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Wigner’s Theorem and geometry of extreme positive maps
, 2009
"... We consider transformation maps on the space of states which are symmetries in the sense of Wigner. By virtue of the convex nature of the space of states, the set of these maps has a convex structure. We investigate the possibility of a complete characterization of extreme maps of this convex body t ..."
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We consider transformation maps on the space of states which are symmetries in the sense of Wigner. By virtue of the convex nature of the space of states, the set of these maps has a convex structure. We investigate the possibility of a complete characterization of extreme maps of this convex body to be able to contribute to the classification of positive maps. Our study provides a variant of Wigner’s theorem originally proved for ray transformations in Hilbert spaces.
Classical tensors and quantum entanglement II: mixed states
, 2011
"... Abstract. Invariant operatorvalued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n) × U(n), may establish a method for the identification of en ..."
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Abstract. Invariant operatorvalued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n) × U(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n = 2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and antisymmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a nonlinear realization of the Lie algebra of SU(2)×SU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables. 1.