  We explicitly exhibit a set of four ququad-ququad orthogonal maximally
entangled states that cannot be perfectly distinguished by means of local
operations and classical communication. Before our work, it was unknown whether
there is a set of $d$ locally indistinguishable $d\otimes d$ orthogonal
maximally entangled states for some positive integer $d$. We further show that
a $2\otimes 2$ maximally entangled state can be used to locally distinguish
this set of states without being consumed, thus demonstrate a novel phenomenon
of "Entanglement Discrimination Catalysis". Based on this set of states, we
construct a new set $\mathrm{K}$ consisting of four locally indistinguishable
states such that $\mathrm{K}^{\otimes m}$ (with $4^m$ members) is locally
distinguishable for some $m$ greater than one. As an immediate application, we
construct a noisy quantum channel with one sender and two receivers whose local
zero-error classical capacity can achieve the full dimension of the input space
but only with a multi-shot protocol.
