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

Abstract:We all know that a density matrix $\rho$ is pure if and only if $\mathrm{Tr} \,\rho^2 = 1$. But is this the only way to prove the purity of $\rho$ 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.

arXiv:2609.09557 [quant-ph]

(or arXiv:2609.09557v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.09557 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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