  Port-based teleportation (PBT) is a variation of regular quantum
teleportation that operates without a final unitary correction. However, its
behavior for higher-dimensional systems has been hard to calculate explicitly
beyond dimension $d=2$. Indeed, relying on conventional Hilbert-space
representations entails an exponential overhead with increasing dimension. Some
general upper and lower bounds for various success measures, such as
(entanglement) fidelity, are known, but some become trivial in higher
dimensions. Here we construct a graph-theoretic algebra (a subset of
Temperley-Lieb algebra) which allows us to explicitly compute the
higher-dimensional performance of PBT for so-called "pretty-good measurements"
with negligible representational overhead. This graphical algebra allows us to
explicitly compute the success probability to distinguish the different
outcomes and fidelity for arbitrary dimension $d$ and low number of ports $N$,
obtaining in addition a simple upper bound. The results for low $N$ and
arbitrary $d$ show that the fidelity asymptotically approaches ${N}/{d^2}$ for
large $d$, confirming the performance of one lower bound from the literature.
