What is meant by a perfect Bell basis? Does a) correspond to doing (|B_00⟩⟨B_00|) |Ψ⟩ for all the corresponding basis?
For b), if Alice sends Bob 01, is the teleported state then |φ⟩ = α|1⟩ + β|0⟩, or is it X * |Ψ⟩? Or is it something else since the entangled pair is corrupted