An equality between entanglement and uncertainty

In their seminal paper, Einstein, Podolsky and Rosen (EPR) show that an observer who is maximally entangled with the system to be measured can perfectly predict the outcome of two incompatible measurements. This feat stands in stark contrast to Heisenberg's uncertainty principle which tell us that if the observer is not entangled with the system at all, then his ability to predict the outcomes of incompatible measurements such as position and momentum is extremely limited. The observations made by EPR and Heisenberg illustrate two extreme cases of the interplay between entanglement and uncertainty. On the one hand, no entanglement means that measurements will be maximally uncertain. Yet on the other hand, maximal entanglement means that there is no more uncertainty at all. Here we show that this apparent rift can be reconciled in that it is indeed possible to have an exact relation - an equality - between the amount of uncertainty and the amount of entanglement.