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

This paper addresses quantum statistical estimation of operators U∈SU(2) acting on CP³ as ψ↦(UI)ψ where ψ∈C²C². 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 {ψ} 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)/±ISO(3) 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.