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Calculating Entanglement Eigenvalues for Nonsymmetric Quantum Pure States Based on the Jacobian Semidefinite Programming Relaxation Method

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Abstract

The geometric measure of entanglement is a widely used entanglement measure for quantum pure states. The key problem of computation of the geometric measure is to calculate the entanglement eigenvalue, which is equivalent to computing the largest unitary eigenvalue of a corresponding complex tensor. In this paper, we propose a Jacobian semidefinite programming relaxation method to calculate the largest unitary eigenvalue of a complex tensor. For this, we first introduce the Jacobian semidefinite programming relaxation method for a polynomial optimization with equality constraint and then convert the problem of computing the largest unitary eigenvalue to a real equality constrained polynomial optimization problem, which can be solved by the Jacobian semidefinite programming relaxation method. Numerical examples are presented to show the availability of this approach.

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Acknowledgements

The authors would like to thank the editors and anonymous referees for their valuable suggestions, which helped us to improve this manuscript. The first and the third authors’ work is partially supported by the Research Programme of National University of Defense Technology (No. ZK16-03-45), and the second author’s work is partially supported by the National Science Foundation of China (No. 11471242).

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Authors and Affiliations

  1. Department of Mathematics, National University of Defense Technology, Changsha, Hunan, People’s Republic of China

    Mengshi Zhang & Guyan Ni

  2. School of Mathematics, Tianjin University, Tianjin, 300072, People’s Republic of China

    Xinzhen Zhang

Authors
  1. Mengshi Zhang
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  2. Xinzhen Zhang
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  3. Guyan Ni
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Correspondence to Guyan Ni.

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Communicated by Liqun Qi.

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Zhang, M., Zhang, X. & Ni, G. Calculating Entanglement Eigenvalues for Nonsymmetric Quantum Pure States Based on the Jacobian Semidefinite Programming Relaxation Method. J Optim Theory Appl 180, 787–802 (2019). https://doi.org/10.1007/s10957-018-1357-7

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  • DOI: https://doi.org/10.1007/s10957-018-1357-7

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