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arXiv:quant-ph/0506229 · Physical Review A

Mixed-state entanglement of assistance and the generalized concurrence

Gilad Gour

Published Oct 19, 2005• 40 citations
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Abstract

We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite mixed states. We show that this assisted entanglement, when measured in terms of the generalized concurrence (called the $G$ concurrence), is (tightly) bounded by an entanglement monotone, which we call the $G$ concurrence of assistance. The $G$ concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the geometric mean of the Schmidt numbers. For a large (nontrivial) class of $(d\ifmmode\times\else\texttimes\fi{}d)$-dimensional mixed states, we are able to generalize Wootters' formula for the concurrence into lower and upper bounds on the $G$ concurrence. Moreover, we have found an explicit formula for the $G$ concurrence of assistance that generalizes the expression for the concurrence of assistance for a large class of $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite pure states.

DOI
10.1103/physreva.72.042318

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We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on (d\ifmmode×\else\texttimes\fid\ifmmode×\else\texttimes\fin)(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)(d\ifmmode×\else\texttimes\fid\ifmmode×\else\texttimes\fin)-dimensional tripartite mixed states. We show that this assisted entanglement, when measured in terms of the generalized concurrence (called the GGG concurrence), is (tightly) bounded by an entanglement monotone, which we call the GGG concurrence of assistance. The GGG concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the geometric mean of the Schmidt numbers. For a large (nontrivial) class of (d\ifmmode×\else\texttimes\fid)(d\ifmmode\times\else\texttimes\fi{}d)(d\ifmmode×\else\texttimes\fid)-dimensional mixed states, we are able to generalize Wootters' formula for the concurrence into lower and upper bounds on the GGG concurrence. Moreover, we have found an explicit formula for the GGG concurrence of assistance that generalizes the expression for the concurrence of assistance for a large class of (d\ifmmode×\else\texttimes\fid\ifmmode×\else\texttimes\fin)(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)(d\ifmmode×\else\texttimes\fid\ifmmode×\else\texttimes\fin)-dimensional tripartite pure states.
Certain quantum-information tasks require entanglement of assistance, namely, a reduction of a tripartite entangled state to a bipartite entangled state via local measurements. We establish that concurrence of assistance (COA) identifies capabilities of and limitations to producing pure bipartite entangled states from pure tripartite entangled states and prove that COA is an entanglement monotone for $(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}n)$-dimensional pure states. Moreover, if the COA for the pure tripartite state is at least as large as the concurrence of the desired pure bipartite state, then the former may be transformed to the latter via local operations and classical communication, and we calculate the maximum probability for this transformation when this condition is not met.
We present a powerful theorem for tripartite remote entanglement distribution protocols, which provides an operational interpretation of concurrence as a type of entanglement capacity, and we establish that concurrence of assistance, which we show is an entanglement monotone, identifies capabilities of and limitations to producing pure bipartite entangled states from pure tripartite entangled states. In addition, we show that, if concurrence of assistance for the pure tripartite state is at least as large as the concurrence of the desired pure bipartite state, then the former may be transformed to the latter via local operations and classical communication, and we calculate the maximum probability for this transformation when this condition is not met.PACS Nos.: 03.67.Mn, 03.67.Hk, 03.65.Ud
An interesting monogamy equation with the form of the Pythagorean theorem is found for $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n$-dimensional pure states, which reveals the relation among bipartite concurrence, concurrence of assistance, and genuine tripartite entanglement. At the same time, a genuine tripartite entanglement monotone as a generalization of 3-tangle is naturally obtained for $(2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n)$-dimensional pure states in terms of a distinct idea. For mixed states, the monogamy equation is reduced to a monogamy inequality. Both results for tripartite quantum states can be employed to multipartite quantum 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.
The Schmidt coefficients capture all entanglement properties of a pure bipartite state and therefore determine its usefulness for quantum information processing. While the quantification of the corresponding properties in mixed states is important both from a theoretical and a practical point of view, it is considerably more difficult, and methods beyond estimates for the concurrence are elusive. In particular this holds for a quantitative assessment of the most valuable resource, the forms of entanglement that can only exist in high-dimensional systems. We derive a framework for lower bounding the appropriate measure of entanglement, the so-called G-concurrence, through few local measurements. Moreover, we show that these bounds have relevant applications also for multipartite states.
Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow us to recognize novel families of multiparticle bound entangled states.