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arXiv:1806.08415 · New Journal of Physics

Entanglement polygon inequality in qubit systems

Xiao-Feng Qian, Miguel Angel Alonso, Joseph H. Eberly

Published May 10, 2018• 39 citations
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Abstract

We prove a set of tight entanglement inequalities for arbitrary N -qubit pure states. By focusing on all bi-partite marginal entanglements between each single qubit and its remaining partners, we show that the inequalities provide an upper bound for each marginal entanglement, while the known monogamy relation establishes the lower bound. The restrictions and sharing properties associated with the inequalities are further analyzed with a geometric polytope approach, and examples of three-qubit GHZ-class and W-class entangled states are presented to illustrate the results.

DOI
10.1088/1367-2630/aac3be

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We prove a set of tight entanglement inequalities for arbitrary N -qubit pure states. By focusing on all bi-partite marginal entanglements between each single qubit and its remaining partners, we show that the inequalities provide an upper bound for each marginal entanglement, while the known monogamy relation establishes the lower bound. The restrictions and sharing properties associated with the inequalities are further analyzed with a geometric polytope approach, and examples of three-qubit GHZ-class and W-class entangled states are presented to illustrate the results.
Entanglement is an important resource for quantum communication tasks. Most results are focused on qubit entanglement. Our goal in this work is to characterize the multipartite high-dimensional entanglement. We first derive an entanglement polygon inequality for the qqq concurrence, which manifests the relationship among all the ``one-to-group'' marginal entanglements in any multipartite qudit system. This implies lower and upper bounds for the marginal entanglement of any three-qudit system. We further extend our study to general entanglement distribution inequalities for high-dimensional entanglement in terms of the unified-(r,s)(r,s)(r,s) entropy entanglement, including Tsallis entropy, R\'enyi entropy, and von Neumann entropy entanglement as special cases. These results provide insights into characterizing bipartite high-dimensional entanglement in quantum information processing.
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.
By introducing an arbitrary-dimensional multipartite entanglement measure, which is defined in terms of the reduced density matrices corresponding to all possible two partitions of the entire system, we prove that multipartite entanglement cannot be freely shared among the parties in both n-qubit systems and three-qutrit systems. Furthermore, our result implies that the satisfaction of the entanglement monogamy is related to the number of particles in the quantum system. As an application of three-qutrit monogamy inequality, we give a condition for the separability of a class of two-qutrit mixed states in a 3 ⊗ 3 system.
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 We present optimized generalized monogamy relations and upper bounds derived from concurrence and concurrence of assistance. We first establish a tighter general upper bound of the αth ( 0 ⩽ α ⩽ 2 ) power of concurrence for N-qubit states. Then for N-qubit systems A B C 1 … C N − 2 , the optimized monogamy relations and upper bounds satisfied by the αth ( 0 ⩽ α ⩽ 2 ) power of concurrence of N-qubit pure states under the partition AB and C 1 … C N − 2 , as well as under the partition ABC 1 and C 2 … C N − 2 are established, which give rise to restrictions on the entanglement distribution and trade offs among the subsystems. Moreover, the utilization of the W-class states demonstrates that our results are tighter compared with the existing results. Similar results are also obtained for negativity.
The entanglement in multipartite quantum system is hard to characterize and quantify, although it has been intensively studied in bipartite systems. The monogamy of entanglement, as a special property of multipartite systems, shows the distribution of entanglement in the system. In this paper, we investigate the monogamy relations for multi-qubit systems. By using two entangled measures, namely the concurrence C and the negativity Nc, we establish tighter monogamy inequalities for their α-th power than those in all the existing ones. We also illustrate the tightness of our results for some classes of quantum states.