  We investigate the effect of common bath decoherence on the qubits of Alice
in the usual teleportation protocol. The system bath interaction is studied
under the central spin model, where the qubits are coupled to the bath spins
through isotropic Heisenberg interaction. We have given a more generalized
representation of the protocol in terms of density matrices and calculated the
average fidelity of the teleported state for different Bell state measurements
performed by Alice. The common bath interaction differentiates the outcome of
various Bell state measurements made by Alice. There will be a high fidelity
teleportation for a singlet measurement made by Alice when both the qubits of
Alice interact either ferromagnetically or antiferromagnetically with bath. In
contrast if one of the Alice's qubits interact ferromagnetically and the other
anti-ferromagnetically then measurement of Bell states belonging to the triplet
sector will give better fidelity. We have also evaluated the average fidelity
when Alice prefers non-maximally entangled states as her basis for measurement.
