Dually flat structures induced from monotone metrics on a two-level quantum state space

The notion of dually flatness is of central importance in information geometry. Nevertheless, little is known about dually flat structures on quantum statistical manifolds except that the Bogoliubov metric admits a global dually flat structure on a quantum state space $${{\mathcal {S}}}({{\mathbb {C}}}^d)$$ for any $$d\ge 2$$ . In this paper, we show that every monotone metric on a two-level quantum state space $${{\mathcal {S}}}({{\mathbb {C}}}^2)$$ admits a local dually flat structure.