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

Richard Cleve1, Benoit Collins2, Li Liu1, and Vern Paulsen3

1Institute for Quantum Computing and Cheriton School of Computer Science, University of Waterloo, Canada.
2Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan.
3Institute for Quantum Computing and Department of Pure Mathematics, University of Waterloo, Canada.

### Abstract

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 $\epsilon_0\gt0$ 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.

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### Cited by

[1] Benoît Collins and Sang-Gyun Youn, "Additivity violation of the regularized Minimum Output Entropy", arXiv:1907.07856.

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