  Quantum teleportation with a two-qubit state can be suitably characterized in
terms of maximal fidelity and fidelity deviation, where the former is the
maximal value of the average fidelity achievable within the standard protocol
and local unitary operations and the latter is the standard deviation of
fidelity over all input states. In this paper, we consider the problem of
characterizing two-qubit states that are optimal for quantum teleportation for
a given value of some state property. The optimal states are defined as those
states that, for a given value of the state property under consideration,
achieve the largest maximal fidelity and also exhibit zero fidelity deviation.
We provide a complete characterization of optimal states for a given linear
entropy, maximum mean value of the Bell-CHSH observable, and concurrence,
respectively. We find that for a given linear entropy or Bell-CHSH violation,
the largest maximal fidelity states are optimal, but for a given concurrence,
the optimal states form a strict subset of the largest maximal fidelity states.
