Quantum Subsystems: Exploring the Complementarity of Quantum Privacy and Error Correction

TL;DR

This paper investigates the relationship between quantum privacy and error correction by establishing algebraic conditions for private quantum subsystems and exploring the concept of complementarity in quantum channels.

Contribution

It introduces algebraic conditions for private quantum subsystems, compares them to error correction conditions, and proposes a broader framework linking quantum cryptography and error correction.

Findings

Derived conditions for private subspaces and subsystems.

Demonstrated when a quantum channel lacks private subspaces.

Explored the concept of complementarity in quantum privacy.

Abstract

This paper addresses and expands on the contents of the recent Letter [Phys. Rev. Lett. 111, 030502 (2013)] discussing private quantum subsystems. Here we prove several previously presented results, including a condition for a given random unitary channel to not have a private subspace (although this does not mean that private communication cannot occur, as was previously demonstrated via private subsystems) and algebraic conditions that characterize when a general quantum subsystem or subspace code is private for a quantum channel. These conditions can be regarded as the private analogue of the Knill-Laflamme conditions for quantum error correction, and we explore how the conditions simplify in some special cases. The bridge between quantum cryptography and quantum error correction provided by complementary quantum channels motivates the study of a new, more general definition of quantum error correcting code, and we initiate this study here. We also consider the concept of complementarity for the general notion of private quantum subsystem.