  We introduce a new notion of a sufficient subalgebra for quantum states: a
subalgebra is 2- sufficient for a pair of states $\{\rho_0,\rho_1\}$ if it
contains all Bayes optimal tests of $\rho_0$ against $\rho_1$. In classical
statistics, this corresponds to the usual definition of sufficiency. We show
this correspondence in the quantum setting for some special cases. Furthermore,
we show that sufficiency is equivalent to 2 - sufficiency, if the latter is
required for $\{\rho_0^{\otimes n},\rho_1^{\otimes}\}$, for all $n$.
