  We develop a formalism for distilling a classical key from a quantum state in
a systematic way, expanding on our previous work on secure key from bound
entanglement [K. Horodecki et. al., Phys. Rev. Lett. 94 (2005)]. More detailed
proofs, discussion and examples are provided of the main results. Namely, we
demonstrate that all quantum cryptographic protocols can be recast in a way
which looks like entanglement theory, with the only change being that instead
of distilling EPR pairs, the parties distill private states. The form of these
general private states are given, and we show that there are a number of useful
ways of expressing them. Some of the private states can be approximated by
certain states which are bound entangled. Thus distillable entanglement is not
a requirement for a private key. We find that such bound entangled states are
useful for a cryptographic primitive we call a controlled private quantum
channel. We also find a general class of states which have negative partial
transpose (are NPT), but which appear to be bound entangled. The relative
entropy distance is shown to be an upper bound on the rate of key. This allows
us to compute the exact value of distillable key for a certain class of private
states.
