General Communication Enhancement via the Quantum Switch

Recent studies have shown that quantum information may be effectively transmitted by a finite collection of completely depolarizing channels in a coherent superposition of different orders, via an operation known as the quantum tt SWITCH. Such results are quite remarkable, as completely depolarizing channels taken in isolation and in a definite order can only output white noise. For general channels however, little is known about the potential communication enhancement provided by the quantum tt SWITCH. In this Letter, we define an easily computable quantity mathcal{P}_n associated with the quantum {tt SWITCH} of n copies of a fixed channel, and we conjecture that mathcal{P}_n>0 is both a necessary and sufficient condition for communication enhancement via the quantum tt SWITCH. In support of our conjecture, we derive a simple analytic expression for the classical capacity of the quantum tt SWITCH of n copies of an arbitrary Pauli channel in terms of the quantity mathcal{P}_n, which we then use to show that our conjecture indeed holds in the space of all Pauli channels. Utilizing such results, we then formulate a communication protocol involving the quantum tt SWITCH which enhances the private capacity of the BB84 channel.