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arXiv:0710.2004 · Journal of Physics A Mathematical and Theoretical

Optimality of private quantum channels

Jan Bouda, Mário Ziman

Published Apr 30, 2007• 10 citations

Abstract

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 $\cal E$ and von Neumann entropy of the ciphertext state $\varrho^{(0)}$. 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.

DOI: 10.1088/1751-8113/40/20/011

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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

\cal E

and von Neumann entropy of the ciphertext state

\varrho^{(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.

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

ρ^{(0)}

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.

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamiołkowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.

This thesis includes a survey of the results known for private and approximate private quantum channels. We develop the best known upper bound for

ε

-randomizing maps,

n+2\log(1/ε)+c

bits required to

ε

-randomize an arbitrary

n

-qubit state by improving a scheme of Ambainis and Smith \cite{AS04} based on small bias spaces \cite{NN90, AGHP92}. We show by a probabilistic argument that in fact the great majority of random schemes using slightly more than this many bits of key are also

ε

-randomizing. We provide the first known non-trivial lower bound for

ε

-randomizing maps, and develop several conditions on them which we hope may be useful in proving stronger lower bounds in the future.

We study in detail the problem of encryption of quantum information. We present an attack on a private quantum channel (PQC) which applies when partial classical description of a ciphertext is known (the so-called known-ciphertext attack) and we show how this situation can be avoided. The quantum analogue of the known plaintext attack is also discussed. We determine how correlations between quantum systems can be encrypted and we conclude that two PQCs on the subsystems form a PQC on the whole composite system. Finally, some applications of the PQC are suggested and a security of a noisy channel is discussed.