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Research in Computing Science

Entanglement in Qutrit Systems

José D. Huerta-Morales

Published Dec 31, 2017• 2 citations
Publisher

Abstract

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.

DOI
10.13053/rcs-131-1-1

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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.
Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a ``universal inverter,'' which acts on quantum systems of arbitrary dimension, and we introduce the corresponding generalized concurrence for joint pure states of ${D}_{1}\ifmmode\times\else\texttimes\fi{}{D}_{2}$ bipartite quantum systems. We call this generalized concurrence the I concurrence to emphasize its relation to the universal inverter. The universal inverter, which is a positive, but not completely positive superoperator, is closely related to the completely positive universal-NOT superoperator, the quantum analogue of a classical NOT gate. We present a physical realization of the universal-NOT superoperator.
Quantum entanglement is of great current interest because of its key role in the modern theory of quantum information. Entanglement in two level bipartite systems is well understood. There exist several useful measures for these systems which include concurrence and reduced entropy. Recently entanglement has been studied for bipartite three‐level atomic systems. These authors have developed interesting measures using extensions of Wootters concurrence and the generators of group SU(3) for the spin operators. In this work we generalize concurrence using the extension of Pauli spin operators to three‐level atomic systems. We explore the above concurrence for three level atomic systems and their extension to n‐level atomic systems. We compare our results with other measures of entanglement.
We derive a classification and a measure of classical- and quantum-correlation of multipartite qubit, qutrit, and in general, nnn-level systems, in terms of SU(n)(n)(n) representations of density matrices. We compare the measure for the case of bipartite correlation with concurrence and the entropy of entanglement. The characterization of correlation is in terms of the number of nonzero singular values of the correlation matrix, but that of mixed state entanglement requires additional invariant parameters in the density matrix. For the bipartite qubit case, the condition for mixed state entanglement is written explicitly in terms of the invariant paramters in the density matrix. For identical particle systems we analyze the effects of exchange symmetry on classical and quantum correlation.
Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow us to recognize novel families of multiparticle bound entangled states.