  Van Dam and Hayden introduced a concept commonly referred to as embezzlement,
where, for any entangled quantum state $\phi$, there is an entangled catalyst
state $\psi$, from which a high fidelity approximation of $\phi \otimes \psi$
can be produced using only local operations. We investigate a version of this
where the embezzlement is perfect (i.e., the fidelity is 1). We prove that
perfect embezzlement is impossible in a tensor product framework, even with
infinite-dimensional Hilbert spaces and infinite entanglement entropy. Then we
prove that perfect embezzlement is possible in a commuting operator framework.
We prove this using the theory of C*-algebras and we also provide an explicit
construction. Next, we apply our results to analyze perfect versions of a
nonlocal game introduced by Regev and Vidick. Finally, we analyze the structure
of perfect embezzlement protocols in the commuting operator model, showing that
they require infinite-dimensional Hilbert spaces.
