  Quantum information theory has generated several interesting conjectures
involving products of completely positive maps on matrix algebras, also known
as quantum channels. In particular it is conjectured that the output state with
maximal p-norm from a product channel is always a product state. It is shown
here that the Lieb-Thirring inequality can be used to prove this conjecture for
one special case, namely when one of the components of the product channel is
of the type known as a diagonal channel, which acts on a state by taking the
Hadamard product with a positive matrix.
