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arXiv:0804.0181 · Journal of Mathematical Physics

Monogamy equality in 2⊗2⊗d quantum systems

Dong Pyo Chi, Jeong Woon Choi, Kabgyun Jeong, Jeong San Kim, Taewan Kim, Soojoon Lee

Published Nov 1, 2008• 8 citations
arXivHTMLPDF

Abstract

There is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman–Kundu–Wootters inequality [Phys. Rev. A 61, 052306 (2000); Coffman–Kundu–WoottersPhys. Rev. Lett. 96, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, CA(BC)2=CAB2+(CACa)2, in the three-qubit system. In this paper, we consider the monogamy equality in 2⊗2⊗d quantum systems. We show that CA(BC)=CAB if and only if CACa=0 and also show that if CA(BC)=CACa, then CAB=0, while there exists a state in a 2⊗2⊗d system such that CAB=0 but CA(BC)>CACa.

DOI
10.1063/1.3020685

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There is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman–Kundu–Wootters inequality [Phys. Rev. A 61, 052306 (2000); Coffman–Kundu–WoottersPhys. Rev. Lett. 96, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, CA(BC)2=CAB2+(CACa)2, in the three-qubit system. In this paper, we consider the monogamy equality in 2⊗2⊗d quantum systems. We show that CA(BC)=CAB if and only if CACa=0 and also show that if CA(BC)=CACa, then CAB=0, while there exists a state in a 2⊗2⊗d system such that CAB=0 but CA(BC)>CACa.
As the quantum entangled states are as a source for many of branches different quantum information theory, such as dense encoding,encryption and teleportation, theoretical study of any relation between the different parts of a entangled quantum state of semi-transparent. One of these relations, inequality monogamy. Although the correlation classical can be freely in the multipartite systems will be shared, the famous that quantum entanglement in this share has restrictions. This restriction on the sharing of entanglement, the famous to monogamy in the multipartite quantum systems. In terms of math, monogamy in the entanglement the first was introduced by Coffman-Kundu-Wootters (CKW) in the three qubit systems. Inequality CKW that obtained for the measure concurrence, mutually unique nature of entanglement two qubits common in the three qubit systems shows. A simple example of the monogamy property can be expressed that the amount of entanglement between A and B to add the amount of entanglement between A and C cannot be greater than the amount of entanglement between A and pair BC. The aim of this thesis reviews the types monogamy inequality defined for the various entangled measures especially measure concurrence.
In this paper, we prove a stronger entanglement monogamy inequality in a 2⊗2⊗3 system's pure state |Ψ⟩ABC. Specifically, we show that the linear entropy of ρA, which is the entanglement between A and BC, is always larger than the sum of the square of concurrence between A and B and the square of concurrence of assistance between A and C. Our proof is based on direct generalizations of the qubit system's results. Our inequality is stronger than the known monogamy inequality of concurrence and shows that the entanglement of assistance always comes from the existing entanglement. However, our inequality also shows that unlike the three-qubit case, in higher dimensional systems the entanglement between A and BC cannot be completely transformed into bipartite entanglement with assistance. Through our proof, we also give some cases when the inequality reduces to an equality.
The monogamy inequality in terms of the concurrence, called the Coffman-Kundu-Wootters inequality [V. Coffman et al., Phys. Rev. A 61, 052306 (2000)], and its generalization [T. J. Osborne and F. Verstraete, Phys. Rev. Lett. 96, 220503 (2006)] hold on general nnn-qubit states including mixed ones. In this Brief Report, we consider the monogamy inequalities in terms of the fully entangled fraction and the teleportation fidelity. We show that the monogamy inequalities do not hold on general mixed states, while the inequalities hold on nnn-qubit pure states.
Abstract Monogamy of entanglement is an interesting property discovered recently in the context of multi-qubit entanglement. Here, we present a new kind of monogamous relations for pure quantum states of N qubits. By comparing the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters, we prove that there exists a strict triangle relation for the three-qubit system. Since the entropy defines the entanglement of a pure state of a pair of quantum systems; therefore, using linear entropy, we conjecture that there is a polygon relation of many-qubit systems. We also formalize the general triangle relation for the three-qubit system.