  Networks of quantum devices with coherent control over their configuration
offer promising advantages in quantum information processing. So far, the
investigation of these advantages assumed that the control system was initially
uncorrelated with the data processed by the network. Here, we explore the power
of quantum correlations between data and control, showing two communication
tasks that can be accomplished with information-erasing channels if and only if
the sender shares prior entanglement with a third party (the ``controller'')
controlling the network configuration. The first task is to transmit classical
messages without leaking information to the controller. The second task is to
establish bipartite entanglement with a receiver, or, more generally, to
establish multipartite entanglement with a number of spatially separated
receivers.
