Constant gap between conventional strategies and those based on C*-dynamics for self-embezzlement

Quantum 6, 755 (2022).

https://doi.org/10.22331/q-2022-07-07-755

We consider a bipartite transformation that we call $self-embezzlement$ 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 $1$).

Self-embezzlement is not a non-local game, hence our results do not impact the celebrated Connes Embedding conjecture. Instead, the significance of these results is to exhibit a reasonably natural quantum information processing problem for which there is a constant gap between the capabilities of the conventional Hilbert space model and the commuting-operator or C*-circuit model.