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arXiv:quant-ph/0607126 · Physical Review Letters

Random Bipartite Entanglement from W and W -Like States

Ben Fortescue, Hoi‐Kwong Lo

Published Jun 28, 2007• 56 citations
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Abstract

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.

DOI
10.1103/physrevlett.98.260501

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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.
We find that the asymptotic entanglement of assistance of a general bipartite mixed state is equal to the smaller of its two local entropies. Our protocol gives rise to the asymptotically optimal Einstein-Podolsky-Rosen (EPR) pair distillation procedure for a given tripartite pure state, and we show that it actually yields EPR and Greenberger-Horne-Zeilinger (GHZ) states; in fact, under a restricted class of protocols, which we call ``one-way broadcasting,'' the GHZ rate is shown to be optimal. This result implies a capacity theorem for quantum channels where the environment helps transmission by broadcasting the outcome of an optimally chosen measurement. We discuss generalizations to mmm parties and show (for m=4m=4m=4) that the maximal amount of entanglement that can be localized between two parties is given by the smallest entropy of a group of parties of which the one party is a member, but not the other. This gives an explicit expression for the asymptotic localizable entanglement and shows that any nontrivial ground state of a spin system can be used as a perfect quantum repeater if many copies are available in parallel. Finally, we provide evidence that any unital channel is asymptotically equivalent to a mixture of unitaries and any general channel to a mixture of partial isometries.
We describe various results related to the random-party distillation of multiparty entangled states---that is, conversion of such states into entangled states shared between fewer parties, where those parties are not predetermined. In previous work we showed that certain output states (namely Einstein-Podolsky-Rosen pairs) could be reliably acquired from a prescribed initial multipartite state [namely the W state ∣W⟩=13(∣100⟩+∣010⟩+∣001⟩)|\text{W}⟩=\frac{1}{\sqrt{3}}(|100⟩+|010⟩+|001⟩)∣W⟩=3​1​(∣100⟩+∣010⟩+∣001⟩)] via random-party distillation that could not be reliably created between predetermined parties. Here we provide a more rigorous definition of what constitutes ``advantageous'' random-party distillation. We show that random-party distillation is always advantageous for W-class three-qubit states (but only sometimes for Greenberger-Horne-Zeilinger class states). We show that the general class of multiparty states known as symmetric Dicke states can be readily converted to many other states in the class via random-party distillation. Finally we show that random-party distillation is provably not advantageous in the limit of multiple copies of pure states.
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.
Bipartite entanglement purification is the conversion of copies of weakly entangled pairs shared between two separated parties into a smaller number of strongly entangled shared pairs using only local operations and classical communication. Choosing between different entanglement purification protocols generally involves weighing up a trade-off between the ratio of strongly entangled pairs produced to weakly entangled pairs consumed, which we call the rate of the protocol, and the degree of the entanglement of the strongly entangled pairs, typically measured by the fidelity of those pairs to maximally entangled states. By randomly choosing a protocol according to a probability distribution over a list of protocols for each pair we want to produce, we can achieve rates and fidelities not achieved by any of the original protocols. Here, we show how to choose this distribution to maximize the rate at which we produce qubit pairs with a given fidelity to a Bell state or, equivalently, to maximize the fidelity to a Bell state of the qubit pairs produced at a given rate. We investigate both the asymptotic case, where the number of initial pairs goes to infinity, and the finite-size regime, where protocols are restricted to a finite number of weakly entangled pairs.
Random party distillation refers to the process by which Einstein-Podolsky-Rosen pairs are probabilistically extracted from a single copy of a multipartite entangled state after multiple rounds of local operations and classical communications (LOCC). In this work, we propose a qubit-based implementation of a random party distillation protocol and demonstrate it on the superconducting hardware device, ibm_aachen. We implement up to 4-rounds of the protocol, achieving a distillation rate of ~0.85 pairs/ W state, surpassing previous experimental demonstrations. Our results demonstrate the utility of multi-round LOCC protocols in an experimental regime that remains comparatively unexplored.