Equivalence between Two-Qubit Entanglement and Secure Key Distribution

We study the problem of secret key distillation from bipartite states in the scenario where Alice and Bob can perform measurements only at the single-copy level and classically process the obtained outcomes. Even with these limitations, secret bits can be asymptotically distilled by the honest parties from any two-qubit entangled state, under any individual attack. Our results point out a complete equivalence between two-qubit entanglement and secure key distribution: a key can be established through a one-qubit channel if and only if it allows one to distribute entanglement. These results can be generalized to a higher dimension for all those states that are one-copy distillable.

Figure 1

Example of measurement attaining the bound of Eq. (11), where the first outcome, $|m_0⟩$, is associated to 0, the second, $|m_1⟩$, symmetric to $|m_0⟩$ with respect to the $z$ axis, to 1, and the third, $|m_{?}⟩=|+z⟩$, to an inconclusive result. The weight of $|m_{?}⟩$ is minimized. One can consider similar three-outcome measurements, just changing the angle between $|m_0⟩$, or $|m_1⟩$, and the $z$ axis. All the measurements such that $|m_0⟩$ is between $|+x⟩$ and $|e_1^{\perp }⟩$ saturate the exponential bound. This does not mean that ${ϵ}_{EN}$ is the same for all of them. The limiting cases, $|m_0⟩=|e_1^{\perp }⟩$ and $|m_0⟩=|+x⟩$, correspond to the optimal measurements for unambiguous discrimination and for maximizing Eve’s information.