A Dynamical Point of View of Quantum Information: Entropy and Pressure

Quantum Information is a new area of research which has been growing rapidly since last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more “dynamical point of view” of this theory. We want to consider the concepts of entropy and pressure for “stationary systems” acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the ThermodynamicFormalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator

ℒ

acting on density matrices ρ ∈ 

ℳ

N

over a finite

N

-dimensional complex Hilbert space

ℒ

(ρ) : =  ∑

i

 = 1

k

tr

(

W

i

ρ

W

i

 ∗ 

)

V

i

ρ

V

i

 ∗ 

, where

W

i

and

V

i

,

i

 = 1, 2, 

…k

are operators in this Hilbert space.

ℒ

is not a linear operator. 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 (acting on the configuration space of density matrices ρ) and the

W

i

play the role of the weights one can consider on the IFS. We suppose that for all ρ we13pc]First author considered as corresponding author. Please check. have that ∑

i

 = 1

k

{ tr}

(

W

i

ρ

W

i

 ∗ 

) = 1. A family

W

: = {

W

i

}

i

 = 1, 

…

, 

k

determines a Quantum Iterated Function System (QIFS)

ℱ

W

,

ℱ

W

 = {

ℳ

N

, 

F

i

, 

W

i

}

i

 = 1, 

…

, 

k

.