Asymptotic correctability of Bell-diagonal quantum states and maximum tolerable bit-error rates

The general conditions are discussed which quantum state purification protocols have to fulfil in order to be capable of purifying Bell-diagonal qubit-pair states, provided they consist of steps that map Bell-diagonal states to Bell-diagonal states and they finally apply a suitably chosen Calderbank-Shor-Steane code to the outcome of such steps. As a main result a condition on asymptotic correctability is presented, which relates this problem to the magnitude of a characteristic exponent governing the relation between bit and phase errors under the purification steps. This condition allows a straightforward determination of maximum tolerable bit-error rates of quantum key distribution protocols whose security analysis can be reduced to the purification of Bell-diagonal states