Pooling quantum states obtained by indirect measurements

We consider the pooling of quantum states when Alice and Bob both have one part of a tripartite system and, on the basis of measurements on their respective parts, each infers a quantum state for the third part S. We denote the conditioned states which Alice and Bob assign to S by alpha and beta respectively, while the unconditioned state of S is rho. The state assigned by an overseer, who has all the data available to Alice and Bob, is omega. The pooler is told only alpha, beta, and rho. We show that for certain classes of tripartite states, this information is enough for her to reconstruct omega by the formula omega \propto alpha rho^{-1} beta. Specifically, we identify two classes of states for which this pooling formula works: (i) all pure states for which the rank of rho is equal to the product of the ranks of the states of Alice's and Bob's subsystems; (ii) all mixtures of tripartite product states that are mutually orthogonal on S.