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A Thermodynamic Formalism for density matrices in Quantum Information
 Applied Mathematics Research Express
"... We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Part of our work is to justify why the definitions and results we describe here are natural generalizations of the classical concepts of Thermodynamic ..."
Abstract

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We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Part of our work is to justify why the definitions and results we describe here are natural generalizations of the classical concepts of Thermodynamic Formalism (in the sense of R. Bowen, Y. Sinai and D. Ruelle). It is wellknown that the concept of density operator should replace the concept of measure for the cases in which we consider a quantum formalism. We consider the operator acting on the space of density matrices MN over a finite Ndimensional complex Hilbert space (ρ):= k∑ i=1 tr(WiρW∗i) ViρV∗i tr(ViρV∗i) where, Wi and Vi, i = 1, 2,..., k are linear operators in this Hilbert space. In some sense, this operator is a version of an Iterated Function System (IFS). Namely, the Vi (.) V∗i =: Fi(.), i = 1, 2,..., k play the role of the inverse branches (i.e., the dynamics on the configuration space of density matrices) and the Wi play the role of the