  Parametrized families of density operators are studied. A generalization of
the lower bound of Cramer and Rao is formulated. It involves escort density
operators. The notion of phi-exponential family is introduced. This family,
together with its escort, optimizes the generalized lower bound. It also
satisfies a maximum entropy principle and exhibits a thermodynamic structure in
which entropy and free energy are related by Legendre transform.
