A Thermodynamic Formalism for density matrices in Quantum Information

Abstract

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 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 operator $\Lambda$ acting on the space of density matrices $\mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $$ \Lambda(\rho):=\sum_{i=1}^k tr(W_i\rho W_i^*)\frac{V_i\rho V_i^*}{tr(V_i\rho V_i^*)}, $$ where $W_i$ and $V_i$, $i=1,2,..., k$ are linear operators in this Hilbert space. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the $V_i\,(.)\,V_i^*=:F_i(.)$, $i=1,2,...,k$, play the role of the inverse branches (i.e., the dynamics on the configuration space of density matrices) and the $W_i$ play the role of the weights one can consider on the IFS. In this way a family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS). We also present some estimates related to the Holevo bound.