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arXiv:quant-ph/0404020

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

J.D. Bulnes, Roberto S. Sarthour, Eduardo Ribeiro de Azevêdo, Fábio Aurélio Bonk, Jair C. C. Freitas, Alberto Passos Guimarães, Tito José Bonagamba, Ivan S. Oliveira

Published Apr 3, 2004• 1 citation
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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.

DOI
10.48550/arxiv.quant-ph/0404020

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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 ρε=(1−ε)Md+ερ1ρ_ε= (1-ε)M_d+ερ_1ρε​=(1−ε)Md​+ερ1​. 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≤cα1,...,αN≤1-1\le c_{α_1,...,α_N}\le 1−1≤cα1​,...,αN​​≤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 εεε depend on the coefficients of the expansion of ρ1ρ_1ρ1​ in the Pauli basis. If ρ1ρ_{1}ρ1​ is hermitian with trace equal to 1, but is allowed to have {\em negative} eigenvalues, it is shown that ρερ_ερε​ can be entangled, even for two qubits.
We give a constructive proof that all mixed states of NNN qubits in a sufficiently small neighborhood of the maximally mixed state are separable (unentangled). The construction provides an explicit representation of any such state as a mixture of product states. We give upper and lower bounds on the size of the neighborhood, which show that its extent decreases exponentially with the number of qubits. The bounds show that no entanglement appears in the physical states at any stage of present NMR experiments. Though this result raises questions about NMR quantum computation, further analysis would be necessary to assess the power of the general unitary transformations, which are indeed implemented in these experiments, in their action on separable states.
The general expression with the physical significance and positive definite condition of the eigenvalues of 4×44\times 44×4 Hermitian and trace-one matrix are obtained. This implies that the eigenvalue problem of the 4×44\times 44×4 density matrix is generally solved. The obvious expression of Peres' separability condition for an arbitrary state of two qubits is then given out and it is very easy to use. Furthermore, we discuss some applications to the calculation of the entanglement, the upper bound of the entanglement, and a model of the transfer of entanglement in a qubit chain through a noisy channel.
By considering the decomposition of a generic two qubit density matrix presented by Wootters [W. K. Wootters, Phys. Rev. Lett. {\bf 80} 2245 (1998)], the robustness of entanglement for any mixed state of two qubit systems is obtained algebraically. It is shown that the robustness of entanglement is proportional to concurrence and in Bell decomposable density matrices it is equal to the concurrence. We also give an analytic expression for two separable states which wipe out all entanglement of these states. Since thus obtained robustness is function of the norm of the vectors in the decomposition we give an explicit parameterization for the decomposition.
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