  We introduce a generalized framework for private quantum codes using von
Neumann algebras and the structure of commutants. This leads naturally to a
more general notion of complementary channel, which we use to establish a
generalized complementarity theorem between private and correctable subalgebras
that applies to both the finite and infinite-dimensional settings. Linear
bosonic channels are considered and specific examples of Gaussian quantum
channels are given to illustrate the new framework together with the
complementarity theorem.
