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PreprintarXiv licencearXiv

From Certifying Rank k Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers

  • H. F. Chau
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Abstract

We all know that a density matrix ρ is pure if and only if Tr ρ2 = 1. But is this the only way to prove the purity of ρ using the trace of its powers? Here I systematically study the necessary and sufficient conditions of the more general question of guaranteeing that all eigenvalues of a Hermitian matrix belong to a specified set through its trace of powers. More importantly, these characterization results are automatically witnesses certifying a quantum state as entangled, a Hermitian operator has least one negative eigenvalue and a linear operator as non-Hermitian. I demonstrate the effectiveness of these witnesses, analyze their performance, and study their strength, weakness together with resource requirement through numerical simulation as well as analytical work. I also discuss briefly the effects of numerical stability, uncertainty in measurement and rounding errors on these problems.

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Cite

BibTeXRIS
BibTeX
@misc{chau2026from,
  title = {{From Certifying Rank $k$ Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers}},
  author = {H. F. Chau},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09557},
  eprint = {2609.09557},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09557},
}
RIS
TY  - UNPB
TI  - From Certifying Rank k Projectors To Non-Positivity, Entanglement, And Non-Hermitian Witnesses Through Traces Of Matrix Powers
AU  - H. F. Chau
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.09557
UR  - https://arxiv.org/abs/2609.09557
AB  - We all know that a density matrix ρ is pure if and only if Tr ρ^2 = 1. But is this the only way to prove the purity of ρ using the trace of its powers? Here I systematically study the necessary and sufficient conditions of the more general question of guaranteeing that all eigenvalues of a Hermitian matrix belong to a specified set through its trace of powers. More importantly, these characterization results are automatically witnesses certifying a quantum state as entangled, a Hermitian operator has least one negative eigenvalue and a linear operator as non-Hermitian. I demonstrate the effectiveness of these witnesses, analyze their performance, and study their strength, weakness together with resource requirement through numerical simulation as well as analytical work. I also discuss briefly the effects of numerical stability, uncertainty in measurement and rounding errors on these problems.
ER  -