Semigroup of positive maps for qudit states and entanglement in tomographic probability representation |
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Authors: | V.I. Man'ko G. Marmo F. Ventriglia |
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Affiliation: | a P.N.Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia b Dipartimento Sc. Fisiche dell'Università Federico II e Sez. INFN di Napoli, Compl. Università di Monte S.Angelo, I-80126 Naples, Italy |
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Abstract: | Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups IGL(n,R) and GL(n,R) respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties. |
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Keywords: | 03.65.-w 03.65.Wj |
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