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Journal of Anqing Teachers College

Research on the Sufficient and Necessary Condition of Hermitian Operator in the Limitless Space

LI Hui-ling

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Studies have shown, in the limitless space, condition F=F~+ is the necessary but insufficient condition of the operator F∧,which is corresponding to the matrix F,being a Hermitian operator.Only when F=F~+ is satisfied and the surface term is zero, the operator F∧ is a Hermitian operator.

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We all know that a density matrix ρρρ is pure if and only if Tr ρ2=1\mathrm{Tr} \,ρ^2 = 1Trρ2=1. But is this the only way to prove the purity of ρρρ 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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