Quantum key distribution

Definition

A quantum key distribution protocol is a protocol between two parties, Alice and Bob. They are linked by a quantum channel and by an authenticated public classical channel. Authenticated means that others can read its messages but cannot alter them. A third party, Eve, may act arbitrarily on the quantum channel and may read the classical channel. At the end the protocol either aborts or outputs bit strings $K_A$ to Alice and $K_B$ to Bob, of a fixed length $\ell$. Let $p$ be the probability that the protocol does not abort. Let $\rho_{K_A E}$ be the joint state of Alice's key and Eve's system, given no abort. Let $\varepsilon \ge 0$. The protocol is $\varepsilon$-secure if, for every strategy of Eve, the following two conditions hold.

1. Correctness: the probability that the protocol does not abort and $K_A \ne K_B$ is at most $\varepsilon$.
2. Secrecy: $p \cdot \frac12 \norm{\rho_{K_A E} - \omega_\ell \otimes \rho_E}_1 \le \varepsilon$.

Here $\rho_E$ is the partial trace of $\rho_{K_A E}$ over the key, and $\omega_\ell = 2^{-\ell} \sum_{k \in \set{0,1}^\ell} \lvert k \rangle \langle k \rvert$ is the uniform distribution on keys.