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New considerations on the separability of very noisy mixed states and implications for NMR quantum computing

J. D. Bulnes·R. S. Sarthour·E. R. de Azevedo·F. A. Bonk·J. C. C. Freitas·A. P. Guimarães·T. J. Bonagamba·I. S. Oliveira
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Abstract

We revise the problem first addressed by Braunstein and co-workers (Phys. Rev. Lett. {\bf 83} (5) (1999) 1054) concerning the separability of very noisy mixed states represented by general density matrices with the form $ρ_ε= (1-ε)M_d+ερ_1$. From a detailed numerical analysis, it is shown that: (1) there exist infinite values in the interval taken for the density matrix expansion coefficients, $-1\le c_{α_1,...,α_N}\le 1$, which give rise to {\em non-physical density matrices}, with trace equal to 1, but at least one {\em negative} eigenvalue; (2) there exist entangled matrices outside the predicted entanglement region, and (3) there exist separable matrices inside the same region. It is also shown that the lower and upper bounds of $ε$ depend on the coefficients of the expansion of $ρ_1$ in the Pauli basis. If $ρ_{1}$ is hermitian with trace equal to 1, but is allowed to have {\em negative} eigenvalues, it is shown that $ρ_ε$ can be entangled, even for two qubits.

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J. D. Bulnes, R. S. Sarthour, E. R. de Azevedo, F. A. Bonk, J. C. C. Freitas, A. P. Guimarães, T. J. Bonagamba, I. S. Oliveira. 2004-04-03. New considerations on the separability of very noisy mixed states and implications for NMR quantum computing. https://arxiv.org/abs/quant-ph/0404020

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