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arXiv:1510.00578 · Discrete Analysis

Dvoretzky's theorem and the complexity of entanglement detection

Stanisław J. Szarek, Guillaume Aubrun

Published Jan 1, 2017• 7 citations
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Abstract

Dvoretzky's theorem and the complexity of entanglement detection, Discrete Analysis 2017:1, 20 pp. Let $H$ be a Hilbert space. A _state_ on $H$ is a linear operator $\rho:H\to H$ such that tr$(\rho)=1$ and tr$(\rho P)\geq 0$ for every orthogonal projection $P$. A linear operator satisfying just the second condition is called _positive_. If $H=H_1\otimes H_2$ then an important piece of information about $\rho$ is the extent to which it can be split up into a part that acts on $H_1$ and a part that acts on $H_2$. An appropriate formal definition is the following: we say that $\rho$ is _separable_ if it can be approximated in the trace-class norm by operators of the form $\sum_ic_i\rho^1_i\otimes\rho^2_i$, where each $\rho^1_i$ is a state on $H_1$, each $\rho^2_i$ is a state on $H_2$, and the $c_i$ are positive constants. An operator that is not separable is called _entangled_. Owing to the importance of entanglement, it is useful to have a criterion that detects it. A simple necessary condition is that if $\Phi$ is a linear map from operators to operators that takes positive operators to positive operators, and if $\rho$ is separable, then $(\Phi\otimes I)\rho$ is a positive operator: that is simply because $(\Phi\otimes I)(\sum_ic_i\rho^1_i\otimes\rho^2_i)=\sum_ic_i\Phi\rho^1_i\otimes\rho^2_i$ and each $\Phi\rho^1_i$ is positive. Interestingly, however, the converse is true: if $\rho$ is entangled, then there exists a positive $\Phi$ such that $(\Phi\otimes I)(\rho)$ is _not_ positive. This equivalence is called the _Horodecki criterion_ for entanglement. The Horodecki criterion gives us a witness $\Phi$ for each entangled state $\rho$. The starting question for this paper is how many different witnesses one needs to detect all entangled states (as a function of the dimensions of $H_1$ and $H_2$). More precisely, since this number is known to be infinite, the paper considers the stronger notion of _robustly_ entangled states, which are states that remain entangled even when you average them with a suitable multiple of the identity. The main result of the paper is that the number needed is large: the authors obtain a bound of $\exp(cd^3/\log d)$. A key observation that enables them to prove this is that there is a resemblance between the problem they are considering and a result from a famous paper of Figiel, Lindenstrauss and Milman concerning Dvoretzky's theorem [1]. It follows from the analysis in the FLM paper that the product of the logarithm of the number of faces of a symmetric convex body with the logarithm of the number of vertices has to be at least $cn$, so it is not possible for a symmetric convex body to have few faces and few vertices (in strong contrast with a general convex body, since an $n$-dimensional simplex has $n+1$ of each). Something like that occurs here. Roughly speaking, the set of separable states can be regarded as having "few" extreme points, and therefore "many" faces. A single witness $\Phi$ cannot help with too many faces, so the number of witnesses needed must be large. These interesting ideas are new to the field of quantum information. [1] T. Figiel, J. Lindenstrauss and V. D. Milman, _The dimension of almost spherical sections of convex bodies,_ Acta Math. 139 (1-2) (1977), pp. 53-94.

DOI
10.19086/da.1242

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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) [10], [12], [13]. Let Vi, Wi, Hi be linear operators, ⁠, with and let 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. Let A = (aij) be a matrix with positive entries and consider the problem of finding its eigenvalues and eigenvectors. Then from A natural question is to ask whether the maximum among normalized Wi, 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 Hi, This problem will be considered in this section and also in the next one. We call inequality (149) the classic inequality associated to the matrix A with positive entries, and stochastic matrix Q. As before, ρβ is an eigenstate of ℒH(ρ) and ρW is the barycenter of the unique attractive, invariant measure for the Markov operator associated to the QIFS ℱW. Given the classic inequality (149), we want to compare it to the basic inequality (150). More precisely, we would like to obtain operators Vi that satisfy the following: given a matrix A with positive entries and a stochastic matrix Q, there are Hi and Wi such that inequality (150) becomes inequality (149). We have the following proposition. We use the following lemma. 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. A. O. Lopes: Partially supported by CNPq, PRONEX – Sistemas Dinâmicos, INCT, and beneficiary of CAPES financial support. C. F. Lardizabal: Supported by CAPES PhD scholarship. M. Terra Cunha: Partially supported by CNPq and FAPEMIG.

arXiv:0911.0179•Applied Mathematics Research eXpress•Apr 15, 2010•9 citations
Dvoretzky's theorem and the complexity of entanglement detection, Discrete Analysis 2017:1, 20 pp. Let HHH be a Hilbert space. A state on HHH is a linear operator ρ:H→H\rho:H\to Hρ:H→H such that tr(ρ)=1(\rho)=1(ρ)=1 and tr(ρP)≥0(\rho P)\geq 0(ρP)≥0 for every orthogonal projection PPP. A linear operator satisfying just the second condition is called positive. If H=H1⊗H2H=H_1\otimes H_2H=H1​⊗H2​ then an important piece of information about ρ\rhoρ is the extent to which it can be split up into a part that acts on H1H_1H1​ and a part that acts on H2H_2H2​. An appropriate formal definition is the following: we say that ρ\rhoρ is separable if it can be approximated in the trace-class norm by operators of the form ∑iciρi1⊗ρi2\sum_ic_i\rho^1_i\otimes\rho^2_i∑i​ci​ρi1​⊗ρi2​, where each ρi1\rho^1_iρi1​ is a state on H1H_1H1​, each ρi2\rho^2_iρi2​ is a state on H2H_2H2​, and the cic_ici​ are positive constants. An operator that is not separable is called entangled. Owing to the importance of entanglement, it is useful to have a criterion that detects it. A simple necessary condition is that if Φ\PhiΦ is a linear map from operators to operators that takes positive operators to positive operators, and if ρ\rhoρ is separable, then (Φ⊗I)ρ(\Phi\otimes I)\rho(Φ⊗I)ρ is a positive operator: that is simply because (Φ⊗I)(∑iciρi1⊗ρi2)=∑iciΦρi1⊗ρi2(\Phi\otimes I)(\sum_ic_i\rho^1_i\otimes\rho^2_i)=\sum_ic_i\Phi\rho^1_i\otimes\rho^2_i(Φ⊗I)(∑i​ci​ρi1​⊗ρi2​)=∑i​ci​Φρi1​⊗ρi2​ and each Φρi1\Phi\rho^1_iΦρi1​ is positive. Interestingly, however, the converse is true: if ρ\rhoρ is entangled, then there exists a positive Φ\PhiΦ such that (Φ⊗I)(ρ)(\Phi\otimes I)(\rho)(Φ⊗I)(ρ) is not positive. This equivalence is called the Horodecki criterion for entanglement. The Horodecki criterion gives us a witness Φ\PhiΦ for each entangled state ρ\rhoρ. The starting question for this paper is how many different witnesses one needs to detect all entangled states (as a function of the dimensions of H1H_1H1​ and H2H_2H2​). More precisely, since this number is known to be infinite, the paper considers the stronger notion of robustly entangled states, which are states that remain entangled even when you average them with a suitable multiple of the identity. The main result of the paper is that the number needed is large: the authors obtain a bound of exp⁡(cd3/log⁡d)\exp(cd^3/\log d)exp(cd3/logd). A key observation that enables them to prove this is that there is a resemblance between the problem they are considering and a result from a famous paper of Figiel, Lindenstrauss and Milman concerning Dvoretzky's theorem [1]. It follows from the analysis in the FLM paper that the product of the logarithm of the number of faces of a symmetric convex body with the logarithm of the number of vertices has to be at least cncncn, so it is not possible for a symmetric convex body to have few faces and few vertices (in strong contrast with a general convex body, since an nnn-dimensional simplex has n+1n+1n+1 of each). Something like that occurs here. Roughly speaking, the set of separable states can be regarded as having "few" extreme points, and therefore "many" faces. A single witness Φ\PhiΦ cannot help with too many faces, so the number of witnesses needed must be large. These interesting ideas are new to the field of quantum information. [1] T. Figiel, J. Lindenstrauss and V. D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1-2) (1977), pp. 53-94.
Let ${H}^{[N]}{=H}^{[{d}_{1}]}\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}{H}^{[{d}_{n}]}$ be a tensor product of Hilbert spaces and let ${\ensuremath{\tau}}_{0}$ be the closest separable state in the Hilbert-Schmidt norm to an entangled state ${\ensuremath{\rho}}_{0}.$ Let ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ denote the closest separable state to ${\ensuremath{\rho}}_{0}$ along the line segment from I/NI/NI/N to ${\ensuremath{\rho}}_{0}$ where I is the identity matrix. Following A. O. Pittenger and M. H. Rubin [Linear Algebr. Appl. 346, 75 (2002)] a witness W0{W}_{0}W0​ detecting the entanglement of ${\ensuremath{\rho}}_{0}$ can be constructed in terms of I, ${\ensuremath{\tau}}_{0},$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ If representations of ${\ensuremath{\tau}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ as convex combinations of separable projections are known, then the entanglement of ${\ensuremath{\rho}}_{0}$ can be detected by local measurements. G\"uhne et al. [Phys. Rev. A 66, 062305 (2002)] obtain the minimum number of measurement settings required for a class of two-qubit states. We use our geometric approach to generalize their result to the corresponding two-qudit case when d is prime and obtain the minimum number of measurement settings. In those particular bipartite cases, ${\ensuremath{\tau}}_{0}={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}.$ We illustrate our general approach with a two-parameter family of three-qubit bound entangled states for which ${\ensuremath{\tau}}_{0}\ensuremath{\ne}{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\tau}}}_{0}$ and we show that our approach works for n qubits. We elaborated earlier [A. O. Pittenger, Linear Algebr. App. 359, 235 (2003)] on the role of a ``far face'' of the separable states relative to a bound entangled state ${\ensuremath{\rho}}_{0}$ constructed from an orthogonal unextendible product base. In this paper the geometric approach leads to an entanglement witness expressible in terms of a constant times I and a separable density ${\ensuremath{\mu}}_{0}$ on the far face from ${\ensuremath{\rho}}_{0}.$ Up to a normalization this coincides with the witness obtained by G\"uhne et al. for the particular example analyzed there.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
Entanglement detection in high dimensional systems is a NP-hard problem since it is lacking an efficient way. Given a bipartite quantum state of interest free entanglement can be detected efficiently by the PPT-criterion (Peres-Horodecki criterion), in contrast to detecting bound entanglement, i.e. a curious form of entanglement that can also not be distilled into maximally (free) entangled states. Only a few bound entangled states have been found, typically by constructing dedicated entanglement witnesses, so naturally the question arises how large is the volume of those states. We define a large family of magically symmetric states of bipartite qutrits for which we find [Formula: see text] to be free entangled, [Formula: see text] to be certainly separable and as much as [Formula: see text] to be bound entangled, which shows that this kind of entanglement is not rare. Via various machine learning algorithms we can confirm that the remaining [Formula: see text] of states are more likely to belonging to the set of separable states than bound entangled states. Most important we find via dimension reduction algorithms that there is a strong two-dimensional (linear) sub-structure in the set of bound entangled states. This revealed structure opens a novel path to find and characterize bound entanglement towards solving the long-standing problem of what the existence of bound entanglement is implying.
Abstract This chapter gives a detailed overview of entanglement for pure and mixed states. We begin with the Schmidt-decomposition theorem, which allows us to associate the subsystem mixedness of pure state with entanglement. We discuss the purification of mixed quantum states, the convexity of the set of separable states, and entanglement as a form of correlation. As an example we examine the Bloch decomposition for two qubits. For the detection of entanglement the Peres-Horodecki criterion is introduced, before we analyse the relation between entanglement and non-locality via the CHSH-Operator Criterion and the family of Werner states. We further consider Tsirelson’s bound and hidden non-locality. We then discuss important mathematical tools for entanglement detection, including the positive-map theorem, the Størmer-Woronowicz theorem, and the reduction criterion. In this context we examine the family of isotropic states. Finally, the Weyl states provide an example for which we study the geometry of two-qubit entanglement
We show how to design families of operational criteria that distinguish entangled from separable quantum states. The simplest of these tests corresponds to the well-known Peres-Horodecki positive partial transpose (PPT) criterion, and the more complicated tests are strictly stronger. The new criteria are tractable due to powerful computational and theoretical methods for the class of convex optimization problems known as semidefinite programs. We successfully applied the results to many low-dimensional states from the literature where the PPT test fails. As a by-product of the criteria, we provide an explicit construction of the corresponding entanglement witnesses.
We show that such operator admits an eigenvalue β and an associated eigenstate ρβ, that is, one satisfying ℒH(ρβ) = βρβ. 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 [3]. 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 and 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. Sections 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). A QIFS is homogeneous if pi and Fipi are affine mappings, ⁠. We recall the definition of the integral above in Section 5. If a QIFS (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 ∈ℳN, given by (13). See [16], [21] for the proof. 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 Now, we consider the following problem. Let A different analysis in the quantum setting which is related to Perron’s theorem is presented in [6]. 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); on 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. There exists ρ ∈ℳN and β > 0 such that ℒW, V(ρ) = βρ. The value β is obtained explicitly: β = tr(ℒW, V(ρ)). 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 ℋN and X will be the space of density matrices ℳN. 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 is compact then the integral of f in X exists and belongs to ⁠. We will also use the following well-known result, the barycentric formula: In the context of QIFS, we can take ⁠. 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 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 ⁠, 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 ⁠. An operator is submarkovian if Q(αx) = αQ(x) for all x, y ∈ V+, α > 0. Every submarkovian operator can be extended in a unique way to a positive linear contraction on V , see [21]. Let a Markov operator and let ⁠, be submarkovian operators such that ⁠. We say that (P, {Pi}i = 1k) is a Markov pair. From [21], we know that there is a 1–1 correspondence between homogeneous IFS and Markov pairs. Let be an IFS, Ψ a linear functional on X. Then, ⁠, where is given by (43). Let be an IFS and let ρ0 ∈ X. Then, Λ(ρ0) = ρ0 if and only if ⁠, for all Ψ linear functional. Let be an IFS. Let ρ0 ∈ X such that Fi(ρ0) = ρ0, ⁠. Then, ⁠. Let ρ0 ∈ X such that ⁠, then Λ(ρ0) = ρ0. Let be a homogeneous IFS, ⁠. Let ρν be the barycenter of a probability measure ν. Then, Λ(ρν) is the barycenter of ⁠, where is the associated Markov operator. Let μ be an invariant probability measure for ⁠. Then, the barycenter of μ, denoted by ρμ, is a fixed point of Λ. 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] [9]. The present setting contains the classical case and therefore in general there is no uniqueness. 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 ⁠. Let be an IFS which admits an attractive invariant measure μ for ⁠. Then limn→∞Λn(ρ0) = ρμ, for every ρ0 ∈ℳN, where ρμ is the barycenter of μ. 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. Let ℱ be an IFS and let g : B → ℝ. Then, for n ∈ ℕ, If g is concave (resp. convex, affine) then, (resp. ⁠, ⁠). If is a fixed point for Λ then, the sequence is decreasing (resp. increasing, constant) ifg is concave (resp. convex, affine). Also suppose that ℱ is homogeneous. Then, If g is concave (resp. convex, affine), then, is concave (resp. convex, affine) . Let ⁠. Then the sequences and are nonnegative, decreasing, and have the same limit. The following result gives us an integral formula for entropy and also a relation between the entropies defined before. We write where is the convex hull of the set of accumulation points of and 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. hV(W) ≥ 0, for every family Wi of linear operators satisfying ⁠. To conclude that hV(W) ≥ 0, we have to show that tr(WjViρWVi∗Wj∗) ≤tr(ViρWVi∗). From ⁠, we get Remark For any fixed dynamics V , if we have that Wm∗Wm = I for some m then the remaining pi must be zero, because of the condition ⁠. In this case, we have hV(W) = 0. We also have that hV(W) ≤ log k and for any given dynamics V , hV(W) attains the maximum if we choose ⁠, for each i, where I denotes the identity operator.□ Note that by the calculations made in Section 9, we have ⁠, where ⁠. Let Vi, Wi be linear operators, ⁠, ⁠. Suppose the Vi are fixed and that they determine a dynamics given by Fi : ℳN →ℳN, ⁠. Define Under the lemma hypothesis, we have limn→∞ΛPn(ρ) = Λπ(ρ), where π = limn→∞Pn is the stochastic matrix which has all columns equal to the stationary vector for P. In [19], is it shown that the additivity conjecture is equivalent to the minimum output entropy conjecture and in [11], a counterexample is obtained for this last conjecture. The number ξ(ℰ) is the Holevo information of the ensemble given by ⁠. 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) [10], [12], [13]. Let Vi, Wi, Hi be linear operators, ⁠, with and let 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. Let A = (aij) be a matrix with positive entries and consider the problem of finding its eigenvalues and eigenvectors. Then from A natural question is to ask whether the maximum among normalized Wi, 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 Hi, This problem will be considered in this section and also in the next one. We call inequality (149) the classic inequality associated to the matrix A with positive entries, and stochastic matrix Q. As before, ρβ is an eigenstate of ℒH(ρ) and ρW is the barycenter of the unique attractive, invariant measure for the Markov operator associated to the QIFS ℱW. Given the classic inequality (149), we want to compare it to the basic inequality (150). More precisely, we would like to obtain operators Vi that satisfy the following: given a matrix A with positive entries and a stochastic matrix Q, there are Hi and Wi such that inequality (150) becomes inequality (149). We have the following proposition. We use the following lemma. 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. A. O. Lopes: Partially supported by CNPq, PRONEX – Sistemas Dinâmicos, INCT, and beneficiary of CAPES financial support. C. F. Lardizabal: Supported by CAPES PhD scholarship. M. Terra Cunha: Partially supported by CNPq and FAPEMIG.