Optimality of private quantum channels

TL;DR

This paper investigates the optimality conditions of private quantum channels, establishing entropy bounds, characterizing channels for various plaintext sets, and deriving minimal key entropy for secure quantum communication.

Contribution

It provides a comprehensive analysis of private quantum channels, including entropy bounds, optimal decompositions, and characterizations for single qubit cases with arbitrary plaintext sets.

Findings

Entropy of classical key is bounded by entropy exchange and ciphertext entropy.

Decomposition into orthogonal unitaries optimizes entropy.

One-bit key suffices unless all states are plaintexts.

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.