Complementarity of Private and Correctable Subsystems in Quantum Cryptography and Error Correction

TL;DR

This paper establishes a fundamental link between private and correctable subsystems in quantum channels, showing they are equivalent for complementary channels, which advances understanding in quantum cryptography and error correction.

Contribution

It proves that a subsystem is private if and only if it is correctable for the complementary channel, extending the result to approximate cases using the diamond norm.

Findings

Private subsystems are correctable for complementary channels.

The equivalence holds even for approximate privacy and correctability.

Extends the fundamental understanding of quantum cryptography and error correction.

Abstract

We make an explicit connection between fundamental notions in quantum cryptography and quantum error correction. Error-correcting subsystems (and subspaces) for quantum channels are the key vehicles for contending with noise in physical implementations of quantum information-processing. Private subsystems (and subspaces) for quantum channels play a central role in cryptographic schemes such as quantum secret sharing and private quantum communication. We show that a subsystem is private for a channel precisely when it is correctable for a complementary channel. This result is shown to hold even for approximate notions of private and correctable defined in terms of the diamond norm for superoperators.