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 $S$.

Figure 1

(Color online) Schematic of the pooling problem we consider. All parties seek to update the quantum state they assign to system $S$ based on the information they acquire. Each party has access to only certain information, represented by the contents of the circle in which he or she is enclosed. All information about $S$ is ultimately derived from indirect measurements on $S$, that is, measurements upon systems ($A$ and $B$) that are correlated with $S$. The prior knowledge about $A$, $B$, and $S$ of the different parties derives from a single tripartite state, but each party has the reduced state for only those systems contained in his or her circle. Alice performs a measurement on $A$ (represented by a POVM $\left\{E_a\right\}$) with outcome $a$, and updates her state for $S$ to $\alpha$. Bob does likewise, mutatis mutandis. Oswald, the overseer, learns everything known to Alice and Bob and updates his state to $\omega$. Penelope, the pooler, learns only the manner in which Alice and Bob update their states. The question we address is: what state should she assign to $S$?