Tripartite Haar random state has no bipartite entanglement

Abstract

We show that no EPR-like bipartite entanglement can be distilled from a tripartite Haar random state $|\hat{I}¨{⟩}_{ABC}$ 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 $\hat{I}\mu$-net argument and concentration of measure. Viewing $|\hat{I}¨{⟩}_{ABC}$ as a bipartite quantum error-correcting code $C\to AB$, this implies that neither output subsystem $A$ nor $B$ supports any non-trivial logical operator. We also establish general constraints on the structure of tripartite entanglement in Haar random states, showing that W- or GHZ-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.

(v1) 14 pages, 5 figures; (v2) an author (Zhi Li) added. Improved bounds with new results on local operation distillation and logical operators; (v3) Added Sections 1.2.4 and 5 discussing generalizations, Section 1.4 for intuitions of the proofs, a comment in relation to QFT entanglement in Section 6.1, and an additional comment in Section 7. 36 pages, 5 figures