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arXiv:0802.0270 · The European Physical Journal D

Two-qutrit entanglement witnesses and Gell-Mann matrices

M. A. Jafarizadeh, Y. Akbari, Naghi Behzadi

Published Mar 13, 2008• 11 citations
arXivHTMLPDF
DOI
10.1140/epjd/e2008-00041-3

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Wootters concurrence is an entanglement measure for bipartite qubit systems.It is defined in terms of a spin inverter.Here, we try to generalize Wootters concurrence for qutrit systems by defining a naïve inverter analogous to the (2) qubit inverter given in terms of Pauli matrix .For qutrits, the corresponding group is (3), so the inverter has to be given in terms of Gell-Mann matrices.The naïve inverter proposed here is given in terms of the Gell-Mann matrices analogous to and we show that it can deliver equivalent information to that given by the formal inverter in just in certain cases.
We present three different matrix bases that can be used to decompose density matrices of d -dimensional quantum systems, so-called qudits: the generalized Gell–Mann matrix basis , the polarization operator basis and the Weyl operator basis . Such a decomposition can be identified with a vector—the Bloch vector, i.e. a generalization of the well-known qubit case—and is a convenient expression for comparison with measurable quantities and for explicit calculations avoiding the handling of large matrices. We present a new method to decompose density matrices via so-called standard matrices, consider the important case of an isotropic two-qudit state and decompose it according to each basis. In the case of qutrits we show a representation of an entanglement witness in terms of expectation values of spin-1 measurements, which is appropriate for an experimental realization.
Abstract Entanglement witnesses (EWs) based on a restricted set of local measurements are experimentally more accessible. For two-qubit systems, we provide a complete characterization of extremal decomposable EWs constructed from local measurements with an anti-diagonal correlation structure. These witnesses can be employed for the detection of gravitational entanglement. We then extend this characterization to higher-dimensional quantum systems and explicitly construct the corresponding extremal decomposable witnesses using generalized Gell-Mann matrices.