The private capacity of quantum channels is not additive

TL;DR

This paper demonstrates that the classical private capacity of quantum channels is non-additive by constructing specific channels where combined capacities exceed individual limits, revealing fundamental non-linearities in quantum information transmission.

Contribution

The authors construct channels with bounded private capacity that, when combined with certain erasure channels, exhibit superadditivity, proving non-additivity of private capacity for the first time.

Findings

Private capacity is non-additive in quantum channels.

Combining channels can increase capacity beyond individual limits.

Violation of additivity occurs when entanglement-assisted quantum capacity exceeds classical capacity.

Abstract

Recently there has been considerable activity on the subject of additivity of various quantum channel capacities. Here, we construct a family of channels with sharply bounded classical, hence private capacity. On the other hand, their quantum capacity when combined with a zero private (and zero quantum) capacity erasure channel, becomes larger than the previous classical capacity. As a consequence, we can conclude for the first time that the classical private capacity is non-additive. In fact, in our construction even the quantum capacity of the tensor product of two channels can be greater than the sum of their individual classical private capacities. We show that this violation occurs quite generically: every channel can be embedded into our construction, and a violation occurs whenever the given channel has larger entanglement assisted quantum capacity than (unassisted) classical capacity.