  Mixed entangled states are generic resource for quantum teleportation.
Optimal teleportation fidelity measures the success of quantum teleportation.
The relevance of rank in the teleportation process is investigated by
constructing three new maximally entangled mixed states (MEMS) of different
ranks. Linear entropy, concurrence, optimal teleportation fidelity and Bell
function are obtained for each of the state analytically. It is found that
mixed states with higher rank are better resource for teleportation. In order
to achieve a fixed value of optimal teleportation fidelity, we find that low
rank states must have high concurrence. Further, for each of ranks 2, 3 and 4,
we numerically generate 30000 maximally entangled mixed states. The analysis of
these states reveals the existence of a rank dependent upper bound on optimal
teleportation fidelity for a fixed purity. In order to achieve a fixed optimal
teleportation fidelity, we find MEMS exhibit a rank dependent lower bound on
concurrence. MEMS are classified in terms of their degree of nonlocality.
