Stealing entanglement perfectly forces a unique quantum structure

Entanglement embezzlement is the idea that a 'catalyst' quantum state can lend entanglement to another system and emerge perfectly unchanged. This paper proves that any protocol achieving this exactly must be built from a unique mathematical object: a quasi-free state on the Cuntz algebra. As a bonus, the von Neumann algebra it generates is pinned down to a specific rare type (Type III factor), meaning perfect embezzlement is a surprisingly rigid, self-certifying quantum process.