A predictive solution of the EPR paradox

In this work, we examine the paradox proposed by Einstein, Podolsky, and Rosen (EPR). They argued that since one may know the exact momentum of a particle without measurement and subsequently measure its position, a contradiction with the Heisenberg uncertainty principle arises. We demonstrate that there is no paradox by two equivalent approaches: first, by computing the quantum conditional expectation to make predictions after a measurement; and second, using the von Neumann post-measurement state. We establish the equivalence between these two methods. In both cases the predictor is an operator valued function of the observables being measured. This ensures that no violation of the Heisenberg uncertainty principle occurs.