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arXiv:2502.04437 · Physical Review X

Tripartite Haar Random State Has No Bipartite Entanglement

Zhi Li, Takato Mori, Beni Yoshida

Published Jul 28, 2026• 0 citations
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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.

DOI
10.1103/d7b9-ctw2

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We describe a protocol for distilling maximally entangled bipartite states between random pairs of parties from those sharing a tripartite W state |W=(1/sqrt[3])(|100+|010+|001)(ABC), and show that the total distillation rate E(t)(infinity) [the total number of Einstein-Podolsky-Rosen (EPR) pairs distilled per W, irrespective of who shares them] may be done at a higher rate than EPR distillation between specified pairs of parties. Specifically, the optimal rate for distillation to specified parties has been previously shown to be 0.92 EPR pairs per W, while our protocol can asymptotically distill 1 EPR pair per W between random pairs of parties, which we conjecture to be optimal. We thus demonstrate a tradeoff between overall distillation rate and final distribution of EPR pairs. We further show that there exist states with fixed lower-bounded E(t)(infinity), but arbitrarily small distillable entanglement for specified parties.
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Distribution of entanglement is an essential task in quantum information processing and the realization of quantum networks. In our work, we theoretically investigate the scenario where a central source prepares an N-partite entangled state and transmits each entangled subsystem to one ofNNNreceivers through noisy quantum channels. The receivers are then able to perform local operations assisted by unlimited classical communication to distill target entangled states from the noisy channel output. In this operational context, we define the EPR distribution capacity and the GHZ distribution capacity of a quantum channel as the largest rates at which Einstein-Podolsky-Rosen (EPR) states and Greenberger-Horne-Zeilinger (GHZ) states can be faithfully distributed through the channel, respectively. We establish lower and upper bounds on the EPR distribution capacity by connecting it with the task of assisted entanglement distillation. We also construct an explicit protocol consisting of a combination of a quantum communication code and a classical-post-processing-assisted entanglement generation code, which yields a simple achievable lower bound for generic channels. As applications of these results, we give an exact expression for the EPR distribution capacity over two erasure channels and bounds on the EPR distribution capacity over two generalized amplitude damping channels. We also bound the GHZ distribution capacity, which results in an exact characterization of the GHZ distribution capacity when the most noisy channel is a dephasing channel.
We present a family of tripartite entangled states that, in an asymptotical sense, can be reversibly converted into Einstein-Podolsky-Rosen (EPR) states, shared by parties B and C, and tripartite Greenberger-Horne-Zeilinger (GHZ) states. Thus we show that a bipartite and a genuine tripartite entanglement can be reversibly combined in a tripartite state. For such states the corresponding fractions of GHZ and EPR states represent a complete quantification of their (asymptotical) entanglement resources. More generally, we show that AB, AC, and BC EPR entanglement and GHZ entanglement can be reversibly combined in a single tripartite state. Finally, we generalize this result to any number of parties.