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arXiv:0709.4059 · Physical Review A

Random-party entanglement distillation in multiparty states

Ben Fortescue, Hoi-Kwong Lo

Published Jul 25, 2008• 23 citations
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Abstract

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 $|\text{W}⟩=\frac{1}{\sqrt{3}}(|100⟩+|010⟩+|001⟩)$] 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.

DOI
10.1103/physreva.78.012348

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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 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 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.
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.
The phenomenon of quantum entanglement is a fundamental feature of quantum mechanics which, as a counterintuitive and inherently ”quantum” phenomenon (with no classical analogue) has been the subject of much study, especially in quantum information theory. One fruitful approach to the description of entanglement has been in its operational description - that is, in the consideration of what can be achieved using entangled states under certain restrictions, typically the regime of local operations and classical communications.\n\nWe present results here related to the operational characterisation of entanglement in the resource model, in both bipartite and multipartite cases. First, we consider the conversion between pure bipartite entangled states in terms of an often-ignored resource - the classical communication cost. Using prior results for more specific conversions, we derive lower bounds on this cost (and the related quantity of the conversion inefficiency) for general bipartite pure states.\n\nWe also consider pure-state conversions of multipartite entanglement, in particular the class of protocols in which multipartite states are converted to states shared between fewer parties. We have found a previously-unconsidered variety of such conversions, in which the target state of the conversion is a state shared between a random subset of the parties. We find that when such post-selection of parties in the protocol is permitted allows for a wider variety of achievable target states; certain states which can not be reliably obtained between predetermined parties (even some where the probability of doing so is arbitrarily small) can be obtained between random parties. We consider a variety of states in which this phenomenon occurs, as well as bounds on such protocols can achieve.\n\nFinally we consider a practical use of entanglement as a resource, in an experimental implementation of a multipartite QKD protocol. This is based on the tripartite GHZ entangled state, but can be implemented using only bipartite entanglement. We adapt existing QKD results for both the bipartite and multipartite case to derive a secure key rate for this implementation, taking into account the ways in which it differs from the idealised theoretical case.
In this article we obtain results for the task of converting a single NNN-qubit WWW-class state (of the form $\sqrt{{x}_{0}}|00...0\ensuremath{\rangle}+\sqrt{{x}_{1}}|10...0\ensuremath{\rangle}+\ensuremath{\cdots}+\sqrt{{x}_{N}}|00...1\ensuremath{\rangle}$) into maximum entanglement shared between two random parties. Previous studies in random distillation have not considered how the particular choice of target pairs affects the transformation, and here we develop a strategy for distilling into general configurations of target pairs. We completely solve the problem of determining the optimal distillation probability for all three-qubit configurations and most four-qubit configurations when x0=0{x}_{0}=0x0​=0. Our proof involves deriving new entanglement monotones defined on the set of four-qubit WWW-class states. As an additional application of our results, we present new upper bounds for converting a generic WWW-class state into the standard WWW state $|{W}_{N}\ensuremath{\rangle}=\sqrt{\frac{1}{N}}(|10...0\ensuremath{\rangle}+\ensuremath{\cdots}+|00...1\ensuremath{\rangle})$.