Tripartite Haar random state has no bipartite entanglement

• URL: http://arxiv.org/abs/2502.04437v2
• Date: Tue, 15 Apr 2025 18:09:57 GMT
• Title: Tripartite Haar random state has no bipartite entanglement
• Authors: Zhi Li, Takato Mori, Beni Yoshida,
• Abstract summary: We show that no EPR-like bipartite entanglement can be distilled from a tripartite Haar random state.<n>We derive an upper bound on the probability of sampling a state with EPR-like entanglement at a given EPR fidelity tolerance.
• Score: 2.8271134123622064
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 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.

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