On the optimality of quantum encryption schemes Available to Purchase

It is well known that $n$ bits of entropy are necessary and sufficient to perfectly encrypt $n$ bits (one-time pad). Even if we allow the encryption to be approximate, the amount of entropy needed does not asymptotically change. However, this is not the case when we are encrypting quantum bits. For the perfect encryption of $n$ quantum bits, $2n$ bits of entropy are necessary and sufficient (quantum one-time pad), but for approximate encryption one asymptotically needs only $n$ bits of entropy. In this paper, we provide the optimal trade-off between the approximation measure $ϵ$ and the amount of classical entropy used in the encryption of single quantum bits. Then, we consider $n$-qubit encryption schemes which are a composition of independent single-qubit ones and provide the optimal schemes both in the 2- and the $∞$-norm. Moreover, we provide a counterexample to show that the encryption scheme of Ambainis-Smith [Proceedings of RANDOM ’04, pp. 249–260] based on small-bias sets does not work in the $∞$-norm.