  Some beautiful identities involving hook contents of Young diagrams have been
found in the field of quantum information processing, along with a
combinatorial proof. We here give a representation theoretic proof of these
identities and a number of generalizations. Our proof is based on trace
identities for elements belonging to a class of permutation centralizer
algebras. These algebras have been found to underlie the combinatorics of
composite gauge invariant operators in quantum field theory, with applications
in the AdS/CFT correspondence. Based on these algebras, we discuss some
analogies between quantum information processing tasks and the combinatorics of
composite quantum fields and argue that this can be fruitful interface between
quantum information and quantum field theory, with implications for AdS/CFT.
