Quantum Privacy and Schur Product Channels

TL;DR

This paper explores the privacy capabilities of certain quantum channels using Schur product operations, identifying private structures even when direct subspace privacy is impossible.

Contribution

It introduces a novel approach linking quantum privacy of channels with Schur product matrix operations and correlation structures, revealing private algebras and subsystems.

Findings

Channels with commuting unitaries can privatize algebras and subsystems.

New connections between quantum privacy and Schur product matrices.

Application of operator algebra and graph theory tools to quantum privacy.

Abstract

We investigate the quantum privacy properties of an important class of quantum chan-nels, by making use of a connection with Schur product matrix operations and associated correlationmatrix structures. For channels implemented by mutually commuting unitaries, which cannot priva-tise qubits encoded directly into subspaces, we nevertheless identify private algebras and subsystemsthat can be privatised by the channels. We also obtain further results by combining our analysiswith tools from the theory of quasiorthogonal operator algebras and graph theory.