A Thermodynamic Formalism for Density Matrices in Quantum Information

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 denitions and results we describe here are natural generalizations of the classical concepts of Thermo- dynamic Formalism (in the sense of R. Bowen, Y. Sinai and D. Ruelle). It is well-known 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 operatoracting on the space of density matr ices MN over anite N-dimensional complex Hilbert space �( �) := k X i=1 tr(WiW � i ) ViV � i tr(ViV � i ) ,