  We show that there exist sets of three mutually orthogonal $d$-dimensional
maximally entangled states which cannot be perfectly distinguished using
one-way local operations and classical communication (LOCC) for arbitrarily
large values of $d$. This contrasts with several well-known families of
maximally entangled states, for which any three states can be perfectly
distinguished. We then show that two-way LOCC is sufficient to distinguish
these examples. We also show that any three mutually orthogonal $d$-dimensional
maximally entangled states can be perfectly distinguished using measurements
with a positive partial transpose (PPT) and can be distinguished with one-way
LOCC with high probability. These results circle around the question of whether
there exist three maximally entangled states which cannot be distinguished
using the full power of LOCC; we discuss possible approaches to answer this
question.
