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 these states 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. The results are found to be same with logarithmic negativity used as a measure of entanglement.

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Three-qubit mixed states are used as channel for controlled quantum teleportation (CQT) of single qubit states. The connection between different channel parameters to achieve maximum controlled teleportation fidelity is investigated. We show that for a given multipartite entanglement and mixedness, a new class of non-maximally entangled mixed X states (X−NMEMS) achieve optimum controlled quantum teleportation fidelity, interestingly a class of maximally entangled mixed X states (X−MEMS) fails to do so. This demonstrates, for a given spectrum and mixedness, X−MEMS is not sufficient to attain optimum controlled quantum teleportation fidelity, which is in contradiction with the traditional quantum teleportation of single qubits. In addition, we show that biseparable X−NMEMS for certain range of mixedness are useful as a resource to attain high controlled quantum teleportation fidelity, which essentially lower the requirements of quantum channels for CQT.I.