We study private quantum channels on a single qubit, which encrypt given set of plaintext states
P. Specifically, we determine all achievable states
ρ(0) (average output of encryption) and for each particular set
P we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set
P is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional
P varies continuously between 1 and 2 bits depending on the state
ρ(0). Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.
A classical one-time pad allows two parties to send private messages over a public classical channel-an eavesdropper who intercepts the communication learns nothing about the message. A quantum one-time pad is a shared quantum state which allows two parties to send private messages or private quantum states over a public quantum channel. If the eavesdropper intercepts the quantum communication she learns nothing about the message. In the classical case, a one-time pad can be created using shared and partially private correlations. Here we consider the quantum case in the presence of an eavesdropper, and find the single-letter formula for the rate at which the two parties can send messages using a general quantum state as a quantum one-time pad. Surprisingly, the formula coincides with the distillable entanglement assisted by a symmetric channel, an important quantity in quantum information theory, but which lacked a clear operational meaning.
We addressed the question of optimality of private quantum channels. We have shown that the Shannon entropy of the classical key necessary to securely transfer the quantum information is lower bounded by the entropy exchange of the private quantum channel
E and von Neumann entropy of the ciphertext state
ϱ(0). Based on these bounds we have shown that decomposition of private quantum channels into orthogonal unitaries (if exists) is optimizing the entropy. For non-ancillary single qubit PQC we have derived the optimal entropy for arbitrary set of plaintexts. In particular, we have shown that except when the (closure of the) set of plaintexts contains all states, one bit key is sufficient. We characterized and analyzed all the possible single qubit private quantum channels for arbitrary set of plaintexts. For the set of plaintexts consisting of all qubit states we have characterized all possible approximate private quantum channels and we have derived the relation between the security parameter and the corresponding minimal entropy.
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 study the complexity of quantum decomposable randomized encodings (QDRE). We establish a bi-directional connection between the QDRE model and the communication complexity model of quantum private simultaneous message protocols (QPSM), where classical communication is free, and quantum communication is the primary complexity measure. We show the following upper and lower bounds. (1) For every quantum channel mapping
n to
m qubits, we give a QPSM protocol with quantum communication
m, using exponential pre-shared entanglement. In particular, constant-output channels have
O(1) quantum communication, and classical-output channels require no quantum communication. (2) With
O(n) pre-shared entanglement, every one-qubit-output channel has an
O(n)-quantum-communication QPSM protocol. Conversely, there exists a channel and a constant
c>0 for which any protocol with at most
cn pre-shared entanglement requires
Ω(n) quantum communication. (3) In contrast, we identify a structured class of quantum channels, Clifford-induced channels, for which there exist QPSM protocols with
O(n) pre-shared entangled qubits and only
O(1) quantum communication complexity. Through our connection, we obtain analogous upper and lower bounds on the quantum encoding size of QDRE. Our results show that the amount of pre-shared entanglement plays a central role in quantum encoding size and quantum communication complexity.