Small sets of locally indistinguishable orthogonal maximally entangled states

We study the problem of distinguishing quantum states using local operations and classical communication (LOCC). A question of fundamental interest is whether there exist sets of k ≤ d orthogonal maximally entangled states in Cd ⊗ Cd that are not perfectly distinguishable by LOCC. A recent result by Yu, Duan, and Ying [Phys. Rev. Lett. 109 020506 (2012)] gives an affirmative answer for the case k = d. We give, for the first time, a proof that such sets of states indeed exist even in the case k < d. Our result is constructive and holds for an even wider class of operations known as positive-partial-transpose measurements (PPT). The proof uses the characterization of the PPT-distinguishability problem as a semidefinite program.