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

This paper addresses quantum statistical estimation of operators $U\in \mathrm{SU}\left(2\right)\mathrm{}$ acting on $C{P}^3$ as $\psi \mapsto (U\otimes I)\psi$ where $\psi \in C^2\otimes C^2.$ This is regarded as a continuous analog of the dense coding. We first prove that the quantum Cramér-Rao lower bound takes the minimum, and is achievable, if and only if ψ 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 $\mathrm{SU}\left(2\right)\mathrm{}/\{\pm I\}\cong \mathrm{SO}\left(3\right)\mathrm{}$ if and only if ψ is a maximally entangled state. These results provide an alternative view for the optimality of the use of a maximally entangled state.