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

Abstract: 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 unit sphere or, equivalently, by a pair of unit vectors. This paper presents an expression for the squared modulus of the inner product of two spin-1 states in terms of their Majorana vectors and uses it to give a geometrical construction of the MUBs and SIC-POVMs of a spin-1 system. The results are not new and duplicate those obtained earlier by other methods, but the Majorana approach nevertheless illuminates them from an unusual point of view. In particular, it reveals the MUBs and SICs as symmetrical collections of vectors in ordinary three-dimensional space, rather than as rays in a projective Hilbert space. While it does not appear feasible to extend this treatment to higher spin systems, the spin-1 case exhibits sufficient subtlety and complexity to be worth spelling out for its pedagogical and historical interest.