  We revise and extend the algorithm provided in [1] to compute the finite
Connes' distance between normal states. The original formula in [1] contains an
error and actually only provides a lower bound. The correct expression, which
we provide here, involves the computation of the infimum of an expression which
involves the "transverse" component of the algebra element in addition to the
"longitudinal" component of [1]. This renders the formula less user-friendly,
as the determination of the exact transverse component for which the infimum is
reached remains a non-trivial task, but under rather generic conditions it
turns out that the Connes' distance is proportional to the trace norm of the
difference in the density matrices, leading to considerable simplification. In
addition, we can determine an upper bound of the distance by emulating and
adapting the approach of [2] in our Hilbert-Schmidt operatorial formulation. We
then look for an optimal element for which the upper bound is reached. We are
able to find one for the Moyal plane through the limit of a sequence obtained
by finite dimensional projections of the representative of an element belonging
to a multiplier algebra, onto the subspaces of the total Hilbert space,
occurring in the spectral triple and spanned by the eigen-spinors of the
respective Dirac operator. This is in contrast with the fuzzy sphere, where the
upper bound, which is given by the geodesic of a commutative sphere is never
reached for any finite $n$-representation of $SU(2)$. Indeed, for the case of
maximal non-commutativity ($n = 1/2$), the finite distance is shown to coincide
exactly with the above mentioned lower bound, with the transverse component
playing no role. This, however starts changing from $n=1$ onwards and we try to
improve the estimate of the finite distance and provide an almost exact result,
using our new and modified algorithm.
