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Physical Review A

Comment on “Optimal encryption of quantum bits”

Xiangfu Zou, Daowen Qiu

Published Aug 27, 2013• 1 citation
Publisher

Abstract

Boykin and Roychowdhury [Phys. Rev. A 67, 042317 (2003)] discussed the optimal encryption of quantum bits, gave a necessary and sufficient condition (NASC) for the security of quantum encryption, and suggested an optimal quantum one-time pad encryption scheme. In this Comment, we point out that the proof of the NASC for the security of quantum encryption is not correct. Moreover, we reprove the NASC for the security of quantum encryption.

DOI
10.1103/physreva.88.026302

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Boykin and Roychowdhury [Phys. Rev. A 67, 042317 (2003)] discussed the optimal encryption of quantum bits, gave a necessary and sufficient condition (NASC) for the security of quantum encryption, and suggested an optimal quantum one-time pad encryption scheme. In this Comment, we point out that the proof of the NASC for the security of quantum encryption is not correct. Moreover, we reprove the NASC for the security of quantum encryption.
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.
We show that 2n2n2n random classical bits are both necessary and sufficient for encrypting any unknown state of n quantum bits in an informationally secure manner. We also characterize the complete set of optimal protocols in terms of a set of unitary operations that comprise an orthonormal basis in a canonical inner product space. Moreover, a connection is made between quantum encryption and quantum teleportation that allows for a different proof of optimality of teleportation.
The security of quantum key distribution is typically defined in terms of the mutual information between the distributed key S and the outcome of an optimal measurement applied to the adversary's system. We show that even if this so-called accessible information is small, the key S might not be secure enough to be used in applications such as one-time pad encryption. This flaw is due to a locking property of the accessible information: one additional (physical) bit of information can increase the accessible information by more than one bit.