Private Capacity of Quantum Channels is Not Additive

Recently there has been considerable activity on the subject of the additivity of various quantum channel capacities. Here, we construct a family of channels with a sharply bounded classical and, 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 nonadditive. 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 a larger entanglement-assisted quantum capacity than (unassisted) classical capacity.

Figure 1

The channel ${\mathcal{T}}_{\mathcal{N}}^k$: in the lower part it contains $\mathcal{N}$ (with input register $A_2$, output register $B_2$, and environment $E$). It also has another input register $A_1$ of dimension $c=|E{|}^{2k}$, which we view in a fixed way as a tensor product of $k$ $|E{|}^2$-dimensional systems $A_{11},\dots ,A_{1k}$, each coming with a fixed computational basis $\{|j⟩{\}}_{j=1,\dots ,|E{|}^2}$. This big register is subjected to a random unitary rotation $U$, where $U$ is chosen from the Haar measure and subsequently output (a classical description of it) in register $B_0$. All registers $A_{12},\dots ,A_{1k}$ are discarded, only $A_{11}$ is measured in the computational basis, and the result $j$ used to control a unitary transformation (Weyl operator) $W_j$ on the environment $E$, which is then output in the register $B_1$. A formal definition can be found in the supplementary material [22].