How do two observers pool their knowledge about a quantum system?

Abstract

In the theory of classical statistical inference one can derive a simple rule by which two or more observers may combine {\em independently} obtained states of knowledge together to form a new state of knowledge, which is the state which would be possessed by someone having the combined information of both observers. Moreover, this combined state of knowledge can be found without reference to the manner in which the respective observers obtained their information. However, in this note we show that in general this is not possible for quantum states of knowledge; in order to combine two quantum states of knowledge to obtain the state resulting from the combined information of both observers, these observers must also possess information about how their respective states of knowledge were obtained. Nevertheless, we emphasize this does not preclude the possibility that a unique, well motivated rule for combining quantum states of knowledge without reference to a measurement history could be found. We examine both the direct quantum analogue of the classical problem, and that of quantum state-estimation, which corresponds to a variant in which the observers share a specific kind of prior information.