  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)).
