Amortized entanglement of a quantum channel and approximately teleportation-simulable channels

TL;DR

This paper introduces the concept of amortized entanglement for quantum channels, providing bounds on their capacities and analyzing approximately teleportation- and PPT-simulable channels, with implications for resource theories.

Contribution

It defines amortized entanglement for quantum channels, proves key properties, and applies these to bounds on capacities for approximately teleportation- and PPT-simulable channels.

Findings

Amortized entanglement obeys desirable mathematical properties.

Single-letter upper bounds on capacities are established for special channels.

Continuity bounds relate channel differences to entanglement measures.

Abstract

This paper defines the amortized entanglement of a quantum channel as the largest difference in entanglement between the output and the input of the channel, where entanglement is quantified by an arbitrary entanglement measure. We prove that the amortized entanglement of a channel obeys several desirable properties, and we also consider special cases such as the amortized relative entropy of entanglement and the amortized Rains relative entropy. These latter quantities are shown to be single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of a quantum channel, respectively. Of especial interest is a uniform continuity bound for these latter two special cases of amortized entanglement, in which the deviation between the amortized entanglement of two channels is bounded from above by a simple function of the diamond norm of their difference and the output dimension of the channels. We then define approximately teleportation- and positive-partial-transpose-simulable (PPT-simulable) channels as those that are close in diamond norm to a channel which is either exactly teleportation- or PPT-simulable, respectively. These results then lead to single-letter upper bounds on the secret-key-agreement and PPT-assisted quantum capacities of channels that are approximately teleportation- or PPT-simulable, respectively. Finally, we generalize many of the concepts in the paper to the setting of general resource theories, defining the amortized resourcefulness of a channel and the notion of $\nu$-freely-simulable channels, connecting these concepts in an operational way as well.