Does Anti-Parallel Spin Always Contain more Information ?

We show that the Bloch vectors lying on any great circle is the largest set S(L) for which the parallel states |n,n> can always be transformed into the anti-parallel states |n,-n>. Thus more information about the Bloch vector is not extractable from |n,-n> than from |n,n> by any measuring strategy, for the Bloch vector belonging to S(L). Surprisingly, the largest set of Bloch vectors for which the corresponding qubits can be flipped is again S(L). We then show that probabilistic exact parallel to anti-parallel transformation is not possible if the corresponding anti-parallel spins span the whole Hilbert space of the two qubits. These considerations allow us to generalise a conjecture of Gisin and Popescu (Phys. Rev. Lett. 83 432 (1999)).