  The single-letter characterisation of the entanglement-assisted capacity of a
quantum channel is one of the seminal results of quantum information theory. In
this paper, we consider a modified communication scenario in which the receiver
is allowed an additional, `inconclusive' measurement outcome, and we employ an
error metric given by the error probability in decoding the transmitted message
conditioned on a conclusive measurement result. We call this setting
postselected communication and the ensuing highest achievable rates the
postselected capacities. Here, we provide a precise single-letter
characterisation of postselected capacities in the setting of entanglement
assistance as well as the more general nonsignalling assistance, establishing
that they are both equal to the channel's projective mutual information -- a
variant of mutual information based on the Hilbert projective metric. We do so
by establishing bounds on the one-shot postselected capacities, with a lower
bound that makes use of a postselected teleportation-based protocol and an
upper bound in terms of the postselected hypothesis testing relative entropy.
As such, we obtain fundamental limits on a channel's ability to communicate
even when this strong resource of postselection is allowed, implying
limitations on communication even when the receiver has access to postselected
closed timelike curves.
