Converse bounds for private communication over quantum channels

TL;DR

This paper develops new theoretical bounds on the maximum private communication rates over quantum channels using the concept of private states and a privacy test, with applications to quantum key distribution.

Contribution

It introduces a general meta-converse bound for private communication over quantum channels, linking relative entropy of entanglement to strong converse rates.

Findings

Relative entropy of entanglement is a strong converse rate for private communication.

Derived second-order bounds for covariant channels.

Established converse bounds for phase-insensitive bosonic channels.

Abstract

This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here we use this approach along with a "privacy test" to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish several converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.