Self-testing of exact entanglement embezzlement

TL;DR

This paper characterizes exact entanglement embezzlement protocols as self-tests for Cuntz algebra isometries and states, linking quantum entanglement manipulation to operator algebra structures.

Contribution

It proves that exact entanglement embezzlement protocols uniquely correspond to states on tensor products of Cuntz algebras, establishing a novel self-testing framework.

Findings

Protocols correspond to unique states on   Cuntz algebra tensor products.

Exact embezzlement acts as a self-test for Cuntz isometries and states.

Von Neumann algebra generated is a unique Type  III factor, determined by Schmidt coefficients.

Abstract

We consider bipartite exact entanglement embezzlement with a catalyst state vector $\psi$ in a Hilbert space $\mathcal{H}$ using unitaries (or more generally, contractions). If $\mathcal{M} \subseteq \mathcal{B}(\mathcal{H})$ is a von Neumann algebra and $U \in M_d \otimes \mathcal{M}$ and $V \in \mathcal{M}' \otimes M_d$ are unitaries (or more generally contractions), then such a protocol is of the form $(U \otimes I_d)(I_d \otimes V)(e_0 \otimes \psi \otimes e_0)=\sum_{i=0}^{d-1} \alpha_i e_i \otimes \psi \otimes e_i$, where each $\alpha_i>0$ and $\sum_{i=0}^{d-1} \alpha_i^2=1$. We show that any such protocol must arise from a unique state on the tensor product $\mathcal{O}_d \otimes \mathcal{O}_d$ of the Cuntz algebra with itself. As a result, we prove that exact entanglement embezzlement is a self-test for a collection of $d$ Cuntz isometries for each party and a unique quasi-free state on the Cuntz algebra $\mathcal{O}_d$ in the sense of \cite{Iz93}. Moreover, we use modular theory to show that the von Neumann algebra generated by the copy of $\mathcal{O}_d$ is the unique separable approximately finite-dimensional Type $\text{III}_{\lambda}$ factor for some $0<\lambda \leq 1$, where $\lambda$ can be determined by an algebraic condition on the Schmidt coefficients of the state $\varphi=\sum_{i=0}^{d-1} \alpha_i e_i \otimes e_i$.