The generalized Lyapunov theorem and its application to quantum channels

We give a simple and physically intuitive necessary and sufficient condition for a map acting on a compact metric space to be mixing (i.e. infinitely many applications of the map transfer any input into a fixed convergency point). This is a generalization of the ‘Lyapunov direct method’. First we prove this theorem in topological spaces and for arbitrary continuous maps. Finally we apply our theorem to maps which are relevant in open quantum systems and quantum information, namely quantum channels. In this context, we also discuss the relations between mixing and ergodicity (i.e. the property that there exists only a single input state which is left invariant by a single application of the map) showing that the two are equivalent when the invariant point of the ergodic map is pure.


Introduction
Repetitive applications of the same transformation is the key ingredient of many control techniques. In quantum information processing [1] they have been exploited to inhibit the decoherence of a system by frequently perturbing its dynamical evolution [2]- [6] (Bang-Bang control) or to improve the fidelity of quantum gates [7] by means of frequent measurements (quantum Zeno-effect [8]). Recently analogous strategies have also been proposed in the context of state preparation [9]- [15] and quantum communication [16]- [19]. In [11,12] for instance, a homogenization protocol was presented which allows one to transform any input state of a qubit into a some pre-fixed target state by repetitively coupling it with an external bath. A similar thermalization protocol was discussed in [13] to study the efficiency of simulating classical equilibration processes on a quantum computer. In [14,15] repetitive interactions with an externally monitored environment were instead exploited to implement purification schemes which would allow one to extract pure state components from arbitrary mixed inputs. An application to quantum communication of similar strategies has been finally given in [16]- [19] where sequences of repetitive operations were used to boost the efficiency of quantum information transmission along spin chains.
The common trait of the proposals [9]- [19] is the requirement that repeated applications of a properly chosen quantum operation τ converges to a fixed density matrix x * independently from the input state x of the system, i.e.
with '•' representing the composition of maps. Following the notation of [20,21] we call equation (1) the mixing property of τ. It is related with another important property of maps, namely ergodicity (see figure 1). The latter requires the existence of a unique input state x 0 which is left invariant under a single application of the map, 3 i.e., Ergodicity and the mixing property are of great interest not only in the context of the above quantum information schemes. They also occur on a more fundamental level in statistical mechanics [23] and open quantum systems [24,25], where one would like to study irreversibility and relaxation to thermal equilibrium.
In the case of quantum transformations one can show that mixing maps with convergency point x * are also ergodic with fixed point x 0 = x * . The opposite implication however is not generally true since there are examples of ergodic quantum maps which are not mixing (see the following). Sufficient conditions for mixing have been discussed both in the specific case of quantum channel [13,20,21] and in the more abstract case of maps operating on topological spaces [23]. In particular the Lyapunov direct method [23] allows one to prove that an ergodic map τ is mixing if there exists a continuous functional S which, for all points but the fixed one, is strictly increasing under τ. Here we strengthen this criterion by weakening the requirement on S: our generalized Lyapunov functions are requested only to have limiting values S(τ n (x))| n→∞ which differ from S(x) for all x = x 0 . It turns out that the existence of such S is not just a sufficient condition but also a necessary condition for mixing. Exploiting this fact one can easily generalize a previous result on strictly contractive maps [20] by showing that maps which are asymptotic deformations (see definition 6) are mixing. This has, unlike contractivity, the advantage of being a property independent of the choice of metric (see however [21] for methods of finding 'tight' norms). In some cases, the generalized Lyapunov method also permits derivation of an optimal mixing condition for quantum channels based on the quantum relative entropy. Finally a slightly modified version of our approach which employs multi-central Lyapunov functions yields a characterization of (not necessarily mixing) maps which in the limit of infinitely many applications move all points towards a proper sub-set (rather than a single point) of the input space.
The introduction of a generalized Lyapunov method seems to be sound not only from a mathematical point of view, but also from a physical point of view. In effect, it often happens that the information available on the dynamics of a system are only those related to its asymptotic 3 Definition (2) may sound unusual for readers who are familiar with a definition of ergodicity from statistical mechanics, where a map is called ergodic if its invariant sets have measure 0 or 1. The notion of ergodicity used in the case of a discrete time evolution of a quantum system is different [20,22]. Here, the map τ is not acting on a measurable space but on the compact convex set of quantum states. A perhaps more intuitive and equivalent definition of ergodicity based on the time average of observables is given by lemma 5 of the appendix. behaviour (e.g. its thermalization process), its finite time evolution being instead difficult to characterize. Since our method is explicitly constructed to exploit asymptotic features of the mapping, it provides a more effective way to probe the mixing property of the process.
Presenting our results we will not restrict ourselves to the case of quantum operations. Instead, following [23] we will derive them in the more general context of continuous maps operating on topological spaces [26]. This approach makes our results stronger by allowing us to invoke only those hypothesis which, to our knowledge, are strictly necessary for the derivation. It is important to stress however that, as a particular instance, all the theorems and lemmas presented in the paper hold for any linear, completely positive, trace preserving map (i.e. quantum channel) operating on a compact sub-set of normed vectors (i.e. the space of the density matrices of a finite dimensional quantum system). Therefore readers who are not familiar with topological spaces can simply interpret our derivations as if they were just obtained for quantum channels acting on a finite dimensional quantum system.
The paper is organized as follows. In section 2 the generalized Lyapunov method along with some minor results are presented in the context of topological and metric spaces. Then quantum channels are analysed in section 3 providing a comprehensive summary of the necessary and sufficient conditions for the mixing property of these maps. Conclusions and remarks end the paper in section 4.

Topological spaces
In this section, we introduce the notation and derive our main result (the generalized Lyapunov theorem). The properties of Hausdorff, compact and sequentially compact topological spaces will be used [26]. For the sake of readability their definitions and their relations are given in the caption of figure 2. Definition 1. Let X be a topological space and let τ : X → X be a map. The sequence x n ≡ τ n (x), where τ n is a short-hand notation for the n-fold composition of τ, is called the orbit of x. An element x * ∈ X is called a fixed point of τ if and only if τ is called ergodic if and only if it has exactly one fixed point. τ is called mixing if and only if there exists a convergence point x * ∈ X such that any orbit converges to it, i.e.
lim n→∞ x n = x * ∀x ∈ X .   Relations between topological spaces [26]. Hausdorff topological spaces have the property that any two distinct points of the space can be separated by open neighbourhoods: for these sets any convergent sequence converges to a unique point of the set. Compact topological spaces are such that any open cover of the set has a finite sub-cover. Sequentially compact topological spaces are those for which the Bolzano-Weierstrass theorem holds, i.e. every sequence has a convergent subsequence. Any compact metric spaces is Hausdorff, compact, and sequentially compact. The space of density matrices on which quantum channels are defined, is a compact and convex subset of a normed vectors space (the space of linear operators of the system) which, in the above graphical representation fits within the set of compact metric spaces.
A direct connection between ergodicity and mixing can be established as follows.

Lemma 1. Let τ : X → X be a continuous mixing map on a topological Hausdorff space X .
Then τ is ergodic.
Proof. Let x * be the convergence point of τ and let x ∈ X arbitrary. Since τ is continuous we can perform the limit in the argument of τ, i.e.
which shows that x * is a fixed point of τ. To prove that it is unique assume by contradiction that τ possesses a second fixed point y * = x * . Then lim n→∞ τ n (y * ) = y * = x * , so τ could not be mixing (since the limit is unique in a Hausdorff space, see figure 2). Hence τ is ergodic.
Remark. The converse is not true in general, i.e. not every ergodic map is mixing (not even in Hausdorff topological spaces). A simple counterexample is given by τ : ≡ −x and the usual topology of R, which is ergodic with fixed point 0, but not mixing since for x = 0, τ n (x) = (−1) n x is alternating between two points. A similar counterexample will be discussed in the quantum channel section (see example 1).
A well known criterion for mixing is the existence of a Lyapunov function [23].
Remark. At this point it is neither assumed that x * is a fixed point, nor that τ is ergodic. Both follow from the theorem below. The proof of this theorem is given in [23]. We will not reproduce it here, because we will provide a general theorem that includes this as a special case. In fact, we will show that the requirement of the strict monotonicity can be much weakened, which motivates the following definition.

Definition 3.
Let τ : X → X be a map on a topological space X . A continuous map S : X → R is called a generalized Lyapunov function for τ around x * ∈ X if and only if the sequence S (τ n (x)) is point-wise convergent for any x ∈ X and S fulfills In general it may be difficult to prove the point-wise convergence. However if S is monotonic under the action of τ and the space is compact, the situation becomes considerably simpler. This is summarized in the following lemma. and for some fixed x * ∈ X is a generalized Lyapunov function for τ around x * .
Proof. It only remains to show the (point-wise) convergence of S (τ n (x)). Since S is a continuous function on a compact space, it is bounded. By equation (6) the sequence is monotonic. Any bounded monotonic sequence converges.
for some fixed N ∈ N and for some x * ∈ X is a generalized Lyapunov function for τ around x * .
Remark. This implies that a strict Lyapunov function is a generalized Lyapunov function (with N = 1).
We can now state the main result of this section. Proof. Consider the orbit x n ≡ τ n (x) of a given x ∈ X . Because X is sequentially compact, the sequence x n has a convergent subsequence (see figure 2), i.e. lim k→∞ x n k ≡x. Let us assume thatx = x * and show that this leads to a contradiction. By equation (5) we know that there exists a finite N ∈ N such that Since τ N is continuous we can perform the limit in the argument, i.e. lim k→∞ τ N (x n k ) = τ N (x). Likewise, by continuity of S we have and on the other hand where the second equality stems from the continuity of the map S and τ N . Because S is a generalized Lyapunov function, the sequence S (x n ) is convergent. Therefore the subsequences (9) and (10) must have the same limit. We conclude that S(τ Nx ) = S(x) which contradicts equation (8). Hencex = x * . Since we have shown that any convergent subsequence of τ n (x) converges to the same limit x * , it follows by lemma 4 of the appendix that τ n (x) converges to x * . Since this holds for arbitrary x, it follows that τ is mixing.
There is an even more general way of defining Lyapunov functions which we state here for completeness. It requires the concept of the quotient topology [26].
For these functions we cannot hope that the orbit is mixing. We can however show that the orbit is 'converging' to the set F in the following sense.

Metric spaces
We now show that for the particular class of compact topological sets which posses a metric, the existence of a generalized Lyapunov function is also a necessary condition for mixing. In this context the convergence of a sequence is defined with respect to the distance function d(·, ·) : X × X → R on the space, so that for instance equation (3)   Proof. Firstly, in metric spaces compactness and sequential compactness are equivalent, so the requirements of theorem 2 are met. Secondly, for any mixing map τ with fixed point x * , a generalized Lyapunov function around x * is given by S(x) ≡ d(x * , x). In fact, it is continuous because of the continuity of the metric and satisfies where the equality holds if and only if x = x * . We call d(x * , x) the trivial generalized Lyapunov function.
Remark. In the above theorem we have not used all the properties of the metric. In fact a continuous semi-metric (i.e. without the triangle inequality) would suffice.
The trivial Lyapunov function requires knowledge of the fixed point of the map. There is another way of characterizing mixing maps as those which bring elements closer to each other (rather than closer to the fixed point).
Remark. The notation adopted here is slightly different from the definitions adopted by other authors [20,27,28] who use contraction to indicate our non-expansive maps. Our choice is motivated by the need to clearly distinguish between non-expansive transformation and weak contractions.
We can generalize the above definition in the following way. Remark. Let τ : X → X be a non-expansive map on a metric space X , and let lim n→∞ d(τ n (x), τ n (y)) < d(x, y) ∀x, y ∈ X , x = y.
Then τ is an asymptotic deformation. Any weak contraction is an asymptotic deformation.

Theorem 5. (Asymptotic deformations). Let τ : X → X be a continuous map on a compact metric space X with at least one fixed point. Then τ is mixing if and only if τ is an asymptotic deformation.
Proof. Firstly, assume that τ is an asymptotic deformation. Let x * be a fixed point and define hence S(x) is a generalized Lyapunov function. By theorem 2 it follows that τ is mixing. Secondly, if τ is mixing, then so τ is an asymptotic deformation.
Remark. Note that the existence of a fixed point is assured if τ is a weak contraction on a compact space [29], or if the metric space is convex compact [30]. As a special case it follows that any weak contraction τ on a compact metric space is mixing. This result can be seen as an instance of the Banach contraction principle on compact spaces. In the second part of the paper we will present a counterexample which shows that weak contractivity is only a sufficient criterion for mixing (see example 2). In the context of quantum channels an analogous criterion was suggested in [20,22] which applied to strict contractions. We also note that for weak and strict contractions, the trivial generalized Lyapunov function (theorem 4) is a strict Lyapunov function.

Quantum channels
In this section we discuss the mixing properties of quantum channels [1] which account for the most general evolution a quantum system can undergo including measurements and coupling with external environments. In this context solving the mixing problem (1) is equivalent to determining if repetitive application of a certain physical transformation will drive any input state of the system (i.e. its density matrices) into a unique output configuration. The relationship between the different mixing criteria one can obtain in this case is summarized in figure 3.
At a mathematical level quantum channels correspond to linear maps acting on the density operators ρ of the system and satisfying the requirement of being completely positive and trace preserving (CPT). For a formal definition of these properties we refer the reader to [27,31,32]: here we recall only that a necessary and sufficient condition to being CPT is to allow Kraus decomposition [31] or, equivalently, Stinespring dilation [32]. Our results are applicable if the underlying Hilbert-space is finite dimensional. In such regime there is no ambiguity in defining the convergence of a sequence since all operator norms are equivalent (i.e. given two norms one can construct an upper and a lower bound for the first one by properly scaling the second one). Also the set of bounded operators and the set of operators of Hilbert-Schmidt class coincide. For the sake of definiteness, however, we will adopt the trace-norm which, given the linear operator : H → H, is defined as 1 = Tr[ √ † ] with Tr[ · · · ] being the trace over H and † being the adjoint of . This choice is in part motivated by the fact [28] that any quantum channel is non-expansive with respect to the metric induced 4 by · 1 (the same property does not necessarily apply to other operator norms, e.g. the Hilbert-Schmidt norm, also when these are equivalent to · 1 ).
We start by showing that the mixing criteria discussed in the first half of the paper do apply to the case of quantum channels. Then we will analyse these maps by studying their linear extensions in the whole vector space formed by the linear operators of H. Similar questions also arise in the context of finitely correlated states, where one investigates the decay of correlations in space (rather than in time) [34].

Mixing criteria for quantum channels
Let H be a finite dimensional Hilbert space and let S(H) be the set of its density matrices ρ. The latter is a convex and compact subspace of the larger normed vector space L(H) made up of the linear operators : H → H of H. From this and from the fact that CPT maps are continuous (indeed they are linear) it follows that for a quantum channel there always exists at least one density operator which is a fixed point [13]. It also follows that all the results of the previous section apply to quantum channels. In particular lemma 1 holds implying that any mixing quantum channel must be ergodic. The following example shows however that it is possible to have ergodic quantum channels which are not mixing.

Theorem 5 implies that a quantum channel τ : S(H) → S(H) is mixing if and only if
it is an asymptotic deformation. As already pointed out in the introduction, this property is metric independent (as opposed to contractivity). Alternatively, if the fixed point of a quantum channel is known, then one may use the trivial generalized Lyapunov function (theorem 4) to check if it is mixing. However both criteria depend on the metric distance, which usually has no easy physical interpretation. A more useful choice is the quantum relative entropy, which is defined as As discussed in [35], the quantum relative entropy is continuous in finite dimensions and can be used as a measure of 'distance' (though it is not a metric). It is finite if the support of ρ is contained in the support of σ. To ensure that it is a continuous function on a compact space, we consider the case when σ is faithful.

Theorem 6. (Relative entropy criterion). A quantum channel with faithful fixed point ρ * is mixing if and only if the quantum relative entropy with respect to ρ * is a generalized Lyapunov function.
Proof. Because of theorem 2 we only need to prove the second part of the thesis, i.e. that mixing channels admit the quantum relative entropy with respect to the fixed point, S(ρ) ≡ H(ρ, ρ * ), as generalized Lyapunov function. Firstly notice that the quantum relative entropy is monotonic under quantum channels [36]. Therefore the limit S * (ρ) ≡ lim n→∞ S (τ n (ρ)) does exist and satisfies the condition S * (ρ) S(ρ). Suppose now there exists a ρ such that S * (ρ) = S(ρ). Because τ is mixing and S is continuous we have and hence H(ρ, ρ * ) = 0. Since H(ρ, σ) = 0 if and only if ρ = σ it follows that S is a Lyapunov function around ρ * .

Corollary 2. (Unital channels). A unital channel is mixing if and only if the von Neumann entropy is a generalized Lyapunov function.
Another sufficient condition for mixing is weak contractivity. As already mentioned in the previous section, unfortunately this not a necessary condition. Here we present an explicit counterexample based on a quantum channel introduced in [13].

Example 2. Consider a three-level quantum system characterized by the orthogonal vectors
|0 , |1 , |2 and the quantum channel τ defined by the transformations τ(|2 2|) = |1 1|, τ(|1 1|) = τ(|0 0|) = |0 0|, and τ(|i j|) = 0 for all i = j. It's easy to verify that after just two iterations any input state ρ will be transformed into the vector |0 0|. Therefore the map is mixing. On the other hand it is explicitly not a weak contraction with respect to the trace norm since, for instance, one has where in the last identity we used the invariance of · 1 with respect to unitary transformations.

Beyond the density matrix operator space: spectral properties
Exploiting linearity quantum channels can be extended beyond the space S(H) of density operators to become maps defined on the full vector space L(H) of the linear operators of the system, in which basic linear algebra results hold. This allows one to simplify the analysis even though the mixing property (1) is still defined with respect to the density operators of the system.
Mixing conditions for quantum channels can be obtained by considering the structure of their eigenvectors in the extended space L(H). For example, it is easily shown that the spectral radius [37] of any quantum channel is equal to unity [13], so its eigenvalues are contained in the unit circle. The eigenvalues λ on the unit circle (i.e. |λ| = 1) are referred to as peripheral eigenvalues. Also, as already mentioned, since S(H) is compact and convex, CPT maps have always at least one fixed point which is a density matrix [13]. A well-known connection between the mixing properties and the spectrum is given by the following.

Theorem 7. (Spectral gap criterion). A quantum channel is mixing if and only if its only peripheral eigenvalue is 1 and this eigenvalue is simple.
Proof. The 'if' direction can be found in linear algebra textbooks (see for example [37, lemma 8.2.7] Now let us assume that τ is a mixing quantum channel with fixed point ρ * . Let be a generic operator in L(H). Then can be decomposed in a finite set of non-orthogonal density operators 5 If there existed any other eigenvector * of τ with eigenvalue on the unit circle, then lim n→∞ τ n ( * ) would not satisfy equation (12).
The speed of convergence can also be estimated by [13] τ n (ρ) − ρ * 1 C N n N κ n , (13) where N is the dimensionality of the underlying Hilbert space, κ is the modulus of the second largest eigenvalue of τ, and C N is some constant depending only on N and on the chosen norm. Hence, for n N the convergence becomes exponentially fast. As mentioned in [20], the criterion of theorem 7 is in general difficult to check. This is because one has to find all eigenvalues of the quantum channel, which is hard especially in the high dimensional case. Also, if one only wants to check if a particular channel is mixing or not, then the amount of information obtained is much higher than the required amount.

Example 3.
As an application consider the non mixing CPT map of example 1. One can verify that apart from the eigenvalue 1 associated with its fixed point (i.e. the completely mixed state), it possesses another peripheral eigenvalue. This is λ = −1 which is associated with the Pauli operator |0 0| − |1 1|.

Corollary 3. The convergence speed of any mixing quantum channel is exponentially fast for sufficiently high values of n.
Proof. From theorem 7 mixing channels have only one peripheral eigenvalue and it is simple. Therefore the derivation of [13] applies and equation (13) holds.
This result should be compared with the case of strictly contractive quantum channels whose convergence were shown to be exponentially fast along to whole trajectory [20,22].

Ergodic channels with pure fixed points
An interesting class of ergodic quantum channel is formed by those CPT maps whose fixed point is a pure density matrix. Among them we find for instance the maps employed in the communication protocols of [16]- [19] or those of the purification schemes of [14,15]. We now show that within this particular class, ergodicity and mixing are indeed equivalent properties. Theorem 8. (Purely ergodic maps). Let |ψ 1 ψ 1 | be the pure fixed point of an ergodic quantum channel τ. It follows that τ is mixing.
Proof. We will use theorem 7 showing that |ψ 1 ψ 1 | is the only eigenvector of τ with peripheral eigenvalue. Assume in fact that ∈ L(H) is a eigenvector of τ with peripheral eigenvalue, i.e.
Introducing a Kraus set {K n } n of τ and employing the Cauchy-Schwartz inequality one can then write where we used the fact that |ψ 1 is the fixed point of τ. Since τ is CPT the quantity ψ 2 |τ(|ψ 2 ψ 2 |)|ψ 2 is upper bounded by 1. Therefore in the above expression the inequality must be replaced by an identity, i.e.
An application of the previous theorem is obtained as follows.  Thus there exists an orthonormal basis {|u k } k of H A ⊗H B diagonalizing simultaneously both U and ρ⊗|φ B φ|. We express the latter in this basis, i.e. ρ⊗|φ B φ| = k p k |u k u k |, and perform the partial trace over subsystem A to get Hence Tr A [|u k u k |] = |φ B φ| for all k, and |u k must be factorizing, If the factorizing eigenstate of U is unique, it follows that ρ = |ν ν| for some |ν and that τ is ergodic. By theorem 8 it then follows that τ is also mixing. If on the other hand there exists more than one factorizing eigenstate, then all states of the form of equation (16) correspond to a fixed point ρ k = |ν k ν k | and τ is neither ergodic nor mixing.
The case discussed in lemma 3 is a generalization of the CPT map discussed in [16] in the context of spin-chain communication. There H A and H B represented two distinct parts of a chain of spins coupled through Heisenberg-like interactions: the latter including the spins controlled by the receiver of the message, while the former accounting for all the remaining spins. Assuming the system to be initially in the ground state (i.e. all spin down), the sender (located at one of the extremes of the chain) encodes her/his quantum messages (i.e. qubits) into superpositions of spins excitations which will start propagating toward to receiver (located at the other extreme of the chain). In [16] it was shown that, by repetitively swapping the spins which are under her/his control with some ancillary spins prepared in the ground state, the receiver will be able to recover the transmitted messages. The key ingredient of such a result is the fact that by applying the swapping operations the receiver is indeed removing all the excitations (and therefore the corresponding encoded quantum information) from that part of the chain which is not directly accessible to him/her (i.e. the part represented by H A ). In its simplest version, the resulting transformation on H A can be described by equation (15) with U and M AB representing, respectively, the free evolution of the spins among two consecutive swaps and the z-component of the magnetization of the chain. Lemma 3 can then be used to provide an alternative proof of convergence of the protocol [16] showing that indeed repetitive applications of τ will drive H A toward a unique convergence point (i.e. the state with no excitation).

Conclusion
In reviewing some known results on the mixing property of continuous maps, we derived a stronger version of the direct Lyapunov method. For compact metric spaces (including quantum channels operating over density matrices) it provides a necessary and sufficient condition for mixing. Moreover it allows us to prove that asymptotic deformations with at least one fixed point must be mixing.
In the specific context of quantum channels we employed the generalized Lyapunov method to analyse the mixing properties. Here we also analysed different mixing criteria. In particular we have shown that an ergodic quantum channel with a pure fixed point is also mixing.