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Physical Review A

Entanglement monogamy in a three-qubit state

Jie-Hui Huang, Shi-Yao Zhu

Published Jul 14, 2008• 10 citations
Publisher

Abstract

We investigate the monogamy nature of entanglement in a three-qubit system. A monogamy inequality is presented to describe the exclusive relation between the $A\text{\ensuremath{-}}B$ two-qubit concurrence ${C}_{AB}$ and the $AB\text{\ensuremath{-}}C$ three-qubit concurrence ${C}_{(AB)C}$, which represents the entanglement between qubits $\mathit{A}$ and $\mathit{B}$ as a whole and the third qubit $\mathit{C}$. It is found that the entanglement between any two qubits in a three-qubit system is limited by the entanglement between these two qubits and another qubit. As a consequence, we present the upper bounds for the concurrence ${C}_{AB}$, when the concurrence between qubits $\mathit{A}$ and $\mathit{C}$ $({C}_{AC})$ and the concurrence between qubits $\mathit{B}$ and $\mathit{C}$ $({C}_{BC})$ are both given or one of the two is provided.

DOI
10.1103/physreva.78.012325

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We investigate the monogamy relations related to the concurrence and the entanglement of formation. General monogamy inequalities given by the $\ensuremath{\alpha}\text{th}$ power of concurrence and entanglement of formation are presented for NNN-qubit states. The monogamy relation for entanglement of assistance is also established. Based on these general monogamy relations, the residual entanglement of concurrence and entanglement of formation are studied. Some relations among the residual entanglement, entanglement of assistance, and three tangle are also presented.
Quantum entanglement for multiparty system has a unique feature when it comes to sharing its property among various subsystems. This is famously stated as the monogamy of entanglement. The traditional monogamy of concurrence for tripartite system was proved in a sum form. Recently, it was found that concurrence also respects a monogamy in the product form. Here, we prove a tight monogamy relation in the product form for the concurrence of pure tripartite systems. We illustrate our relation with several examples, including the canonical three qubit states, where this monogamy relation is saturated.
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