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 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.


Introduction
In this work we investigate a generalization of the classical Thermodynamic Formalism (in the sense of Bowen, Sinai and Ruelle) for the setting of density matrices. We consider the operator Λ acting on the space of density matrices M N over a finite N -dimensional complex Hilbert space where W i and V i , i = 1, 2, . . . , k are linear operators in this Hilbert space. Note that Λ is not a linear operator. This operator can be seen as 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 ρ) and the W i play the role of the weights one can consider on the IFS. We suppose that for all ρ we have that k i=1 tr(W i ρW * i ) = 1. Note that such trace preserving condition, for any normalized operator ρ (that is, with tr(ρ) = 1), is Supported in part by CAPES and CNPq. equivalent to the explicit condition i W * i W i = I. We say that Λ is a normalized operator.
A family W := {W i } i=1,...,k determines a Quantum Iterated Function System (QIFS) F W , Basic references on QIFS are [13] and [16]. We want to consider a new concept of entropy for stationary systems acting on density matrices which generalizes the usual one in Ergodic Theory. In our setting the V i , i = 1, 2, . . . , k are fixed (i.e. the dynamics of the inverse branches is fixed in the beginning) and we consider the different families W i , i = 1, 2, . . . , k, (also with the attached corresponding eigendensity matrix ρ W ) as possible Jacobians of stationary probabilities. Given a normalized family W i , i = 1, 2, . . . , k, a natural definition of entropy is given by (2) where ρ W denotes the barycenter of the unique invariant, attractive measure for the Markov operator V associated to F W . We show that this generalizes the entropy of a Markov System. We also want to present a concept of pressure for stationary systems acting on density matrices which generalizes the usual one in Ergodic Theory. In addition to the dynamics obtained by the V i , which are fixed, a family of potentials H i , i = 1, 2, . . . k induces a kind of Ruelle operator given by (3) L H (ρ) : We show that such operator admits an eigenvalue β and an associated eigenstate ρ β , that is, one satisfying L H (ρ β ) = β ρ β .
The natural generalization of the concept of pressure for a family H i , i = 1, 2, . . . , k is the problem of maximizing, on the possible normalized families W i , i = 1, 2, . . . k, the expression (4) h We show a relation between the eigendensity matrix ρ β for the Ruelle operator and the set of W i , i = 1, 2, . . . k, which maximizes pressure. In the particular case that each of the V i is unitary, i = 1, 2, . . . k, the maximum value is log β. Our work is inspired by the results presented in [16] and [21]. We would like to thank these authors for supplying us with the corresponding references.
It is well-known that completely positive mappings (operators) acting on density matrices are of great importance in Quantum Computing. These operators can be written in the Stinespring-Kraus form (see section 12). Also a nice exposition on the interplay of Ergodic Theory and Quantum Information is presented in [4].
The initial part of our work aims to present some of the definitions and concepts that are not very well-known (at least for the general audience of people in Dynamical Systems), in a systematic way. We present the main basic definitions which are necessary to understand the theory. However, we do not have the intention of exhausting what is already known. We believe that the theoretical results presented here can be useful as a general tool to understand problems in Quantum Computing.
Several examples are presented in the text. We believe that this will help the reader to understand some of the main issues of the theory. In order to simplify the notation we will present most of our results for the case of matrices of order 2.
In sections 2 and 3 we present some basic definitions, examples and we show some preliminary relations of our setting to the classical Thermodynamic Formalism. In section 4 we present an eigenvalue problem for non-normalized Ruelle operators which will be required later. Some properties and concepts about density matrices and Ruelle operators are presented in sections 6 and 7. Sections 8 and 9 are dedicated to the introduction of some different kinds of entropy that were already known but do not have a stationary character. In section 10 we introduce the concept of stationary entropy for measures defined on the set of density matrices. In section 11 we compare this definition with the usual one for Markov Chains. Section 12 is dedicated to motivate the interest on pressure and the capacity-cost function. Section 13, 14, 15 and 16 are dedicated to the presentation of our main results on pressure, important inequalities, examples and its relation with the classical theory of Thermodynamic Formalism.
In [1] we present a general exposition (describing the setting we consider here) where we omit proofs, but provide many examples. We believe that paper will help to complement the present paper for the reader which is a newcomer in the area. We also present there some basic results concerning the discrete Wigner measure.
In [2] we propose a different concept of entropy which is also a generalization of the classical one. We also describe some properties of the Quantum Stochastic Process associated to the Quantum Iterated Function System. This work is part of the thesis dissertation of C. F. Lardizabal in Prog. Pos-Grad. Mat. UFRGS (Brazil).

Basic definitions
Let M N (C) the set of complex matrices of order n. If ρ ∈ M N (C) then ρ * denotes the transpose conjugate of ρ. A state (or vector) in C n will be denoted by ψ or |ψ , and the associated projection will be written |ψ ψ|. Define the space of hermitian, positive, density operators and pure states, respectively. Density operators are also called mixed states. If a quantum system can be in one of the states {ψ 1 , . . . , ψ k } then a mixed state ρ will be written as where the p i are positive numbers with i p i = 1.
. . , k and such that Suppose that the QIFS considered is such that there are V i and W i linear maps, where M 1 (M N ) denotes the space of probability measures over M N . We also define Λ : The operator defined above has no counterpart in the classical Thermodynamic Formalism. We will also consider the operator defined on the space of density matrices ρ, If for all ρ we have k i=k q i (ρ) = 1, we say the operator is normalized. We are also interested in the non-normalized case. If the QIFS is homogeneous, then We recall the definition of the integral above in section 5.
In order to define hyperbolic QIFS, one has to define a distance on the space of mixed states. For instance, we could choose one of the following: the Hilbert-Schmidt, trace, and Bures distances, respectively. Such metrics generate the same topology on M N . Considering the space of mixed states with one of those metrics we can use a definition of hyperbolicity similar to the one used for IFS. That is, we say a QIFS is hyperbolic if the quantum maps F i are contractions with respect to one of the distances on M N and if the maps p i are Hölder-continuous and positive, see for instance [16]. (6) is homogeneous and hyperbolic then the associated Markov operator admits a unique invariant measure µ. Such invariant measure determines a unique Λ-invariant state ρ ∈ M N , given by (13).

Examples of QIFS
The normalized identity matrix ρ * = I/N is Λ-invariant, for any choice of unitary U 1 and U 2 . Note that we can write (17) where the measure µ, uniformly distributed over P N (the Fubini-Study metric), is V-invariant.

♦
We recall that a mapping Λ is completely positive (CP) if Λ ⊗ I is positive for any extension of the Hilbert space considered H N → H N ⊗ H E . We know that every CP mapping which is trace-preserving can be represented (in a nonunique way) in the Stinespring-Kraus form where the V i are linear operators. Moreover if we have k j=1 V j V * j = I, then Λ(I/N ) = I/N . This is the case if each of the V i are normal.
We call a unitary trace-preserving CP map a bistochastic map. An example of such a mapping is where the U i are unitary operators and i p i = 1. Note that if we write F i (ρ) = U i ρU * i , then example 1 is part of this class of operators. For such operators we have that ρ * is an invariant state for Λ U and also that δ ρ * is invariant for the Markov operator P U induced by this QIFS.
We will present a simple example of the kind of problems we are interested here, namely eigenvalues and eigendensity matrices. Let H N be a Hilbert space of dimension N . As before, let M N be the space of density operators on H N . A natural problem is to find fixed points for Λ : In order to simplify our notation we fix N = 2 and k = 2. Let where V 1 and V 2 are invertible and ρ is a density operator. We would like to find ρ such that where k, l ∈ R, p ∈ (0, 1). Then V * 1 V 1 + V * 2 V 2 = I. A simple calculation shows that ρ 2 = 0, and then Now we make a few considerations about the Ruelle operator L defined before. In particular, we show that Perron's classic eigenvalue problem is a particular case of the problem for the operator L acting on matrices. Let We have that L(ρ) = ρ implies ρ 2 = 0 and Solving (23) and (24) in terms of ρ 1 gives that is, which is a restriction over the q i . For simplicity we assume here that the q i are constant. One can show that be a column-stochastic matrix. Let π = (π 1 , π 2 ) such that P π = π. Then (27) π = ( p 01 p 01 − p 00 + 1 , 1 − p 00 p 01 − p 00 + 1 ) Comparing (27) and (26) suggests that we should fix Then the nonzero entries of ρ are equal to the entries of π and therefore we associate the fixed point of P to the fixed point of some L in a natural way. But note that such a choice of q i is not unique, because (29) q 2 = 1 − q 1 p 2 00 p 01 p 10 , q 4 = 1 − q 3 p 10 p 01 p 2

11
, for any q 1 , q 3 also produces ρ with nonzero coordinates equal to the coordinates of π. We also note that the above calculations can be made by taking the V i matrices with nonzero entries equal to √ p ij instead of p ij . Now we consider the following problem. Let where q i ∈ R. Assume that h ij ∈ R, so we want to obtain λ such that L(ρ) = λρ, λ = 0, and λ is the largest eigenvalue. With a few calculations we obtain ρ 2 = ρ 3 = 0, Solving for λ, we obtain the eigenvalues and the associated eigenfunctions Therefore we obtained that ρ 1 , ρ 4 , q 1 , . . . , q 4 , λ are implicit solutions for the set of equations (31)-(32). Recall that in this case we obtained ρ 2 = ρ 3 = 0.
Now we consider the problem of finding the eigenvector associated to the dominant eigenvalue of H. The eigenvalues are Then we can find v such that Hv = λv from the set of equations we have that the set of equations (31)-(32) and (35)-(36) are the same. Hence we conclude that Perron's classic eigenvalue problem is a particular case of the problem for L acting on matrices.
♦ A different analysis in the quantum setting which is related to Perron's theorem is presented in [6].

A theorem on eigenvalues for the Ruelle operator
The following proposition is inspired in [18]. We say that a hermitian operator We point out that this operator is completely general. In an analogy with the classical case we can say it corresponds to the general Perron Theorem for positive matrices (having positive eigenvalues which can be bigger or smaller than one), by the other hand the setting described in [16], [21] "basically" considers the analogous case of the Perron Theorem for stochastic matrices.
We need a result in this form in order to better understand the Pressure problem which will be described later.
, n ≥ 1 The operator above is well defined. In fact, note that We know that for any positive operator P = 0, if {v 1 , . . . , v N } is a orthonormal base for H N , then Therefore, tr(L W,V (ρ + I n )) > 0, n ≥ 1. Hence L n (ρ) is well defined.
We know that M N is compact and convex, so we can apply Schauder's theorem for each of the mappings L n , n ≥ 1 and get ρ n ∈ M N such that By the compacity of M N , we can choose a point ρ ∈ M N which is limit of the sequence {ρ n } ∞ n=1 and then, by continuity, is positive, and the inequality will be equal to zero if and only if L W,V (ρ) is the zero operator. Hence, we proved that there exists ρ ∈ M N and β > 0 such that L W,V (ρ) = βρ.

Vector integrals and barycenters
We recall here a few basic definitions. For more details, see [16] and [21]. Let X be a metric space. Let (V, +, ·) be a real vector space, and τ a topology on V . We say that (V, +, ·; τ ) is a topological vector space if it is Hausdorff and if the operations + and · are continuous. For instance, in the context of density matrices, we will consider V as the space of hermitian operators H N and X will be the space of density matrices M N .
It is known that if we have a compact metric space X, V is a locally convex space and f : X → V is a continuous function such that cof (X) is compact then the integral of f in X exists and belongs to cof (X). We will also use the following well-known result, the barycentric formula: convex and bounded set, and µ ∈ M 1 (E). Then there is a unique x ∈ E such that In the context of QIFS, we can take V = E = M N .

Example: density matrices
In this section we briefly review how the constructions of the previous section adjust to the case of density matrices. Define V := H N , V + := PH N (note that such space is a convex cone), and let the partial order ≤ on is a regular state space [21]. Also, the set B of unity trace in V + is, of course, the space of density matrices, so B = M N .
Let Z ⊂ V * be a nonempty vector subspace of V * . The smallest topology in V such that every functional defined in Z is continuous on that topology, denoted by σ(V, Z), turns V into a locally convex space. In particular, σ(V, V * ) is the weak topology in V . If (V, · ) is a normed space, then σ(V * , V ) is called a weak * topology in V * (we identify V with a subspace of V * * ). We also have that (C, τ ) = (PH N , τ ), where τ is the weak * topology (and which is equal to the Euclidean, see [21]) is a metrizable compact structure. In this case we have that Definition 4. A Markov operator for probability measures is an operator P : An example of such operator is the one which we have defined before and we denote it by V : where f, µ denotes the integral of f with respect to µ.
Every submarkovian operator Q : V + → V + can be extended in a unique way to a positive linear contraction on V , see [21].
From [21], we know that there is a 1-1 correspondence between homogeneous IFS and Markov pairs.

Some lemmas for IFS
We want to understand the structure of Λ : Such operator is associated in a natural way to an IFS which is not homogeneous. In this section we state a few useful properties which are relevant for our study. The following lemmas hold for any IFS, except for lemma 3, where the proof presented here is valid only for homogeneous IFS.
(1) Let ρ ν be the barycenter of a probability measure ν. Then Λ(ρ ν ) is the barycenter of Vν, where V is the associated Markov operator. (2) Let µ be an invariant probability measure for V. Then the barycenter of µ, denoted by ρ µ , is a fixed point of Λ.
Proof 1. We have, for Ψ linear functional, where the fact that U • Ψ is linear follows from the homogeneity of F.
In order to prove uniqueness in item (2) above it would be necessary to assume hyperbolicity [20]. It is known that without this hypothesis even in the classical case (for transformations for instance) it can happen the phenomena of phase transition (two or more probabilities which are solutions) [23] [15]. The present setting contains the classical case and therefore in general there is no uniqueness.
We can apply lemma 2 and conclude that δ ρ 0 is an invariant measure for the Markov operator V associated to the IFS determined by p i and F i .

♦
The following lemma, a simple variation from results seen in [21], specifies a condition we need in order to obtain a fixed point for Λ from a certain measure which is invariant for the Markov operator V.
In lemma 4, we have a general QIFS and an attractive invariant µ, then µ is the unique invariant measure, an easy consequence of attractivity [21]. In general, we will be interested in QIFS which has an attractive invariant measure. This will follow if we assume hyperbolicity.

Integral formulae for the entropy of IFS
Part of the results we present here in this section are variations of results presented in [21]. Let (X, d) be a complete separable metric space. Then Proposition 7. Let F be an IFS and let g : B → R. Then for n ∈ N, Also suppose that F is homogeneous. Then (3) If g is concave (resp. convex, affine), then Ug is concave (resp. convex, affine).
We recall some well-known definitions and results. Define η : R + → R as Then the Shannon-Boltzmann entropy function is h : Let n ∈ N. Define the partial entropy H n : X → R + as the upper and lower entropy on x. If such limits are equal, we call its common value the entropy on x, denoted by H(x).
The partial entropy of the measure µ is defined by for n ≥ 1 and H 0 (µ) := 0.
We denote the common limit of the sequences mentioned in the proposition above as H(µ) and we call it the entropy of the measure µ, i.e., The following result gives us an integral formula for entropy, and also a relation between the entropies defined before. We write S(µ) where Lim(V n µ) n∈N is the convex hull of the set of accumulation points of (V n µ) n∈N , and S F (µ) is the set S(µ) associated to the Markov operator induced by the IFS F. For the definition of compact structure and (C, τ )-continuity, see [21].
The analogous result for hyperbolic IFS is the following. and

Some calculations on entropy
Let U be a unitary matrix of order mn acting on H m ⊗ H n . Its Schmidt decomposition is The operators V A i and V B i act on certain Hilbert spaces H m and H n , respectively. We also have that Above, recall that the partial trace is where |a 1  Let F be the homogeneous IFS associated to the V A i , that is, and let ρ 0 be a fixed point of Λ = i p i F i . Following [21], we have that ρ 0 is the barycenter of V n δ ρ 0 , n ∈ N. By theorem 2, we can calculate the entropy of such IFS. In this case we have and since h(ρ) = k j=1 η(p j (ρ)), we have, for ι = (i 1 , . . . , i n ), and every ρ 0 ∈ M N , for every n.

An expression for a stationary entropy
In this section we present a definition of entropy which captures a stationary behavior. Let H be a hermitian operator and V i , i = 1, . . . , k linear operators. We can define the dynamics F i : M N → M N : Remember that by lemma 4, we have that ρ W is a fixed point for Proof Note that, by definition, and the function h (Shannon-Boltzmann entropy) is ≥ 0. This proves the lemma. Another elementary proof is the following. Since ρ W is positive, we have that Analogously the expression above holds for the V i ρ W V * i , and therefore also for Remark For any fixed dynamics V , if we have that W * m W m = I for some m then the remaining p i must be zero, because of the condition i W * i W i = I. In this case we have h V (W ) = 0. We also have that h V (W ) ≤ log k and for any given dynamics V , h V (W ) attains the maximum if we choose W i = 1/ √ kI, for each i, where I denotes the identity operator.

♦
Note that by the calculations made in section 9, we have h (60) and (61).
Proof The fact that Λ(ρ 0 ) = ρ 0 follows from lemma 2, item 2. Also, Then if ρ µ is the barycenter of µ we have, for any ρ, Proof The inequality follows from [21], proposition 1.15. Also, by proposition 4 we have the last equality being true because of the weak convergence of (V n δ ρ ) n∈N . This proves the first equality in (64). Since Uhdµ = hdVµ = hdµ, we obtain the second equality. (60)  Proof The proof follows by induction. Let n = 1. We have: And note that U n h(ρ) = U (U n−1 h)(ρ), which concludes the proof.

Entropy and Markov chains
Suppose the V i are fixed and that they determine a dynamics given by Note that a family W : Let P = (p ij ) i,j=1,...,N be a stochastic, irreducible matrix. Let p be the stationary vector of P . The entropy of P is defined as We consider a few examples which will be useful later in this work.  is column-stochastic. We have The fixed point of Λ Let π = (π 1 , π 2 ) such that P π = π. We know that Then the nonzero entries of ρ V are the entries of π and so we associate the fixed point of P to the fixed point of a certain Λ in a natural way. Let us calculate h V (W ). Note that Λ defined above is associated to a homogeneous IFS. Then where H(P ) is the entropy of P , given by (69). This shows that the entropy of Markov chains is a particular case of the QIFS entropy.  Note that the p ij cancel and so we obtain a calculation which is the same as the one obtained in the previous example. Hence , and its nonzero entries are the entries of the fixed point for the stochastic matrix 10 (q 00 log q 00 + q 10 log q 10 ) − q 10 q 01 + q 10 (q 01 log q 01 + q 11 log q 11 ) = H(Q) So we have obtained a calculation which is analogous to the one for the homogeneous case. This result generalizes what we have seen in the previous example.
by an identical calculation made for the equation (72) from the previous example. In other words, the fact that the fixed point of Λ is not diagonal does not change the calculations for the entropy.  is column-stochastic. From where P = (p ij ) i,j=1,...,n . Then for all n, Λ n P (ρ) = Λ P n (ρ).
Proof Note that The general case follows by iterating the above calculation.

Corollary 2.
Under the lemma hypothesis, we have lim n→∞ Λ n P (ρ) = Λ π (ρ), where π = lim n→∞ P n is the stochastic matrix which has all columns equal to the stationary vector for P .

Capacity-cost function and pressure
Recall that every trace preserving, completely positive (CP) mapping can be written in the Stinespring-Kraus form, for V i linear operators. These mappings are also called quantum channels. This is one of the main motivations for considering the class of operators (a generalization of the above ones) described in the present paper. These are natural objets in the analysis of certain problems in quantum computing.
Definition 8. The Holevo capacity for sending classic information via a quantum channel Λ is defined as where S(ρ) = −tr(ρ log ρ) is the von Neumann entropy. The maximum is, therefore, over all choices of p i , i = 1, . . . , n and density operators ρ i , for some n ∈ N. The Holevo capacity establishes an upper bound on the amount of information that a quantum system contains [17].

S(Λ(|ψ ψ|))
Additivity conjecture We have that Minimum output entropy conjecture For any channels Λ 1 and Λ 2 , In [19], is it shown that the additivity conjecture is equivalent to the minimum output entropy conjecture, and in [10] a counterexample is obtained for this last conjecture.
Remark Concerning QIFS, our interest in capacity is motivated by the following observation. Considering expression (75), note that the term is a convex combination of von Neumann entropies, in the same way as the QIFS entropy. So we see that given a QIFS, we can consider capacity functions, and the QIFS entropy arises in a natural way. For an example, we perform the following calculation. If λ i are the eigenvalues of ρ then we can write Then write the QIFS entropy as We see that for ρ W ∈ M N and i fixed, we have i a ij (ρ W ) = 1. Define for each i the density operator Then by (77), By (78), we can write A Positive Operator-Valued Measurement (POVM) is described by a set of positive operators P i (POVM elements) such that i P i = I. If the measurement is performed on a system described by the state vector |ψ , then the probability of obtaining i as the outcome is given by Note that a QIFS F induced by linear V i and W i , contains a POVM by taking W * i W i as POVM elements. If X is a random variable that takes values p 1 , . . . p k then the Shannon entropy is H(X) = − i p i log p i and the joint entropy of variables X and Y is where p(x, y) is the probability that X = x and X = y. The mutual information The number ξ(E) is the Holevo information of the ensemble given by E = {ρ i ; p i } i=1,...,k . We see that (85) holds by applying the Holevo bound for the von Neumann entropy (see [17]) together with (80) and (82).

♦
We are also interested in a different class of problems which concern maximization (and not minimization) of entropy plus a given potential (a cost) [9], [11], [12]. 11 and let H be a hermitian operator. For µ ∈ M F let ρ µ be its barycenter. Define the capacity-cost function C : R + → R + as

Definition 10. Let M F be the set of invariant measures defined in the section
The following analysis is inspired in [8]. There is a relation between the costcapacity function and the variational problem for pressure. In fact, let F : R + → R + be the function given by We have the following fact. There is a unique probability measure ν 0 ∈ M F such that Also, we have the following lemma: Lemma 10. Let λ ≤ 0, andâ = tr(Hρ ν 0 ). Then Proof Let ν ∈ M F , ν = ν 0 , with tr(Hρ ν ) ≤â = tr(Hρ ν0 ). Then 13. Analysis of the pressure problem a hermitian operator. We are interested in obtaining a version of the variational principle of pressure for our context. We will see that the pressure will be a maximum whenever we have a certain relation between the potential H and the probability distribution considered (represented here by the W i ). We begin by fixing a dynamics, given by the V i . From the reasoning described below, it will be natural to consider as definition of pressure the maximization among the possible stationary W i of the expression where ρ β is the eigenstate of a certain Ruelle operator, described below. We begin our analysis by using the following elementary lemma.
The potential given by (89), together with the V i , induces an operator given by By proposition 2 we know that such operator admits an eigenvalue β with its associated eigenstate ρ β . Then L H (ρ β ) = βρ β implies In coordinates, (92) can be written as Remark Comparing the above calculation with the problem of finding an eigenvalue λ of a matrix A = (a ij ), we have that equation (92) can be seen as the analogous of the expression Above, the matrix A plays the role of a potential, E A denotes the matrix with entries e a ij and l j denotes the j-th coordinate of the left eigenvector l associated to the eigenvalue λ. In coordinates, i l i e aij = λl j , i, j = 1, . . . , k ♦ From this point we can perform two calculations. First, considering (92) we will take the trace of such equation in order to obtain a scalar equation. In spite of the fact that taking the trace makes us lose part of the information given by the eigenvector equation, we are still able to obtain a version of what we will call the basic inequality, which can be seen as a QIFS version of the variational principle of pressure. However, there is an algebraic drawback to this approach, namely, that we will not be able to recover the classic variational problem as a particular case of such inequality (such disadvantage is a consequence of taking the trace, clearly). The second calculation begins at equation (93), the coordinate equations associated to the matrix equation for the eigenvectors. In this case we also obtain a basic inequality, but then we will have the classic variational problem of pressure as a particular case.
An important question which is of our interest, regarding both calculations mentioned above, is to ask whether it is possible for a given system to attain its maximum pressure. It is not clear that given any dynamics, we can obtain a measure reaching such a maximum. With respect to our context, we will remark a natural condition on the dynamics which allows us to determine expressions for the measure which maximizes the pressure. Now we perform the calculations mentioned above.
Based on (92), define So we have j r j = 1. Let Then we can apply lemma 11 for r j , q i j , j = 1, . . . k, with i fixed, to obtain and equality holds if and only if for all i, j, Multiplying by tr(W i ρ W W * i ) and summing over the i index, we have and equality holds if and only if for all i, j, Let us rewrite inequality (102). First we use the fact that ρ W is a fixed point of Now we compose both sides of the equality above with the operator and then we obtain Taking the trace on both sides we get Note that the left hand side of (108) is one of the sums appearing in (102). Therefore replacing (108) into (102) gives our main result.
and equality holds if and only if for all i, j, In section 16 we make some considerations about certain cases in which we can reach an equality in (109).

♦
For the calculations regarding expression (93), define Then we have j r jlm = 1. Let and equality holds if and only if for all i, j, l, m,

Revisiting the eigenvalue problem
Consider the operator  And we can also write, for i = 1, 2, Then we get Then the operator L H has a diagonal eigenstate (134) ρ β = ρ 11 0 0 ρ 22 associated to the eigenvalue β, and we have that, defining v = (ρ 11 , ρ 22 ), we get Av = βv.
Then Av = βv leads us to

Some classic inequality calculations
A natural question is to ask whether the maximum among normalized W i , i = 1, . . . , k, for the pressure problem associated to a given potential is realized as the logarithm of the main eigenvalue of a certain Ruelle operator associated to the potential H i , i = 1, . . . , k. This problem will be considered in this section and also in the next one.
We begin by recalling a classic inequality. Consider given by lemma 11. Let A be a matrix. If v denotes the left eigenvector of matrix E A (such that each entry is e a ij ), then vE A = βv can be written as That is, Let Q be a matrix with entries q ij , let π = (π 1 , . . . , π k ) be the stationary vector associated to Q. Since i q ij = 1, Q is column-stochastic so we write Qπ = π. Multiplying the above inequality by π j and summing the j index, we get In coordinates, Qπ = π is j q ij π j = π i , for all i. Then These calculations are well-known and gives us the following inequality: Definition 11. We call inequality (149) the classic inequality associated to the matrix A with positive entries, and stochastic matrix Q.
Definition 12. For fixed k, and l, m = 1, . . . , k we call the inequality the basic inequality associated to the potential Hρ = i H i ρH * i and to the QIFS determined by V i , W i , i = 1, . . . , k. Equality holds if for all i, j, l, m, As before, ρ β is an eigenstate of L H (ρ) and ρ W is the barycenter of the unique attractive, invariant measure for the Markov operator V associated to the QIFS F W . Given the classic inequality (149) we want to compare it to the basic inequality (150). More precisely, we would like to obtain operators V i that satisfy the following: given a matrix A with positive entries and a stochastic matrix Q, there are H i and W i such that inequality (150) becomes inequality (149). We have the following proposition.
and also Then the basic inequality associated to W i , V i , H i , i = 1, . . . , 4, l = m = 1 or l = m = 2, is equivalent to the classic inequality associated to A and Q.
We use the following lemma.
Lemma 12. For V i given by where v ij > 0, we have that the associated QIFS is such that ρ W and ρ β are diagonal density operators, for any choice of W i and H i , i = 1, . . . , 4.

Proof of Lemma 12
We have that ρ W is a fixed point of we have that Λ(ρ) = ρ leads us to Then ρ 12 = 0 and so ρ W is diagonal. In a similar way we prove ρ β is diagonal.
Proof of Proposition 9 Let V i , W i , i = 1, . . . , 4 and H ij , i, j = 1, 2 as in the statement of the proposition. A simple calculation shows that (159) tr(H ij ρ β H * ij ) = e aij (since ρ β is diagonal, by lemma 12). By example 5, the choice of V i and W i we made is such that the entropy h V (W ) reduces to the Markov chain entropy. Then a calculation yields In a similar way, Then from the basic inequality with l = m = 1 or l = m = 2 we get Finally, since tr(H ij ρ β H * ij ) = e aij and Qπ = π, we conclude that (162) becomes (149).

Example 9. Let
If we suppose the V i are the same as from proposition 9, we have that ρ β is diagonal, so i ) Choose, for instance, l = m = 1. Then condition (164) becomes Then a calculation shows that LH (ρ) =βρ gives us the same eigenstate as before, that is ρβ = ρ β . But note that the associated eigenvalue becomesβ = αβ. Now, note that it is possible to renormalize the W i in such a way that we obtainW i with iW * iW i = I, and that these maximize the basic inequality for the H i initially fixed. In fact, given the renormalizedH i , define

Remarks on the problem of pressure and quantum mechanics
One of the questions we are interested in is to understand how to formulate a variational principle for pressure in the context of quantum information theory. An appropriate combination of such theories could have as a starting point a relation between the inequality for positive numbers − i q i log q i + i q i log p i ≤ 0, (lemma 11, seen in certain proofs of the variational principle of pressure), and the QIFS entropy. We have carried out such a plan and then we have obtained the basic inequality, which can be written as where equality holds if and only if for all i, j, i ) As we have discussed before, it is not clear that given any dynamics, we can obtain a measure such that we can reach the maximum value log β. Considering particular cases we can suppose, for instance, that the V i are unitary. In this way we combine in a natural way a problem of classic thermodynamics, with an evolution which has a quantum character. In this particular setting, we have for each i that where I is the identity. The equality condition (178) is satisfied by suchŴ j , so the lemma follows.
Remark The above lemma also holds for the basic inequality in coordinates, given by (150). Also, it is immediate to obtain a similar version of the above lemma for any QIFS such that the V i are multiples of the identity, and also for QIFS such that ρ W fixes each branch of the QIFS, that is, satisfying, for each i,

Concluding remarks
Considering the QIFS setting, we defined a concept of entropy and a Ruelle operator in such a way that we are able to get some analogous results to the classical Thermodynamic Formalism. Such Ruelle operator admits a positive eigenvalue, which gives us an upper bound for the pressure (entropy plus a potential) associated to the QIFS. Our configuration space is the set of density matrices. We did not consider the usual space of symbols or a shift operator, as it is assumed in the Ruelle-Perron-Frobenius theory. We have replaced the dynamics given by the shift with the one given by the inverse branches of the iterated functions (which are defined by a set of operators).
The references [16] and [21] are of fundamental importance in our investigation. A starting point for further investigation could be to study more properties of the QIFS entropy, such as convexity and subadditivity. Also, a natural question is to ask whether it is possible to consider a QIFS acting in an infinite tensor product of finite Hilbert spaces which would be the analogous of considering the full Bernoulli space.
In a forthcoming paper we are going to consider relative entropies and quantum conditional expectations.