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

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 ${\rho }_{\epsilon }=(1-\epsilon )M_d+\epsilon {\rho }_1$. From a detailed numerical analysis, it is shown that: (1) there exist infinite values in the interval taken for the density matrix expansion coefficients, $-1\le c_{{\alpha }_1,...,{\alpha }_N}\le 1$, 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 $\epsilon$ depend on the coefficients of the expansion of ${\rho }_1$ in the Pauli basis. If ${\rho }_1$ is hermitian with trace equal to 1, but is allowed to have {\em negative} eigenvalues, it is shown that ${\rho }_{\epsilon }$ can be entangled, even for two qubits.

12 pages