  In quantum information theory, the Schmidt rank is a fundamental measure for
the entanglement dimension of a pure bipartite state. Its natural definition
uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which
does not exist (or at least is not canonically given) if the observable
algebras of the local systems are allowed to be general C*-algebras. In this
work, we generalize the Schmidt rank to the commuting operator framework where
the joint system is not necessarily described by the minimal tensor product but
by a general bipartite algebra. We give algebraic and operational definitions
for the Schmidt rank and show their equivalence. We analyze bipartite states
and compute the Schmidt rank in several examples: The vacuum in quantum field
theory, Araki-Woods-Powers states, as well as ground states and translation
invariant states on spin chains which are viewed as bipartite systems for the
left and right half chains. We conclude with a list of open problems for the
commuting operator framework.
