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Symmetric Hermitian decomposability criterion, decomposition, and its applications

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Abstract

The Hermitian tensor is an extension of Hermitian matrices and plays an important role in quantum information research. It is known that every symmetric tensor has a symmetric CP-decomposition. However, symmetric Hermitian tensor is not the case. In this paper, we obtain a necessary and sufficient condition for symmetric Hermitian decomposability of symmetric Hermitian tensors. When a symmetric Hermitian decomposable tensor space is regarded as a linear space over the real number field, we also obtain its dimension formula and basis. Moreover, if the tensor is symmetric Hermitian decomposable, then the symmetric Hermitian decomposition can be obtained by using the symmetric Hermitian basis. In the application of quantum information, the symmetric Hermitian decomposability condition can be used to determine the symmetry separability of symmetric quantum mixed states.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions, which helped them to improve this manuscript. This work was supported by the National Natural Science Foundation of China (Grant No. 11871472).

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Authors and Affiliations

  1. Department of Mathematics, National University of Defense Technology, Changsha, 410073, China

    Guyan Ni & Bo Yang

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  1. Guyan Ni
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  2. Bo Yang
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Correspondence to Guyan Ni.

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Ni, G., Yang, B. Symmetric Hermitian decomposability criterion, decomposition, and its applications. Front. Math 17, 961–986 (2022). https://doi.org/10.1007/s11464-021-0927-4

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