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arXiv:0904.2809 · Physical Review A

Bound of entanglement of assistance and monogamy constraints

Zong-Guo Li, Shao-Ming Fei, Sergio A Albeverio, Wan-Yu Liu

Published Sep 25, 2009• 41 citations
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Abstract

We investigate the entanglement of assistance which quantifies capabilities of producing pure bipartite entangled states from a pure tripartite state. The lower bound and upper bound of entanglement of assistance are obtained. In the light of the upper bound, monogamy constraints are proved for arbitrary $n$-qubit states.

DOI
10.1103/physreva.80.034301

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The newfound importance of ``entanglement as a resource'' in quantum computation and quantum communication compels us to quantify it in as many distinct ways as possible. Here we explore a new measure of entanglement for mixed quantum states of bipartite systems, which we name the Entanglement of Assistance. We show it to be the maximum average entanglement of all pure-state ensembles consistent with the given density matrix. In this sense, the Entanglement of Assistance is a quantity directly dual to the more standard Entanglement of Formation. With the help of lower and upper bounds, we calculate the Entanglement of Assistance for a few cases and use these results to show that it possesses the surprising property of superadditivity. We believe that this may shed some light on the question of additivity for the Entanglement of Formation.

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We investigate the entanglement of assistance which quantifies capabilities of producing pure bipartite entangled states from a pure tripartite state. The lower bound and upper bound of entanglement of assistance are obtained. In the light of the upper bound, monogamy constraints are proved for arbitrary nnn-qubit 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.
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.
We provide a detailed proof of an analytical lower bound of entanglement quantified by concurrence for arbitrary bipartite quantum states. It is shown that though the bound does not allow one to detect bound entanglement, it is tight for some mixed states and can detect most of the free entanglement. On the other hand, it is known that the entanglement monogamy inequality proposed by Coffman, Kundu, and Wootters is, in general, not true for higher-dimensional quantum states. Inducing from the lower bound of concurrence, we find a proper form of entanglement monogamy inequality for arbitrary pure quantum states.
Abstract Monogamy and polygamy relations quantify complementary restrictions on the distribution of entanglement in multipartite quantum systems. We establish two endpoint-matched scalar inequalities that retain a hierarchy-dependent mixed term discarded by standard weighted estimates. Conditional on a valid base-power monogamy or polygamy relation for the chosen entanglement quantity and state class, these inequalities yield refined tripartite bounds and recursive extensions to arbitrary numbers of parties. Analytical comparison with earlier coefficients proves that the resulting monogamy lower bounds are no smaller and the polygamy upper bounds are no larger throughout the common parameter domain. Concurrence and concurrence-of-assistance examples verify the ordering and identify the equality conditions.
We establish duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, as quantified by linear entropy, we prove that the same quantity (namely, linear entropy) provides a lower bound for distribution of bipartite entanglement in a multipartite system. Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest.
The newfound importance of ``entanglement as a resource'' in quantum computation and quantum communication compels us to quantify it in as many distinct ways as possible. Here we explore a new measure of entanglement for mixed quantum states of bipartite systems, which we name the Entanglement of Assistance. We show it to be the maximum average entanglement of all pure-state ensembles consistent with the given density matrix. In this sense, the Entanglement of Assistance is a quantity directly dual to the more standard Entanglement of Formation. With the help of lower and upper bounds, we calculate the Entanglement of Assistance for a few cases and use these results to show that it possesses the surprising property of superadditivity. We believe that this may shed some light on the question of additivity for the Entanglement of Formation.