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

This paper addresses quantum statistical estimation of operators $U\ensuremath{\in}\mathrm{SU}(2)$ acting on $C{P}^{3}$ as $\ensuremath{\psi}\ensuremath{\mapsto}(U\ensuremath{\bigotimes}I)\ensuremath{\psi}$ where $\ensuremath{\psi}\ensuremath{\in}{C}^{2}\ensuremath{\bigotimes}{C}^{2}.$ This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cram\'er-Rao lower bound takes the minimum, and is achievable, if and only if \ensuremath{\psi} is a maximally entangled state. We next show that an SU(2) orbit on $C{P}^{3}$ equipped with the standard Riemannian structure is isometric to $\mathrm{SU}(2)/{\ifmmode\pm\else\textpm\fi{}I}\ensuremath{\cong}\mathrm{SO}(3)$ if and only if \ensuremath{\psi} is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally entangled state.