On the optimality of quantum encryption schemes

It is well known that n bits of entropy are necessary and sufficient toperfectly encrypt n bits (one-time pad). Even if we allow the encryption to beapproximate, the amount of entropy needed doesn't asymptotically change.However, this is not the case when we are encrypting quantum bits. For theperfect encryption of n quantum bits, 2n bits of entropy are necessary andsufficient (quantum one-time pad), but for approximate encryption oneasymptotically needs only n bits of entropy. In this paper, we provide theoptimal trade-off between the approximation measure epsilon and the amount ofclassical entropy used in the encryption of single quantum bits. Then, weconsider n-qubit encryption schemes which are a composition of independentsingle-qubit ones and provide the optimal schemes both in the 2- and theoperator-norm. Moreover, we provide a counterexample to show that theencryption scheme of Ambainis-Smith based on small-bias sets does not work inthe operator-norm.Comment: 15 page