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In the Majorana or stellar representation of quantum states, an arbitrary(pure) state of a spin-1 system is represented by a pair of points on the unitsphere or, equivalently, by a pair of unit vectors. This paper presents anexpression for the squared modulus of the inner product of two spin-1 states interms of their Majorana vectors and uses it to give a geometrical constructionof the MUBs and SIC-POVMs of a spin-1 system. The results are not new andduplicate those obtained earlier by other methods, but the Majorana approachnevertheless illuminates them from an unusual point of view. In particular, itreveals the MUBs and SICs as symmetrical collections of vectors in ordinarythree-dimensional space, rather than as rays in a projective Hilbert space.While it does not appear feasible to extend this treatment to higher spinsystems, the spin-1 case exhibits sufficient subtlety and complexity to beworth spelling out for its pedagogical and historical interest.

MUBs and SIC-POVMs of a spin-1 system from the Majorana approach