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Extremal Geometric Measure of Entanglement and Riemannian Optimization Methods

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Abstract

In this paper, we study the extremal geometric measure of quantum entanglement of superposition states of several symmetric pure states with parameters. This problem is equivalent to calculate their minimum entanglement value or maximum entanglement value. We propose the block coordinate descent (BCD) method to calculate the minimum entanglement value and establish its convergence analysis. We propose a gradient projection ascent with min-oracle(GPAM) method to calculate the maximum entanglement value. The subproblems of these two problems are optimization problems with a complex unit spherical constraint. The complex unit sphere is a Riemannian manifold. We propose an inexact Riemannian Newton-CG method to solve this Riemannian optimization problem. The numerical examples are presented to demonstrate the effectiveness of the proposed methods.

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Authors and Affiliations

  1. School of Mathematics, Hunan University, Changsha, 410082, Hunan, China

    Min-Ru Bai, Shan-Shan Yan & Qi Zeng

Authors
  1. Min-Ru Bai
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  2. Shan-Shan Yan
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  3. Qi Zeng
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Contributions

M.-R. Bai is in charge of conceptualization, methodology, supervision and the correctness of the paper. Q. Zeng is in charge of model and algorithm design and analysis of extremal geometric measure of entanglement and paper writing. S.-S. Yan is in charge of the algorithm and theory of Riemannian optimization subproblem.

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Correspondence to Min-Ru Bai.

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The authors declare no conflict of interest.

Additional information

This work was supported by the National Natural Science Foundation of China (Nos. 11971159 and 12071399).

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Bai, MR., Yan, SS. & Zeng, Q. Extremal Geometric Measure of Entanglement and Riemannian Optimization Methods. J. Oper. Res. Soc. China (2023). https://doi.org/10.1007/s40305-023-00504-1

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  • DOI: https://doi.org/10.1007/s40305-023-00504-1

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