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arXiv:1009.2467 · Physical Review A

Optimal entanglement transformations among N -qubit W -class states

Wei Cui, Eric Chitambar, Hoi‐Kwong Lo

Published Dec 14, 2010• 9 citations
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Abstract

We investigate the physically allowed probabilities for transforming one $N$-partite $W$-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 $W$-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 $W$-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.

DOI
10.1103/physreva.82.062314

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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.
The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit states cannot be obtained from nor transformed to any state (i.e., they are isolated), which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that generic nnn-qubit symmetric states are also isolated, we consider particular families for which we can determine to be, on the contrary, endowed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. In the course of the study of LOCC transformations, we also characterize the local symmetries of symmetric states.
The transformations of W-type entangled states by using local operations assisted with classical communication are investigated. For this purpose, a parametrization of the W-type states which remains invariant under local unitary transformations is proposed and a complete characterization of the local operations carried out by a single party is given. These are used for deriving the necessary and sufficient conditions for deterministic transformations. A convenient upper bound for the maximum probability of distillation of arbitrary target states is also found.
This thesis studies the manipulation of entanglement in three-qubit quantum systems. I consider the operational setting in which the qubits are distributed to three spatially separated parties. The parties act locally on their quantum systems and share classical communication with one another, a scenario commonly called local operations and classical communication (LOCC). In the LOCC setting, there are two different classes of entanglement in multipartite systems, called the GHZ and W classes, which are inequivalent in the sense that states from one class cannot be transformed into the other without the consumption of additional entanglement. In this thesis, I first show that the LOCC conversion of certain GHZ and W-class states becomes possible by using only one additional ebit (“entangled bit”) of shared entanglement. In many cases, this can be proven as the minimal amount of needed entanglement. I then consider the problem of one-way communication transformations from general three-qubit states into two-qubit maximally entangled states, known as EPR states. An optimal protocol in terms of success probability is provided for W-class states. The success probability of this protocol coincides with the optimal success probability if two of the parties are allowed to act jointly within the same laboratory. In other words, forcing the locality constraint on all three parties does not weaken their capabilities for obtaining bipartite entanglement when starting from a W-class state. I also present that this property holds for certain types of GHZ-class states as well.
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 $|{W}_{N}\ensuremath{\rangle}=\sqrt{\frac{1}{N}}(|10...0\ensuremath{\rangle}+\ensuremath{\cdots}+|00...1\ensuremath{\rangle})$.
We analytically obtain the maximum probability of converting a finite number of copies of an arbitrary two-qubit pure state to a single copy of a maximally entangled two-qubit pure state via entanglement assisted local operations and classical communications using a two-qubit catalyst state. We show that the optimal catalyst for this transformation is always more entangled than the initial state but any two-qubit state can act as a (non-optimal) catalyst. Interestingly, the entanglement of the optimal two-qubit catalyst state is shown to decrease with that of the initial state. Entanglement assisted strategies for obtaining multiple Bell states are discussed.