Tripartite Haar random state has no bipartite entanglement

TL;DR

This paper demonstrates that tripartite Haar random states cannot be distilled into bipartite EPR-like entanglement when subsystems are smaller than half the total qubits, with implications for quantum error correction and holography.

Contribution

It provides a rigorous upper bound on bipartite entanglement distillation from tripartite Haar random states and explores the structure of tripartite entanglement.

Findings

No bipartite EPR-like entanglement can be distilled from such states.

Sampling states with high EPR fidelity is doubly-exponentially suppressed.

Tripartite Haar random states lack W- or GHZ-like entanglement and global symmetries.

Abstract

We show that no EPR-like bipartite entanglement can be distilled from a tripartite Haar random state $|\Psi\rangle_{ABC}$ 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 $\epsilon$-net argument and concentration of measure. Viewing $|\Psi\rangle_{ABC}$ 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.