Postselected communication over quantum channels

TL;DR

This paper introduces the concept of postselected communication over quantum channels, providing a single-letter capacity characterization under entanglement and nonsignalling assistance, and establishing fundamental limits on quantum communication with postselection.

Contribution

It offers the first precise single-letter characterization of postselected capacities in quantum channels, linking them to the projective mutual information and bounding one-shot capacities.

Findings

Postselected capacities equal the channel's projective mutual information.

Bounds on one-shot postselected capacities are established using teleportation and hypothesis testing.

Fundamental limits are identified for quantum communication with postselection, even with advanced resources.

Abstract

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.