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Entanglement and mixedness of a bipartite mixed state resource are crucialfor the success of quantum teleportation. Upper bounds on measures ofmixedness, namely, von Neumann entropy and linear entropy beyond which thebipartite state ceases to be useful for quantum teleportation are known in theliterature. In this work, we generalize these bounds and obtain rank dependentupper bounds on von Nuemann entropy and linear entropy for an arbitrarybipartite mixed state resource. We observe that the upper bounds on measures ofmixedness increase with increase in the rank. For two qubit mixed states, weobtain rank dependent lower bound on the concurrence, a measure ofentanglement, below which the state is useless for quantum teleportation.Werner state, which is a fourth rank state, exhibits the theoretical upperbound on mixedness among two qubit mixed states. We construct Werner likestates of lower ranks and show that these states possess the theoretical rankdependent upper bounds obtained on the measures of mixedness and theoreticalrank dependent lower bounds on the concurrence.

Rank dependent bounds on mixedness and entanglement for quantum teleportation