A Thermodynamic Formalism for Density Matrices in Quantum Information
Alexandre Baraviera, Carlos F. Lardizabal, Artur Oscar Lopes +1 more
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.
arXiv:0911.0179•Applied Mathematics Research eXpress•Apr 15, 2010•9 citations