Pooling quantum states obtained by indirect measurements

We consider the pooling of quantum states when Alice and Bob both have onepart of a tripartite system and, on the basis of measurements on theirrespective parts, each infers a quantum state for the third part S. We denotethe conditioned states which Alice and Bob assign to S by alpha and betarespectively, while the unconditioned state of S is rho. The state assigned byan overseer, who has all the data available to Alice and Bob, is omega. Thepooler is told only alpha, beta, and rho. We show that for certain classes oftripartite states, this information is enough for her to reconstruct omega bythe formula omega \propto alpha rho^{-1} beta. Specifically, we identify twoclasses of states for which this pooling formula works: (i) all pure states forwhich the rank of rho is equal to the product of the ranks of the states ofAlice's and Bob's subsystems; (ii) all mixtures of tripartite product statesthat are mutually orthogonal on S.Comment: Corrected a mistake regarding the scope of our original result. This version to be published in Phys. Rev. A. 6 pages, 1 figur