Tripartite Haar random state has no bipartite entanglement

Takato Mori, Beni Yoshida (Feb 10 2025).

Abstract: We show that no EPR-like bipartite entanglement can be distilled from a tripartite Haar random state $|\Psi\rangle_{ABC}$ by local unitary rotations when each subsystem has fewer than half of the total qubits. Specifically, we derive an upper bound on the probability of sampling a state with EPR-like entanglement for a given EPR fidelity tolerance, showing a doubly-exponential suppression in the number of qubits. Our proof relies on a simple volume argument supplemented by an $\epsilon$-net construction. We also discuss a physical interpretation in the AdS/CFT correspondence, indicating that a connected entanglement wedge does not necessarily imply bipartite entanglement as opposed to a previous belief.

Arxiv: https://arxiv.org/abs/2502.04437