(2007):

ABSTRACT:
Gisin and Popescu (Gisin N and Popescu S 1999 Phys. Rev. Lett. 83 432)
have shown that more information about their direction can be obtained from
a pair of anti-parallel spins compared to a pair of parallel spins, where the
first member of the pair (which we call the pointer member) can point equally
along any direction in the Bloch sphere. They argued that this was due to the
difference in dimensionality spanned by these two alphabets of states. Here
we consider similar alphabets, but with the first spin restricted to a fixed
small circle of the Bloch sphere. In this case, the dimensionality spanned
by the anti-parallel versus parallel alphabet is now equal. However, the
anti-parallel alphabet is found to still contain more information in
general. We generalize this to having *N* parallel spins and *M*
anti-parallel spins. When the pointer member is restricted to a small
circle these alphabets again span spaces of equal dimension, yet in
general, more directional information can be found for sets with smaller
|*N* - *M*| for any fixed total number of spins. We find
that the optimal POVMs for extracting directional information in these cases
can always be expressed in terms of the Fourier basis. Our results show that
dimensionality alone cannot explain the greater information content in
antiparallel combinations of spins compared to parallel combinations.
In addition, we describe an LOCC protocol which extracts optimal directional
information when the pointer member is restricted to a small circle and
a pair of parallel spins are supplied.