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Thermodynamic Formalism for Quantum Channels: Entropy, Pressure and Gibbs channels

Jader E. Brasil, Josué Knorst, Artur Oscar Lopes

Published Jan 28, 2019• 0 citations
OpenAlex

Abstract

Denote $M_k$ the set of complex $k$ by $k$ matrices. We will analyze here quantum channels $\phi_L$ of the following kind: given a measurable function $L:M_k\to M_k$ and the measure $\mu$ on $M_k$ we define the linear operator $\phi_L:M_k \to M_k$, via the expression $\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho L(v)^\dagger \, \dm(v).$ A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where $L$ was the identity. Under some assumptions on the quantum channel $\phi_L$ we analyze the eigenvalue property for $\phi_L$ and we also define entropy for the channel $\phi_L$. For a fixed $\mu$ (the {\it a priori} measure) given an Hamiltonian $H: M_k \to M_k$ (which defines $\phi_H$) we present a variational principle of pressure and relate it to the eigenvalue problem. We introduce the concept of Gibbs channel. We describe a related process $X_n$, $n\in \mathbb{N}$, taking values on the projective space $ P(\C^k)$ and analyze the question of existence of invariant probabilities. We also consider an associated process $\rho_n$, $n\in \mathbb{N}$, with values on $D_k$ ($D_k$ is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator fixed for $\phi_L$.

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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). 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arXiv:0911.0179•Applied Mathematics Research eXpress•Apr 15, 2010•9 citations

A dynamical point of view of Quantum Information: entropy, pressure and Wigner measures

Alexandre Baraviera, Carlos F. Lardizabal, Artur Oscar Lopes +1 more

Quantum Information is a new area of research which has been growing rapidly since the 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 Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator $\mathcal{L}$ acting on density matrices $ρ\in \mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,$ where $W_i$ and $V_i$, $i=1,2,... k$ are operators in this Hilbert space. $\mathcal{L}$ 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 also analyze the discrete Wigner function. We suppose that for all $ρ$ we have that $\sum_{i=1}^k tr(W_iρW_i^*)=1$. A family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS) $\mathcal{F}_{W}$, $\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.$

arXiv:0911.0193•Nov 1, 2009•3 citations

Lyapunov Exponents for Quantum Channels: An Entropy Formula and Generic Properties

Jader E. Brasil, Josué Knorst, Artur Oscar Lopes

We denote by Mk the set of k by k matrices with complex entries. We consider quantum channels φL of the form: given a measurable function L: Mk → Mk and a measure µ on Mk we define the linear operator φL : Mk → Mk, by the law ρ → φL(ρ) = ∫ Mk L(v)ρL(v)† dµ(v).In a previous work, the authors show that for a fixed measure µ the Φ-Erg property is generic on the function L (also irreducibility). Here we will show that the purification property is also generic on L for a fixed µ.Given L and µ there are two related stochastic processes: one takes values on the projective space P (ℂk) and the other on matrices in Mk. The Φ-Erg property and the purification condition are the nice hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically on L, if ∫ |L(v)|2 log |L(v)| dµ(v) < ∞, the Lyapunov exponents ∞ > γ1 ≥ γ2 ≥ … ≥ γk ≥ −∞ are well defined.In a previous work, the concepts of entropy of a channel and Gibbs channel were presented; and also an example (associated to a stationary Markov chain) in which this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the mentioned example and we show that it is equal to −1/2 h, where h is the entropy of the associated Markov invariant probability.

arXiv:1908.08942•Journal of Dynamical Systems and Geometric Theories•Jul 3, 2021•1 citation

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Diane Christine Pelejo

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QP-PQ, quantum probability and white noise analysis•Jul 1, 1992•2 citations

Note on entropies for quantum dynamical systems

Noboru Watanabe

Quantum entropy and channel are fundamental concepts for quantum information theory progressed recently in various directions. We will review the fundamental aspects of mean entropy and mean mutual entropy and calculate them for open system dynamics.

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Apr 19, 2016•9 citations
Denote MkM_kMk​ the set of complex kkk by kkk matrices. We will analyze here quantum channels ϕL\phi_LϕL​ of the following kind: given a measurable function L:Mk→MkL:M_k\to M_kL:Mk​→Mk​ and the measure μ\muμ on MkM_kMk​ we define the linear operator ϕL:Mk→Mk\phi_L:M_k \to M_kϕL​:Mk​→Mk​, via the expression ρ → ϕL(ρ)=∫MkL(v)ρL(v)† \dm(v).\rho \,\to\,\phi_L(\rho) = \int_{M_k} L(v) \rho L(v)^\dagger \, \dm(v).ρ→ϕL​(ρ)=∫Mk​​L(v)ρL(v)†\dm(v).
A recent paper by T. Benoist, M. Fraas, Y. Pautrat, and C. Pellegrini is our starting point. They considered the case where LLL was the identity.
Under some assumptions on the quantum channel ϕL\phi_LϕL​ we analyze the eigenvalue property for ϕL\phi_LϕL​ and we also define entropy for the channel ϕL\phi_LϕL​. For a fixed μ\muμ (the {\it a priori} measure) given an Hamiltonian H:Mk→MkH: M_k \to M_kH:Mk​→Mk​ (which defines ϕH\phi_HϕH​) we present a variational principle of pressure and relate it to the eigenvalue problem. We introduce the concept of Gibbs channel.
We describe a related process XnX_nXn​, n∈Nn\in \mathbb{N}n∈N, taking values on the projective space P(\Ck) P(\C^k)P(\Ck) and analyze the question of existence of invariant probabilities.
We also consider an associated process ρn\rho_nρn​, n∈Nn\in \mathbb{N}n∈N, with values on DkD_kDk​ (DkD_kDk​ is the set of density operators). Via the barycenter we associate the invariant probabilities mentioned above with the density operator fixed for ϕL\phi_LϕL​.
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.
Quantum Information is a new area of research which has been growing rapidly since the 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 Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator L\mathcal{L}L acting on density matrices ρ∈MNρ\in \mathcal{M}_Nρ∈MN​ over a finite NNN-dimensional complex Hilbert space L(ρ):=∑i=1ktr(WiρWi∗)ViρVi∗,\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,L(ρ):=∑i=1k​tr(Wi​ρWi∗​)Vi​ρVi∗​, where WiW_iWi​ and ViV_iVi​, i=1,2,...ki=1,2,... ki=1,2,...k are operators in this Hilbert space. L\mathcal{L}L is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the Vi(.)Vi∗=:Fi(.)V_i (.) V_i^*=:F_i(.)Vi​(.)Vi∗​=:Fi​(.), i=1,2,...,ki=1,2,...,ki=1,2,...,k, play the role of the inverse branches (acting on the configuration space of density matrices ρρρ) and the WiW_iWi​ play the role of the weights one can consider on the IFS. We also analyze the discrete Wigner function. We suppose that for all ρρρ we have that ∑i=1ktr(WiρWi∗)=1\sum_{i=1}^k tr(W_iρW_i^*)=1∑i=1k​tr(Wi​ρWi∗​)=1. A family W:={Wi}i=1,...,kW:=\{W_i\}_{i=1,..., k}W:={Wi​}i=1,...,k​ determines a Quantum Iterated Function System (QIFS) FW\mathcal{F}_{W}FW​, FW={MN,Fi,Wi}i=1,...,k.\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.FW​={MN​,Fi​,Wi​}i=1,...,k​.
We denote by Mk the set of k by k matrices with complex entries. We consider quantum channels φL of the form: given a measurable function L: Mk → Mk and a measure µ on Mk we define the linear operator φL : Mk → Mk, by the law ρ → φL(ρ) = ∫ Mk L(v)ρL(v)† dµ(v).In a previous work, the authors show that for a fixed measure µ the Φ-Erg property is generic on the function L (also irreducibility). Here we will show that the purification property is also generic on L for a fixed µ.Given L and µ there are two related stochastic processes: one takes values on the projective space P (ℂk) and the other on matrices in Mk. The Φ-Erg property and the purification condition are the nice hypothesis for the discrete time evolution given by the natural transition probability. In this way it will follow that generically on L, if ∫ |L(v)|2 log |L(v)| dµ(v) < ∞, the Lyapunov exponents ∞ > γ1 ≥ γ2 ≥ … ≥ γk ≥ −∞ are well defined.In a previous work, the concepts of entropy of a channel and Gibbs channel were presented; and also an example (associated to a stationary Markov chain) in which this definition of entropy (for a quantum channel) matches the Kolmogorov-Shanon definition of entropy. We estimate here the larger Lyapunov exponent for the mentioned example and we show that it is equal to −1/2 h, where h is the entropy of the associated Markov invariant probability.
In this dissertation, we present several matrix-related problems and results motivated by quantum information theory. Some background material of quantum information science will be discussed in chapter 1, while chapter 7 gives a summary of results and concluding remarks. In chapter 2, we look at 2n×2n2^n\times 2^n2n×2n unitary matrices, which describe operations on a closed nnn-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general 2n×2n2^n\times 2^n2n×2n unitary matrix to the product of no more than 2n−1(2n−1)2^{n-1}(2^n-1)2n−1(2n−1) single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element Φ\PhiΦ among a given set of quantum channels S\mathcal{S}S that will produce the optimal value of a scalar function D(ρ1,Φ(ρ2))D(\rho_1,\Phi(\rho_2))D(ρ1​,Φ(ρ2​)), on two fixed quantum states ρ1\rho_1ρ1​ and ρ2\rho_2ρ2​. Some of the functions we considered for D(⋅,⋅)D(\cdot,\cdot)D(⋅,⋅) are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when S\mathcal{S}S is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers a1≥…≥a2n≥0a_1\geq\ldots\geq a_{2n}\geq 0a1​≥…≥a2n​≥0, we want to determine if there exist a 2n×2n2n\times 2n2n×2n density matrix ρ\rhoρ having eigenvalues a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on a1,…,a2na_1,\ldots,a_{2n}a1​,…,a2n​ for n≤6n\leq 6n≤6 and state some observations on general values of nnn. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k)\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}Φ(ρ(1))=σ(1),…,Φ(ρ(k))=σ(k) for given ρ(1),…,ρ(k)∈Dn\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_nρ(1),…,ρ(k)∈Dn​ and σ(1),…,σ(k)∈Dm\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_mσ(1),…,σ(k)∈Dm​, (b) constructing a multipartite state ρ\rhoρ having a prescribed set of reduced states ρ1,…,ρr\rho_1,\ldots, \rho_rρ1​,…,ρr​ on rrr of its subsystems, (c) constructing a multipartite stateρ\rhoρ having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix AAA can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets K1K_1K1​ and K2K_2K2​ of $\IC$ given by K1K2={ab:a∈K1,b∈K2}K_1K_2 = \{ab: a \in K_1, b\in K_2\}K1​K2​={ab:a∈K1​,b∈K2​}, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that K1K2K_1K_2K1​K2​ is star-shaped when K1K_1K1​ and K2K_2K2​ are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set K1K2K_1K_2K1​K2​ is star-shaped if K1K_1K1​ is a line segment or a circular disk.
The notion of entropy was introduced by Clausius in order to discuss the thermal behavior of physical systems. After the work of Clausius, Boltzmann, Gibbs and others, the beginning of the exact treatment of entropy was due to von Neumann and Shannon. Their starting point was very di erent, von Neumann was motivated by quantum mechanics and Shannon founded communication theory. The entropy of a state describing a physical system is a quantity expressing the uncertainty or randomness of the system. Shannon regarded this uncertainty attached to a physical system as the amount of information carried by the system, so that the entropy of a state can be read as the information carried by the state. His idea comes from the following consideration: If a physical system has a large uncertainty and one receives information on the system by some procedure, then the so obtained information is more valuable than that received from a system having less uncertainty. It is understood in probability theory that the notion of (Shannon or measure theoretic) entropy has successful applications in a variety of subjects because it determines the asymptotic behavior of certain probabilities in the course of independent trials. First we will discuss this phenomena for nite quantum systems. By a nite quantum system we mean an algebra of matrices which is stable under taking adjoint. (In other words, a nite quantum system is a nite dimensional C*algebra.) If A is such an algebra then there is a linear functional Tr which takes the value 1 at each minimal projection. It is "tracial " in the sense that Tr ab = Tr ba (a; b 2 A) Every functional! on A is determined by a density operator D! 2 A in the form!(a) = Tr D! a (a 2 A): The entropy S(!) of a functional! is de ned by means of its density operator as S(!) = Tr (D!): This notion was introduced by von Neumann in 1927 and we term it von Neumann's entropy or shortly entropy (cf. [1]). In fact, von Neumann de ned the entropy such