We consider a bipartite transformation that we call emph{self-embezzlement} and use it to prove a constant gap between the capabilities of two models of quantum information: the conventional model, where bipartite systems are represented by tensor products of Hilbert spaces; and a natural model of quantum information processing for abstract states on C*-algebras, where joint systems are represented by tensor products of C*-algebras. We call this the C*-circuit model and show that it is a special case of the commuting-operator model (in that it can be translated into such a model). For the conventional model, we show that there exists a constant $epsilon_0 > 0$ such that self-embezzlement cannot be achieved with precision parameter less than $epsilon_0$ (i.e., the fidelity cannot be greater than $1 - epsilon_0$); whereas, in the C*-circuit model---as well as in a commuting-operator model---the precision can be $0$ (i.e., fidelity~$1$).
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