. In all cases the optimal
figure of merit, averaged over the unknown states, increases with growing
dimensionality d of the qudits.
