  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.
