Estimation of SU(2) operation and dense coding: An information geometric approach

This paper addresses quantum statistical estimation of operators U is an element of SU(2) acting on CP3 as psi -->(U x I) psi where psi is an element of C-2 x C-2. This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cramer-Rao lower bound takes the minimum, and is achievable, if and only if psi is a maximally entangled state. We next show that an SU(2) orbit on CP3 equipped with the standard Riemannian structure is isometric to SU(2)/{+/-I}=SO(3) if and only if psi is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally entangled state.