Distillation of secret key and entanglement from quantum states

We study and solve the problem of distilling a secret key from quantum states representing correlation between two parties (Alice and Bob) and an eavesdropper (Eve) via one-way public discussion: we prove a coding theorem to achieve the 'wire-tapper' bound, the difference of the mutual information Alice-Bob and that of Alice-Eve, for so-called classical-quantum-quantum-correlations, via one-way public communication. This result yields information-theoretic formulae for the distillable secret key, giving 'ultimate' key rate bounds if Eve is assumed to possess a purification of Alice and Bob's joint state. Specializing our protocol somewhat and making it coherent leads us to a protocol of entanglement distillation via one-way LOCC (local operations and classical communication) which is asymptotically optimal: in fact we prove the so-called 'hashing inequality', which says that the coherent information (i.e. the negative conditional von Neumann entropy) is an achievable Einstein-Podolsky-Rosen rate. This result is known to imply a whole set of distillation and capacity formulae, which we briefly review.