Tripartite Haar Random State Has No Bipartite Entanglement

Abstract

We show that no Einstein-Podolsky-Rosen (EPR)–like bipartite entanglement can be distilled from a tripartite Haar random state | Ψ ⟩ A B C by local unitaries or local operations when each subsystem A , B , or C 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 at 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 ε -net argument and concentration of measure. Viewing | Ψ ⟩ A B C as a bipartite quantum error-correcting code C → A B , this implies that neither output subsystem A nor B supports any nontrivial logical operator. We also establish general constraints on the structure of tripartite entanglement in Haar random states, showing that W- or Greenberger-Horne-Zeilinger–like entanglement cannot be distilled and that nontrivial global symmetries are absent. Finally, we discuss a physical interpretation in the AdS/CFT correspondence, indicating that a connected entanglement wedge does not necessarily imply bipartite entanglement, contrary to a previous belief.