Monotone metrics on statistical manifolds of density matrices by geometry of non-commutative $L^2$-spaces,

Using an integral decomposition of non-commutative monotone metrics we show that each monotone metric described by the Petz classification theorem is related to the geometry of a suitable non-commutative L^2-space. This exactly reproduces and generalizes the commutative case where the unique monotone metric (Chentsov theorem about Fisher-Rao metic) is classically related to the commutative L^2 geometry.