Non-locality breaking qubit channels: the case for CHSH inequality

Abstract

Entanglement breaking channels play a significant role in quantum information theory. In this work we investigate qubit channels through their property of ‘non-locality breaking’, which is defined in a natural way but within the purview of CHSH non-locality. This also provides a different perspective on the relationship between entanglement and non-locality through the dual picture of quantum channels instead of through states. For a channel to be entanglement breaking, it is sufficient to ‘break’ the entanglement of maximally entangled states. We provide examples to show that for CHSH nonlocality breaking such a property does not hold in general, although for certain channels and for a restricted class of states for all channels this holds. We also consider channels whose output remains local under SLOCC and call them ‘strongly non-locality breaking’. We provide a closed-form necessary-sufficient condition for any two-qubit state to show hidden CHSH non-locality, which is likely to be useful for other purposes as well. This in turn allows us to characterize all strongly non-locality breaking qubit channels. It turns out that unital qubit channels breaking non-locality of maximally entangled states are strongly non-locality breaking while extremal qubit channels cannot be so unless they are entanglement breaking.