Abstract
In quantum physics, it is very important to give the theoretical lower and upper bounds for the geometric measure of entanglement of a multipartite pure state with nonnegative amplitudes. Existing literature shows that the theoretical bounds can be obtained by the bounds of \(Z_2\)-spectral radius of a nonnegative tensor. In this paper, a part of conclusions on Perron-Frobenius Theorem of the \(Z_2\)-eigenpairs for a nonnegative tensor are extended to its \(Z_p\)-eigenpairs, where p is any positive integer. Subsequently, an upper bound of any \(Z_p\)-eigenvalue of a tensor is derived. And then, a lower bound of the ratio of the largest and smallest components of a positive \(Z_p\)-eigenvector of an irreducible and nonnegative tensor is provided. Finally, two numerical examples are given to show the effectiveness of the obtained bounds in estimating the geometric measure of entanglement.
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Acknowledgements
The authors are very grateful to the editor and the anonymous reviewers for their insightful comments and constructive suggestions, which considerably improve our manuscript.
Funding
This work is supported by Guizhou Provincial Science and Technology Projects, China (Grant No. QKHJC-ZK[2022]YB215) and Natural Science Research Project of Department of Education of Guizhou Province, China (Grant Nos. QJJ[2023]012; QJJ[2023]061; QJJ[2023]062).
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Zhao, J., Shi, Q. Bounds for \(Z_p\)-eigenpairs of a tensor with application to geometric measure of entanglement. Comp. Appl. Math. 44, 173 (2025). https://doi.org/10.1007/s40314-025-03126-w
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DOI: https://doi.org/10.1007/s40314-025-03126-w
Keywords
- Geometric measure of entanglement
- Nonnegative tensors
- \(Z_p\)-eigenpairs
- Perron–Frobenius Theorem
- Spectral radius