Private communication from Pauli channels with no privacy

Abstract

A channel capacity quantifies the communication capability of a noisy physical process. In contrast to communication channels in the classical world, quantum theory makes this capability contextual. We show that two Pauli two-qubit channels, each with zero private classical capacity, can be used to transmit private information when used together. One is a two-qubit Pauli channel whose environment can reconstruct the receiver output up to matrix transposition; the other is an antidegradable channel. We obtain a similar result when the second channel is the 50% qubit erasure channel. A simple binary code built from rank-three mixtures of Bell states activates private communication. The main ingredient in our construction is a transpose-antidegradable channel that is not antidegradable.