Perfect Embezzlement of Entanglement

TL;DR

This paper investigates the possibility of perfect embezzlement of entanglement in quantum systems, proving its impossibility in tensor product models but feasibility in the commuting operator framework, with implications for nonlocal games.

Contribution

It establishes the impossibility of perfect embezzlement in tensor product models and demonstrates its feasibility in the commuting operator framework, providing explicit constructions and analysis.

Findings

Perfect embezzlement impossible in tensor product models.

Perfect embezzlement possible in commuting operator models.

Perfect protocols require infinite-dimensional Hilbert spaces.

Abstract

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.