Quantum Encryption and Generalized Quantum Shannon Impossibility

TL;DR

This paper extends Shannon's classical impossibility result to quantum encryption, analyzing the trade-offs between secrecy, correctness, and key length in quantum systems with imperfect security.

Contribution

It introduces a quantum analogue of Shannon's impossibility theorem, studying secure quantum encryption under different secrecy definitions and their security implications.

Findings

Weaker secrecy implies stronger secrecy with a security loss proportional to system dimension.

Quantum encryption can achieve imperfect secrecy and correctness with shorter keys under certain conditions.

The security loss is acceptable if the secrecy error is smaller than the inverse of the system dimension.

Abstract

The famous Shannon impossibility result says that any encryption scheme with perfect secrecy requires a secret key at least as long as the message. In this paper we provide its quantum analogue with imperfect secrecy and imperfect correctness. We also give a systematic study of information-theoretically secure quantum encryption with two secrecy definitions. We show that the weaker one implies the stronger but with a security loss in $d$, where $d$ is the dimension of the encrypted quantum system. This is good enough if the target secrecy error is of $o(d^{-1})$.