Structure of sufficient quantum coarse-grainings

Let H and K be finite-dimensional Hilbert spaces, T:B(H)-->(K) be a coarse-graining and D-1, D-2 be density matrices on H. In this Letter the consequences of the existence of a coarse-graining beta:B(K)-->B(H) satisfying betaT(D-s)=D-s are given. ( This means that T is sufficient for D-1 and D-2.) It is shown that D-s=Sigma(p=1)(r) lambda(s)(p)S-s(H)(p)R-H(p) (s=1,2) should hold with pairwise orthogonal summands and with commuting factors and with some probability distributions lambda(s)(p) for 1less than or equal topless than or equal tor (s=1,2). This decomposition allows to deduce the exact condition for equality in the strong subadditivity of the von Neumann entropy.