Limiting one-way distillable secret key via privacy testing of extendible states

TL;DR

This paper establishes new, efficiently computable upper bounds on the one-way distillable secret key and private capacity of quantum states and channels, using privacy tests and extendibility concepts, improving upon previous bounds.

Contribution

It introduces a novel method linking privacy tests with $k$-extendibility to derive tighter, single-letter bounds on quantum key distillation and private capacity.

Findings

Maximum privacy test passing probability equals fidelity to maximally entangled state.

Derived upper bounds are tighter than existing bounds for key and capacity.

Bounds are efficiently computable and applicable to i.i.d. scenarios.

Abstract

The notions of privacy tests and $k$-extendible states have both been instrumental in quantum information theory, particularly in understanding the limits of secure communication. In this paper, we determine the maximum probability with which an arbitrary $k$-extendible state can pass a privacy test, and we prove that it is equal to the maximum fidelity between an arbitrary $k$-extendible state and the standard maximally entangled state. Our findings, coupled with the resource theory of $k$-unextendibility, lead to an efficiently computable upper bound on the one-shot, one-way distillable key of a bipartite state, and we prove that it is equal to the best-known efficiently computable upper bound on the one-shot, one-way distillable entanglement. We also establish efficiently computable upper bounds on the one-shot, forward-assisted private capacity of channels. Extending our formalism to the independent and identically distributed setting, we obtain single-letter efficiently computable bounds on the $n$-shot, one-way distillable key of a state and the $n$-shot, forward-assisted private capacity of a channel. For some key examples of interest, our bounds are significantly tighter than other known efficiently computable bounds.