  It has previously been shown that quantum nonlocality offers no benefit over
classical correlations for performing a distributed task known as nonlocal
computation. This is where separated parties must compute the value of a
function without individually learning anything about the inputs. We show that
giving the parties some knowledge of the inputs, however small, is sufficient
to unlock the power of quantum mechanics to out-perform classical mechanics.
This role of information held locally gives new insight into the general
question of when quantum nonlocality gives an advantage over classical physics.
Our results also reveal a novel feature of the nonlocality embodied in the
celebrated task of Clauser, Horne, Shimony and Holt.
