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arXiv:1106.1209 · Physical Review A

Randomly distilling W -class states into general configurations of two-party entanglement

Wei Cui, Eric Chitambar, Hoi‐Kwong Lo

Published Nov 1, 2011• 15 citations
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Abstract

In this article we obtain results for the task of converting a single $N$-qubit $W$-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 ${x}_{0}=0$. Our proof involves deriving new entanglement monotones defined on the set of four-qubit $W$-class states. As an additional application of our results, we present new upper bounds for converting a generic $W$-class state into the standard $W$ state $|{W}_{N}\ensuremath{\rangle}=\sqrt{\frac{1}{N}}(|10...0\ensuremath{\rangle}+\ensuremath{\cdots}+|00...1\ensuremath{\rangle})$.

DOI
10.1103/physreva.84.052301

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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.
In this article, we extend recent results concerning random-pair Einstein-Podolsky-Rosen distillation and the operational gap between separable operations (SEPs) and local operations with classical communication (LOCC). In particular, we consider the problem of obtaining bipartite maximal entanglement from an NNN-qubit WWW-class state (i.e., that of the form $\sqrt{{x}_{0}}|00\ensuremath{\cdots}0\ensuremath{\rangle}+\sqrt{{x}_{1}}|10\ensuremath{\cdots}0\ensuremath{\rangle}+\ensuremath{\cdots}+\sqrt{{x}_{n}}|00\ensuremath{\cdots}1\ensuremath{\rangle}$) when the target pairs are a priori unspecified. We show that when x0=0{x}_{0}=0x0​=0, the optimal probabilities for SEPs can be computed using semidefinite programming. On the other hand, to bound the optimal probabilities achievable by LOCC, we introduce entanglement monotones defined on the NNN-qubit WWW class of states. The LOCC monotones we construct can be increased by SEPs, and in terms of transformation success probability, we are able to quantify a gap as large as 37% between the two classes. Additionally, we demonstrate transformations ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}n}\ensuremath{\rightarrow}{\ensuremath{\sigma}}^{\ensuremath{\bigotimes}n}$ that are feasible by SEP for any nnn but impossible by LOCC.
Entanglement is one of the fundamental features of quantum information science.\nThough bipartite entanglement has been analyzed thoroughly in theory and shown to\nbe an important resource in quantum computation and communication protocols, the\ntheory of entanglement shared between more than two parties, which is called multipartite\nentanglement, is still not complete. Specifically, the classification of multipartite\nentanglement and the transformation property between different multipartite states by\nlocal operators and classical communications (LOCC) are two fundamental questions in\nthe theory of multipartite entanglement.\nIn this thesis, we present results related to the LOCC transformation between multipartite\nentangled states. Firstly, we investigate the bounds on the LOCC transformation\nprobability between multipartite states, especially the GHZ class states. By analyzing\nthe involvement of 3-tangle and other entanglement measures under weak two-outcome\nmeasurement, we derive explicit upper and lower bound on the transformation probability\nbetween GHZ class states. After that, we also analyze the transformation between\nN-party W type states, which is a special class of multipartite entangled states that\nhas an explicit unique expression and a set of analytical entanglement monotones. We\npresent a necessary and sufficient condition for a known upper bound of transformation\nprobability between two N-party W type states to be achieved.\nWe also further investigate a novel entanglement transformation protocol, the random\ndistillation, which transforms multipartite entanglement into bipartite entanglement\nii\nshared by a non-deterministic pair of parties. We find upper bounds for the random distillation\nprotocol for general N-party W type states and find the condition for the upper\nbounds to be achieved. What is surprising is that the upper bounds correspond to entanglement\nmonotones that can be increased by Separable Operators (SEP), which gives\nthe first set of analytical entanglement monotones that can be increased by SEP.\nFinally, we investigate the idea of a new class of multipartite entangled states, the\nAbsolutely Maximal Entangled (AME) states, which is characterized by the fact that\nany bipartition of the states would give a maximal entangled state between the two sets.\nThe relationship between AME states and Quantum secret sharing (QSS) protocols is\nexhibited and the application of AME states in novel quantum communication protocols\nis also explored.
We investigate the physically allowed probabilities for transforming one NNN-partite WWW-class state to another by means of local operations assisted with classical communication. Recently, S. Kinta\ifmmode \mbox{\c{s}}\else \c{s}\fi{} and S. Turgut [J. Math. Phys. 51, 092202 (2010)] obtained an upper bound for the maximum probability of transforming two such states. Here, we provide a simple sufficient and necessary condition for when this upper bound can be satisfied and, thus, when optimality of state transformation can be achieved. Our discussion involves obtaining lower bounds for the transformation of arbitrary WWW-class states and showing precisely when this bound saturates the bound of Kinta\ifmmode \mbox{\c{s}}\else \c{s}\fi{} and Turgut. Finally, we consider the question of transforming symmetric WWW-class states and find that, in general, the optimal one-shot procedure for converting two symmetric states requires a nonsymmetric filter by all the parties.
For a tripartite pure state of three qubits, it is well known that there are two inequivalent classes of genuine tripartite entanglement, namely the Greenberger-Horne-Zeilinger (GHZ) class and the W class. Any two states within the same class can be transformed into each other with stochastic local operations and classical communication with a nonzero probability. The optimal conversion probability, however, is only known for special cases. Here, lower and upper bounds are derived for the optimal probability of transformation from a GHZ state to other states of the GHZ class. A key idea in the derivation of the upper bounds is to consider the action of the local operations and classical communications (LOCC) protocol on a different input state, namely $1/\sqrt{2}[|000\ensuremath{\rangle}\ensuremath{-}|111\ensuremath{\rangle}]$, and to demand that the probability of an outcome remains bounded by 1. We also find an upper bound for more general cases by using the constraints of the so-called interference term and 3-tangle. Moreover, some of the results are generalized to the case in which each party holds a higher dimensional system. In particular, the GHZ state generalized to three qutrits; that is, $|{\mathrm{GHZ}}_{3}\ensuremath{\rangle}=1/\sqrt{3}[|000\ensuremath{\rangle}+|111\ensuremath{\rangle}+|222\ensuremath{\rangle}]$ shared among three parties can be transformed to any tripartite three-qubit pure state with probability 1 via LOCC. Some of our results can also be generalized to the case of a multipartite state shared by more than three parties.
We consider a single copy of a pure four-partite state of qubits and investigate its behavior under the action of stochastic local quantum operations assisted by classical communication (SLOCC). This leads to a complete classification of all different classes of pure states of four qubits. It is shown that there exist nine families of states corresponding to nine different ways of entangling four qubits. The states in the generic family give rise to Greenberger-Horne-Zeilinger-like entanglement. The other ones contain essentially two-or three-qubit entanglement distributed among the four parties. The concept of concurrence and 3-tangle is generalized to the case of mixed states of four qubits, giving rise to a seven-parameter family of entanglement monotones. Finally, the SLOCC operations maximizing all these entanglement monotones are derived, yielding the optimal single-copy distillation protocol.