  To explore the properties of a two-qubit mixed state, we consider quantum
teleportation. The fidelity of a teleported state depends on the resource state
purity and entanglement, as characterized by concurrence. Concurrence and
purity are functions of state parameters. However, it turns out that a state
with larger purity and concurrence, may have comparatively smaller fidelity. By
computing teleportation fidelity, concurrence and purity for two-qubit
X-states, we show it explicitly. We further show that fidelity changes
monotonically with respect to functions of parameters - other than concurrence
and purity. A state with smaller concurrence and purity, but larger value of
one of these functions has larger fidelity. These functions, thus characterize
nonlocal classical and/or quantum properties of the state that are not captured
by purity and concurrence alone. In particular, concurrence is not enough to
characterize the entanglement properties of a two-qubit mixed state.
