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arXiv:0803.2954 · Physical Review A

Entanglement monogamy of tripartite quantum states

Chang‐shui Yu, He‐Shan Song

Published Mar 19, 2008• 70 citations
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Abstract

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.

DOI
10.1103/physreva.77.032329

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An interesting monogamy equation with the form of the Pythagorean theorem is found for 2\ensuremath⨂2\ensuremath⨂n2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n2\ensuremath⨂2\ensuremath⨂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⨂2\ensuremath⨂n)(2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}n)(2\ensuremath⨂2\ensuremath⨂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.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
Monogamy inequalities of entanglement of assistance in 2⊗2⊗d2\otimes 2\otimes d2⊗2⊗d systems
The monogamy relations characterize the distribution of quantum correlations among the multipartite quantum systems. We study the monogamy relations of the entanglement of assistance in 2⊗2⊗d2\otimes 2\otimes d2⊗2⊗d systems. We present explicitly the relations satisfied by the concurrence, the tangle and the concurrence of assistance, which can be used to derive rigorous monogamy relations. Detailed examples are given to illustrate our results.
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 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 $(d\ifmmode\times\else\texttimes\fi{}d)$-dimensional mixed states, we are able to generalize Wootters' formula for the concurrence into lower and upper bounds on the 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 $(d\ifmmode\times\else\texttimes\fi{}d\ifmmode\times\else\texttimes\fi{}n)$-dimensional tripartite pure states.
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they are monogamous on pure states, and show that for any quantum state that satisfies the disentangling condition, its entanglement of formation equals the entanglement of assistance. We characterize all bipartite mixed states with this property, and use it to show that the G-concurrence is monogamous. In the case of two qubits, we show that the equality between entanglement of formation and assistance holds if and only if the state is a rank 2 bipartite state that can be expressed as the marginal of a pure 3-qubit state in the W class.
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.