Quantum Privacy and Schur Product Channels

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.