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Distribution of spin correlation strengths in multipartite systems

Yu, B., Naihuan Jing, Li-Jost, X.

Published Jan 1, 2019• 2 citations
OpenAlex

Abstract

For a two-qubit state, the isotropic strength measures the degree of isotropic spin correlation. The concept of isotropic strength is generalized to multipartite qudit systems, and the strength distributions for tripartite and quadripartite qudit systems are thoroughly investigated. We show that the sum of relative isotropic strengths of any three-qudit state over d-dimensional Hilbert space cannot exceed $$d-1$$ , which generalizes the case $$d=2$$ . The trade-off relations and monogamy-like relations of the sum of spin correlation strengths for pure three- and four-partite systems are derived. Moreover, the bounds of spin correlation strengths among different subsystems of a quadripartite state are used to analyze quantum entanglement.

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For a two-qubit state, the isotropic strength measures the degree of isotropic spin correlation. The concept of isotropic strength is generalized to multipartite qudit systems, and the strength distributions for tripartite and quadripartite qudit systems are thoroughly investigated. We show that the sum of relative isotropic strengths of any three-qudit state over d-dimensional Hilbert space cannot exceed d−1d-1d−1 , which generalizes the case d=2d=2d=2 . The trade-off relations and monogamy-like relations of the sum of spin correlation strengths for pure three- and four-partite systems are derived. Moreover, the bounds of spin correlation strengths among different subsystems of a quadripartite state are used to analyze quantum entanglement.
We explore the quantum correlation distribution in multipartite quantum states based on the square of quantum discord (SQD). For tripartite quantum systems, we derive the necessary and sufficient condition for the SQD to satisfy the monogamy relation. Particularly, we prove that the SQD is monogamous for three-qubit pure states, based on which a genuine tripartite quantum correlation measure is introduced. In addition, we also address the quantum correlation distributions in four-qubit pure states. As an example, we investigate multipartite quantum correlations in the dynamical evolution of multipartite cavity-reservoir systems.
Based on the quantitative complementarity relations, we analyze thoroughly the properties of multipartite quantum correlation and entanglement in four-qubit pure states. It is found that, unlike the three-qubit case, the single residual correlation and the genuine correlations of three and four qubits are unable to quantify entanglement appropriately. More interestingly, from our qualitative and numerical analysis, it is conjectured that the sum of all residual correlations is a good quantity for characterizing the multipartite entanglement in the system.
The key ingredient of the approach presented in this paper is the factorization property of SU (2) coherent states upon the splitting or decay of a quantum spin system. In this picture, the even and odd spin coherent states are viewed as comprising two, three or more spin sub-systems. From this perspective, we investigate the multipartite quantum correlations defined as the sum of the correlations of all possible bi-partitions. The pairwise quantum correlations are quantified by entanglement of formation and quantum discord. Special attention is devoted to tripartite splitting schemes. We explicitly derive the sum of the entanglement of formation for all possible bi-partitions. It coincides with the sum of all possible occurrences of pairwise quantum discord. The conservation relation between the distribution of entanglement of formation and quantum discord in the tripartite splitting scheme is discussed. We show that entanglement of formation and quantum discord possess the monogamy property for even spin coherent states, in contrast to odd ones which violate the monogamy relation when the overlap of the coherent states approaches unity.
It is a well-known fact that both quantum entanglement and nonlocality (implied by the violation of Bell inequalities) constitute quantum correlations that cannot be arbitrarily shared among subsystems. They are both monogamous, albeit in a different fashion. In the present contribution we focus on nonlocality monogamy relations such as the Toner-Verstraete, the Seevinck, and a derived monogamy inequality for three parties and compare them with multipartite nonlocality measures for the whole set of pure states distributed according to the Haar measure. In this numerical endeavor, we also see that, although monogamy relations for nonlocality cannot exist for more than three parties, in practice the exploration of the whole set of states for different numbers of qubits will return effective bounds on the maximum value of all bipartite Bell violations among subsystems. Hence, we shed light on the effective nonlocality monogamy bounds in the multiqubit case.
Abstract The distribution of entanglement in a multiparty system can be described by the principle of monogamy. Monogamy is a fundamental characteristic of entanglement that restricts its distribution among more than two parties. In this work, our aim is to explore how quantum entanglement is distributed in accordance with monogamy relations, which utilize both genuine multipartite entanglement measures and bipartite entanglement measures. Specifically, we treat the source entanglement or the accessible entanglement as genuine multipartite entanglement measures and use the entanglement of formation for bipartite cases. For GHZ class states, we analytically demonstrate that the square of the source entanglement serves as an upper bound for the sum of the squares of the entanglement of formation of the reduced subsystems, with some exceptions in specific non-generic GHZ states. We will also present numerical evidence supporting this result for W-class states. In addition, we also explore the monogamy relationship by using accessible entanglement as an upper bound. Our analysis reveals an opposite but consistent pattern in the behavior of monogamy depending on whether source entanglement or accessible entanglement is used as the upper bound. This pattern emphasizes the dual nature of these two measures.