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Journal of Physics A Mathematical and Theoretical

A stronger entanglement monogamy inequality in a 2⊗2⊗3 system

Xi-Jun Ren, Yanni Shi

Published Aug 23, 2010• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8113/43/39/395301

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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.
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.
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.
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.
Abstract We investigate the polygamy relations related to the concurrence of assistance for any multipartite pure states. General polygamy inequalities given by the th power of concurrence of assistance is first presented for multipartite pure states in arbitrary-dimensional quantum systems. We further show that the general polygamy inequalities can even be improved to be tighter inequalities under certain conditions on the assisted entanglement of bipartite subsystems. Based on the improved polygamy relations, lower bound for distribution of bipartite entanglement is provided in a multipartite system. Moreover, the th ( ) power of polygamy inequalities are obtained for the entanglement of assistance as a by-product, which are shown to be tighter than the existing ones. A detailed example is presented.
We investigate the polygamy relations related to the concurrence of assistance for any multipartite pure states. General polygamy inequalities given by the α\alphaαth (0≤α≤2)(0\leq \alpha\leq 2)(0≤α≤2) power of concurrence of assistance is first presented for multipartite pure states in arbitrary-dimensional quantum systems. We further show that the general polygamy inequalities can even be improved to be tighter inequalities under certain conditions on the assisted entanglement of bipartite subsystems. Based on the improved polygamy relations, lower bound for distribution of bipartite entanglement is provided in a multipartite system. Moreover, the β\betaβth (0≤β≤10\leq \beta \leq 10≤β≤1) power of polygamy inequalities are obtained for the entanglement of assistance as a by-product, which are shown to be tighter than the existing ones. A detailed example is presented.