Ergodic and Mixing Quantum Channels in Finite Dimensions

The paper provides a systematic characterization of quantum ergodic and mixing channels in finite dimensions and a discussion of their structural properties. In particular, we discuss ergodicity in the general case where the fixed point of the channel is not a full-rank (faithful) density matrix. Notably, we show that ergodicity is stable under randomizations, namely that every random mixture of an ergodic channel with a generic channel is still ergodic. In addition, we prove several conditions under which ergodicity can be promoted to the stronger property of mixing. Finally, exploiting a suitable correspondence between quantum channels and generators of quantum dynamical semigroups, we extend our results to the realm of continuous-time quantum evolutions, providing a characterization of ergodic Lindblad generators and showing that they are dense in the set of all possible generators.


Introduction
In the study of signal processing, a stochastic process is said to be ergodic if its statistical properties can be deduced from a single, sufficiently long realization of the process. Ergodicity plays a fundamental role in the study of the irreversible dynamics associated with the relaxation to the thermal equilibrium [1][2][3][4][5][6][7][8]. In quantum mechanics the evolution of an open system, interacting with an external (initially uncorrelated) environment, is fully characterized in terms of special linear maps M (known as quantum channels) operating on the space of density matrices ρ of the system of interest, under certain structural constraints (quantum channels and all their extensions should preserve the positivity and the trace of the operators on which they act upon, see e.g. [9][10][11][12]). In this framework the rigorous definition of quantum ergodic channels can be traced back to a series of works that appeared in the late 1970s, which set the proper mathematical background and clarified the main aspects of the problem. For a review on the subject see e.g. [5][6][7]13]. A convenient definition of ergodic quantum channels can be given as follows: a quantum channel M is ergodic if and only if it admits a unique fixed point in the space of density matrices, that is, if there is only one density matrix ρ * that is unaltered by the action of M [1,[14][15][16]. The rationale behind such formulation is clear when we consider the discrete trajectories associated with the evolution of a generic input state ρ, evolving under iterated applications of the transformation M: in this case, the mean value of a generic observable A, averaged over the trajectories, converges asymptotically to the expectation value Tr[Aρ * ] of the observable on the fixed point [5,17,18] 7 . A related, but stronger property of an ergodic quantum channel, is the ability of transforming a generic input state into the fixed point ρ * after a sufficiently large number of repeated applications. In the context of dynamical semigroups [13] this property is often called relaxing, here instead we follow the notation of [14] and dubbed it mixing.
Beyond the study of relaxation processes, ergodicity and mixing have found important applications in several fields of quantum information theory. Most notably in quantum control [20][21][22][23][24][25][26], quantum estimation [27], quantum communication [28,29] and in the study of efficient tensorial representation of critical many-body quantum systems [30][31][32][33][34][35]. A detailed analysis of ergodicity in the quantum domain appears hence to be mandatory. Here we contribute to this goal by providing a systematic characterization of the structural properties of ergodic and mixing channels on finite-dimensional quantum systems. The results are presented in a systematic way, starting from the case of general channels and then specializing to particular classes of channels (such as random-unitary channels). For completeness we also include some alternative proofs of existing results, which come out quite naturally in our approach and which, unlike most results in the previous literature, are derived without making explicit use of the faithfulness of the fixed point unless such assumption is strictly necessary. This allows us to guide the reader through some key aspects of the structure of ergodic and mixing channels, presenting both new and old results in a compact, self-contained form (still providing, whenever possible, proper references to the original works). A further advantage of our presentation with respect to the original literature of the 1970s is that here all proofs are elementary, due to our focus on finite dimensions, and often what is presented here as a single theorem (with a rather straightforward proof) was the subject of a full paper in the original von Neumann algebraic setting, thus preventing a comprehensive overlook on the subject.
Summary of the main results. The paper starts with a characterization of ergodicity for channels without a faithful fixed point, provided in theorem 1: a quantum channel is ergodic if and only if it possesses a minimal invariant subspace (a subspace of the Hilbert space that is left invariant by the action of the channel and does not contain any smaller, non-trivial subspace with the same property). Later, this characterization is used to prove a fundamental property of ergodic channels, namely the fact that ergodicity is stable under randomizations (theorem 4). Precisely, we show that a convex combination of an ergodic channel with a generic (not necessarily ergodic) channel yields a new transformation which is always ergodic. In addition, we show that the fixed point of the new ergodic map is related with the fixed point of the original ergodic channel via a convex combination. These results extend to the case of ergodic channels a property that was previously known to hold for the restricted subset of mixing channels [21,22,36]. In addition, we provide a series of conditions under which ergodicity can be upgraded to the stronger property of mixing: for example, we show that, rather counterintuitively, convex combinations of ergodic channels with the identity channel are always mixing. Another remarkable property is that a random-unitary channel M (a channel that is a convex combination of unitaries) in dimension d is mixing if and only if the channel M d obtained by applying M on the system d times is ergodic. Finally, we discuss the case of quantum dynamical semigroups [13]. In this context we show that the ergodicity of a quantum dynamical semigroup is equivalent to the ergodicity of a suitable quantum channel (theorem 14) and, exploiting the properties of the latter, we prove a stability property under randomization of the generators of ergodic quantum semigroups (theorem 15).
The paper is organized as follows. We start in section 2 by reviewing some basic definitions and properties. In section 3 we provide a characterization of ergodicity in terms of the invariant subspaces of the channel and discuss how ergodicity and mixing properties of the latter are connected with analogous properties of its adjoint map. The convexity properties of ergodic channel are analysed in section 4. Here we provide two alternative proofs of the stability of ergodicity under randomization, and analysing the relations between ergodicity and mixing, we show that a generic convex combination of the identity map with an ergodic channel is mixing. Section 5 is instead specialized on the case of channels which admit faithful fixed points, providing a characterization of the peripheral spectrum of the maps and introducing a necessary and sufficient condition which ergodic maps have to fulfil in order to be mixing. In section 6 we discuss the case of continuous-time dynamical semigroups, providing a characterization of ergodicity and using it to prove the stability property of ergodic semigroups under randomizations. Final remarks are presented in section 7. The paper also contains an appendix dedicated to the more technical aspects of the proofs.

Definitions and basic properties
Consider a quantum system with associated Hilbert space H of finite dimension d < ∞. In what follows we will use the symbols B(H) and S(H) [⊂B(H)] to represent the set of linear operators on H and the set of the density matrices, respectively. A quantum channel operating on the system is then defined as a linear mapping M : B(H) → B(H) which is completely positive and trace-preserving (CPTP). While referring the reader to [9][10][11][12] for an exhaustive review on the subject, we find it useful to recall a few basic properties of CPTP maps, which will be exploited in the following.
(ii) The set C(H) formed by the quantum channels on the system is closed under convex combination and multiplication, i.e. given M 1 , M 2 ∈ C(H) and p ∈ [0, 1], the transformations defined by the mappings are also elements of C(H).
(iii) Any CPTP map M is non-expansive: when M is applied to a couple of input states ρ, σ ∈ S(H), it produces output density matrices M(ρ), M(σ) whose relative distance is not greater than the original one, i.e. where Since M is a linear operator defined on a linear space of dimension d 2 , it admits up to d 2 distinct (complex) eigenvalues λ which solve the equation for some A ∈ B(H), A = 0. Such eigenvalues can be determined as the zeros of the associated characteristic polynomial, i.e.
Poly (M) where {M i } i∈X is a set of Kraus operators for M andM i is the operator obtained by taking the entry-wise complex conjugate of M i with respect to a selected basis of H.
(iv) The spectrum of M is invariant under complex conjugation: if λ ∈ C is an eigenvalue with eigenvector A, then its complex conjugateλ is an eigenvalue with eigenvector given by the adjoint operator A † , namely withλ being the complex conjugate of λ and A † being the adjoint of A. This property holds not only for quantum channels, but also for all linear maps that are Hermitian-preserving (that is, they send Hermitian operators to Hermitian operators).
(v) The eigenvalues of a CPTP map M are confined in the unit circle on the complex plane. In other words, if there exists a non-zero A ∈ B(H) such that (6) holds, then we must have |λ| 1 (this property is indeed a direct consequence of (iii)).
The eigenvalues of M which lie at the boundary of the permitted region, i.e. which have unit modulus |λ| = 1, are called peripheral. Of particular interest for us is the unit eigenvalue λ = 1: the associated eigenvectors A ∈ B(H) are called fixed points of M to stress the fact that they are left unchanged by the action of the map M.
(vi) Every CPTP map admits at least one fixed point state, i.e. a solution of (6) for λ = 1, which belongs to the set S(H) of the density matrices of the system.
As a matter of fact, a generic quantum channel possesses more than just one density matrix that fulfils the requirement (vi) (for instance unitary transformations admit infinitely many fixed point states). In the rest of the paper however we will focus on the special subset of CPTP maps which have exactly a single element of S(H) that is stable under the transformation: Definition 1. A CPTP map M is said to be ergodic if there exists a unique state ρ * ∈ S(H) which is left unchanged by the channel M, i.e. which solves (6) for λ = 1. We introduce the symbol C E (H) to represent the subset containing all the ergodic elements of C(H).
As it will be explicitly shown in the next section (see corollary 2), the ergodicity is strong enough to guarantee that ρ * is not only the unique λ = 1 solution of (6) on S(H) but also (up to a multiplicative factor) the only solution for the same problem in the larger set of the operators B(H). An equivalent (and possibly more intuitive) way to define ergodicity can be obtained by posing a constraint on the effective discrete-time evolution generated by the repetitive applications of M [5,7,[17][18][19]. Specifically, given ρ ∈ S(H) a generic input state of the system, consider the series (M) where M 0 = I stands for the identity superoperator, while for n 1, M n is a CPTP map associated with n recursive applications of M, i.e.
(see e.g. [14,37] for an explicit proof of this fact). Equation (9) Accordingly, this implies that for ergodic channels the average of the expectation values A proper subset of C E (H) is constituted by mixing/relaxing maps [14]: We call the special state ρ * the fixed point state of M and introduce the symbol C M (H) to represent the subset containing all the mixing elements of C(H).
As anticipated all mixing maps are ergodic (with their fixed point states provided by the stable density matrices of the mixing channels), i.e. C M (H) ⊂ C E (H). Notably however the opposite is not true [14]: for instance the qubit channel is ergodic with fixed point state ρ * = (|0 0| + |1 1|)/2, but it does not fulfil the mixing condition (13) (in fact, M n (|0 0|) keeps oscillating between |0 0| and |1 1|). Interestingly in the case of continuous time Markovian evolution, mixing and ergodicity are equivalent (see section 6), which means that the above channel cannot be obtained as the result of a Markovian time evolution (cf [38]).
The mixing property of a channel M can be described in terms of its spectral properties.
Indeed a necessary and sufficient condition for mixing is the fact that (up to a multiplicative factor) the fixed point state ρ * of M is the unique peripheral eigenvector of M. More specifically a channel is mixing if and only if it is ergodic and no solutions exists in B(H) for the eigenvalue equation (6) with both |λ| = 1 and λ = 1.

Remark 1.
Requiring λ = 1 to be the only peripheral eigenvalue of the channel is not sufficient to enforce the mixing property (or even ergodicity). As a counterexample consider for instance the case of the identity channel I. If however λ = 1 is the only peripheral eigenvalue and has a multiplicity one, then the channel is mixing (see e.g. [12]).

Characterization of ergodicity in terms of invariant subspaces
In this section we present a characterization of the ergodicity of a channel M in terms of the subspaces that are left invariant by its action. We start in section 3.1 with the discussion on the linear space generated by the fixed points. This introductory subsection collects some facts outlined in the seminal works by Davies [19], Morozova andČencov [18] and Evans and Høegh-Krohn [17]. Building on these facts, we give a characterization of ergodicity for general channels (without the assumption that the fixed point be a faithful state): a channel is ergodic if and only if it has a minimal invariant subspace (section 3.2). An equivalent characterization is provided in section 3.3, stating that a channel is ergodic if and only if its adjoint map has a maximal invariant subspace.

The linear space spanned by fixed points
The linear subspace of B(H) generated by the fixed points of a channel M can be shown to be spanned by positive operators. This fact can be easily established for instance by exploiting the following property (see e.g. theorem 7.5 of [18] or proposition 6.8 of [12]): Proof. The proof is provided in the appendix.
From this it immediately follows that non-ergodic channels always admit at least two fixed point states that are 'not overlapping': can be expressed as Proof. If all the solutions of (15) can be expressed as (16), then the map M is clearly ergodic, the converse instead follows by contradiction from lemma 1.
It is worth noticing that an alternative proof of corollary 2 can also be obtained from lemma 6 of [14], which states that if A is a peripheral eigenvector of a channel M then |A| := √ A † A must be a fixed point of M (see the appendix for details).

Fixed points and invariant subspaces
Here we link the ergodicity property of a channel to the structure of its invariant subspaces, that is, of those subspaces of H that are left invariant by the action of the Kraus operators of the channel. The study of invariant subspaces and their relation to fixed points was previously used in [39] as a tool to engineer stable discrete-time quantum dynamics and in [40] as a tool to characterize the algebraic structure of the fixed points of a given quantum channel. It is worth reminding that the condition that S is an invariant subspace under M can be equivalently expressed by the following properties: (a) M i P = P M i P for every i ∈ X, where P is the projector on S; The invariant subspaces of a channel are related to its fixed point states in the following way.
Consider then the case in which ρ is a fixed point state for M, i.e. M(ρ) = ρ. Equation (17) then implies that Supp[M(|ϕ ϕ|)] ⊆ Supp(ρ) for every |ϕ ∈ Supp(ρ), namely that Supp(ρ) is an invariant subspace. Conversely, let S be an invariant subspace for M. Then the restriction of M to S is a channel in C(S), and, as such, has a fixed point ρ S 0 with Supp(ρ S ) ⊆ S.
Using the relation between fixed point states and invariant subspaces we can obtain a first characterization of ergodicity in terms of the invariant subspaces of the channel:   Proof. If all the unitaries U i commute, they can be jointly diagonalized, and every joint eigenvector is a fixed point, implying that the channel is not ergodic. Conversely, if at least two unitaries of the set do not commute, the only invariant subspace S = 0 is S = H. Hence, S is clearly minimal and theorem 1 guarantees that M is ergodic.

Remark 2.
We remind that in the case of qubit channels the set of random-unitaries coincides with the set of unital maps (i.e. with the set of CPTP maps which admit the identity operator as fixed point) [41]. Corollary 3 hence provides a complete characterization of ergodicity for qubit unital maps. A (partial) generalization of this result to the case of unital maps operating on higher dimensional Hilbert spaces is given in section 5.1.

Ergodicity in terms of the adjoint map
A fully equivalent description of the ergodicity property of a quantum channel M can be obtained by considering its adjoint map M † . As a matter of fact the original works on ergodicity for quantum stochastic processes were mostly discussed in this context, the ergodicity of the original channels been typically discussed as a derived property, see e.g. [5,[17][18][19].
where A, B := Tr[A † B] is the Hilbert-Schmidt product.
Going from a quantum channel to its adjoint is the same as going from the Schrödinger As a consequence, the spectra of the two maps are identical, i.e. they share the same eigenvalues. Indeed, (22) implies that the characteristic polynomials (7) of the two maps coincide up to complex conjugation, i.e.
Since the spectrum of a completely positive map is invariant under complex conjugation (cf property (iv) in section 2), this proves that M and M † have the same spectrum.
In view of the above result it makes sense to extend the definition of ergodicity and mixing also for the adjoints of CPTP channels: Proof. Consider the quantum channels (M) N defined in (9). As discussed in section 2, if M is ergodic with fixed point ρ * , then for each bounded operator A and for all density matrices ρ we must have By linearity the adjoint channel of (M) N is given by ( In this case (24) can be written as which, to be true for all ρ, implies A = A, ρ * I . Conversely suppose that all constants of motion of M are proportional to the identity. Assume then by contradiction that M is not ergodic. By corollary 1 we know that M must have two orthogonal invariant states ρ 0 and ρ 1 .
Take then two orthogonal projectors P 0 = 0 and P 1 = 0 such that P i , ρ j = δ i j , and define the operators P i,∞ := lim N →∞ . One can easily verify that P 0,∞ and P 1,∞ are constants of motion of M (indeed they verify the identities M † (P i,∞ ) = P i,∞ ). However since ρ 0 and ρ 1 are fixed point states of the map, we also have P i,∞ , ρ j = lim N →∞ P i , (M) N (ρ j ) = P i , ρ j = δ i j , which is in contradiction to the fact that P 0,∞ and P 1,∞ should be multiples of the identity. Proof. Recall that M † and M share the same spectrum. Having M mixing implies that λ = 1 is the unique peripheral eigenvalue of M † . The first implication then follows from theorem 2 by recalling that any mixing channel M is also ergodic. Conversely, if M † admits the identity operator as unique eigenvector associated with peripheral eigenvalues then by theorem 2 M is ergodic and no other peripheral eigenvalues can exist, i.e. it is mixing.
We have already noticed that the spectrum of a quantum channel M coincides with that of its adjoint M † . Here we strengthen this result by showing that, at least for peripheral eigenvalues, there is a simple relation which connects the associated eigenvectors: where we have used the fact that ρ * is a fixed point state for M, the cyclicity of the trace, and the Cauchy-Schwarz inequality. Since the equality can only be obtained when the Cauchy-Schwarz inequality is saturated, we must have ωM i Aρ 1/2 on the right and summing over i, we obtain the identity ωM( = Aρ * , which shows thatÃ = Aρ * is indeed the eigenoperator of M belonging to the eigenvalueω. Conversely, suppose that M has a strictly positive fixed point ρ * > 0 and that M(Ã) =ωÃ. Defining A =Ãρ −1 * we have Again, to attain the equality in the Cauchy-Schwarz inequality, we must have It is worth noticing that, for ω Using this fact, the characterization of theorem 1 becomes

Ergodicity and mixing under randomization
It is known that when we prepare a (non-trivial) convex combination of a mixing channel with a generic (not necessarily mixing) quantum channel, the resulting transformation is also mixing [21,22,36]. This implies that mixing channels are stable under randomization, or in a more formal language, that C M (H) constitutes a convex subset, which is dense in C(H).
Aim of this section is to extend this analysis showing that the same property holds for the larger set of ergodic channels C E (H) (notice that this fact cannot be established by simple geometric arguments based upon the fact that C E (H) includes C M (H)). We also prove a rather counterintuitive fact, namely that the mere action of randomizing an ergodic (not necessarily mixing) channel with the identity mapping is capable of introducing mixing into the system.

Theorem 4 (Stability of ergodicity under randomization
is also ergodic. Moreover, denoting by ρ * and ρ * , p the fixed point states of M and M p , respectively, we have that for some probability π p ∈ (0, 1] and for some state σ p ∈ S(H). where Proof. If (31) holds then M is clearly ergodic. Indeed if this is not the case M must have at least two different fixed points ρ * = ρ * , which verify the identity contradicting the assumption. Conversely, assume M ∈ C E (H). For any pair of states ρ = ρ and for any N > 0 and λ > 0, we have where the first inequality follows from the triangular inequality of the trace distance, while showing that it can be transformed into an equality if and only if µ 1 = 1. Replacing this into the parallelism constraint we can conclude that the upper bound of (34) can be saturated if and only if M(ρ − ρ ) = ρ − ρ . However, since M is ergodic, from corollary 2 we must have ρ − ρ = ρ * Tr[ρ − ρ ] = 0, which contradicts the assumption ρ = ρ . Therefore, the upper bound of (34), i.e. of (31), cannot be saturated for any pair of states ρ = ρ .
We are now in a position to present our alternative proof that M p of (29) is ergodic. Proof of theorem 4. Note that for every integer n one can write M n p = p n M n + (1 − p n )S n , where S n is a CPTP map. Therefore, for N > 0 and λ > 0 we can invoke the triangular inequality of the trace norm to state that Since M is ergodic, we can use corollary 5 to bound the first term as follows: On the contrary using again the triangular inequality and the non-expansiveness of S n we have N n=0 Substituting this and (37) in (36) we arrive at which according to corollary 5 is sufficient to claim the ergodicity of M p .

From ergodicity to mixing
Once established that a channel M is ergodic, one may further ask whether it is also mixing. To answer to this question we have to study the peripheral eigenvalues of M and to see whether or not ω = 1 is the only peripheral eigenvalue. A useful observation in this direction is the following results which provide a refinement of lemma 6 of [14]: Proof. The thesis immediately follows by observing that if (40) holds then we have where in the second to last passage we have used the fact that |A| is a fixed point of M.
Quite interestingly, when considering peripheral eigenvalues of a trace-preserving CP map the sufficient condition of lemma 5 can be transformed into a necessary one: Since the Cauchy-Schwarz inequality is saturated, we necessarily have for every i (in turn, this is equivalent to (42)). Hence, we have namely |A| is a fixed point of M. The converse is just the statement of lemma 5. Finally, when M is ergodic, the fixed point |A| must be proportional to ρ * , by corollary 2.
Remark 3. If M is ergodic and ω = 1 then corollary 2 implies that the unitary U in theorem 5 must be the identity operator.

Remark 4.
Note that theorem 5 contains the statement that |A| is a fixed point of M whenever A is an eigenvector of M for some peripheral eigenvalue ω with |ω| = 1. This is the statement of lemma 6 of [14], of which theorem 5 provides an alternative derivation.
Theorem 5 implies a rather counterintuitive fact: whenever we mix an ergodic channel with the identity channel we necessarily obtain a mixing channel! Corollary 6 (The mixture of an ergodic channel and the identity is mixing). Let M∈C(H) be an ergodic channel. If the linear span Span C {M i } i∈X contains the identity, then M is mixing. In particular, if M is an ergodic channel and I is the identity channel, then is mixing for every p ∈ (0, 1). Proof. If ω = e iβ is a generic peripheral eigenvalue of M then by construction its associated eigenoperator A must satisfy the relation

Proof. Let
where we have used the fact that p = 0. This in particular implies that A must also be an eigenoperator of M belonging to an eigenvalue λ = [ω − (1 − p)]/ p. Since M is CPTP, however, we must have |λ| 1, i.e.
which, since p = 1, can only be true if β = 0, i.e. ω = 1 and λ = 1. Moreover in case M is ergodic we can invoke corollary 2 to claim that A must be proportional to its fixed point state ρ * . It then follows that (up to a proportionality constant) the only peripheral eigenvector of M coincides with ρ * : the channel is hence mixing, with its stable point being ρ * .
We conclude the section by observing that theorem 5 allows one to give an alternative proof of the well-known fact about the stability of the property of mixing under convex randomizations: where Q p is the projector on the support of ρ p, * . Now, since by theorem 4 the support of ρ * is contained in the support of ρ p, * , we have Q p ρ * = ρ * , and therefore, Since there is a Kraus form for M p that includes all the Kraus operators of M, this implies in particular M i Uρ * = ωU M i ρ * , ∀i ∈ X. From the fact that M is mixing we conclude that ω = 1. Hence, M p is mixing.

Ergodicity and mixing for channels with faithful fixed point
The characterization of ergodic channels, provided by theorem 1, becomes more specific in the case of channels with a faithful fixed point, namely a fixed point state ρ * with Supp(ρ * ) = H, or, equivalently ρ * > 0. Ergodic channels with faithful fixed point are also known as irreducible quantum channels [12,19] (in the same references, mixing channels with faithful fixed point state are referred to as primitive). As already mentioned in the introduction, the majority of the results obtained in the field were explicitly derived for this specific maps (see e.g. [5,8]).

Theorem 7 (Ergodicity and proper invariant subspaces). If a channel
N ∈ N} spans the whole Hilbert space. Further information about the structure of an ergodic map with faithful fixed point M can be extracted from the analysis of its peripheral eigenvalues and eigenvectors. This analysis is the subject of a quantum generalization of the Perron-Frobenius theory of classical Markov chains-see e.g. [5,6,30,42,43]. The following theorem summarizes some of these results and takes advantage of the complete positivity of quantum channels to give a convenient condition for the peripheral eigenvectors in terms of the Kraus operators:  A ω = U ω ρ * for some unitary U ω such that M i U ω ρ * = ωU ω M i ρ * for every i ∈ X. Since ρ * is invertible, this condition is equivalent to (50). (ii) Equation (50) allows us to show that if ω is an eigenvalue with eigenvector A ω = U ω ρ * , then also its inverseω is an eigenvalue of M, with eigenvector Aω := U † ω ρ * . In addition, if ω 1 and ω 2 are two eigenvalues, with unitaries U ω 1 and U ω 2 , respectively, then also ω 1 ω 2 is an eigenvalue, with unitary U ω 1 U ω 2 (multiplicative rules). This proves that the eigenvalues of M must form a group F. Clearly, the group has order |F| d 2 , because the eigenvectors belong to the d 2 -dimensional vector space B(H). Moreover, since the peripheral eigenvalues lie on the unit circle, F must be a cyclic group, consisting of powers of some generator ω 1 = e 2πi/L for some integer number L d 2 .

Proof. (i) ⇔ (ii) By definition, A M is reducible if and only if it has a proper invariant subspace S ⊂ H. (i) ⇔ (iii) The span of the vectors {M
(iii) The multiplicative rules in the previous point imply that the operator A ω = U ω ρ * with U ω defined as in (50) is the only eigenvector of M with eigenvalue ω. Indeed if there were two of them, we would have two unitaries U ω and V ω satisfying (50). Therefore by the multiplicative properties discussed above also the operator B = V † ω U ω ρ * would be a fixed point for M. Since M is ergodic, we must have B = λρ * for some proportionality constant λ ∈ C. Hence, V ω = λU ω .
(iv) Let A ω 1 = U ω 1 ρ * be the eigenvector corresponding to the eigenvalue ω 1 := e 2π i/L . Then, by the multiplicative rules of point (i) it follows that A ω l := U l ω 1 ρ * is an eigenvector corresponding to the eigenvalue ω l := e 2π il/L , for every l ∈ N. Since the eigenvalues are non-degenerate, U l ω 1 ρ * is actually the eigenvector corresponding to the eigenvalue e 2πil/L (up to a multiplicative constant). In particular, since ω L = 1, we must have U L ω 1 = e iα I , for some phase α ∈ [0, 2π ). Now, without loss of generality the phase α can be chosen to be 0: indeed, we can always re-define U ω 1 to be U ω 1 := e iα/L U ω 1 . With this choice, the correspondence ω l → U ω l is a unitary representation of the group F.
(v) The thesis follows by noticing that ifω is a peripheral eigenvalue of M † with eigenvector B then lemma 3 implies that ω must be an eigenvalue of M with eigenvector Bρ * . Since ω is peripheral by construction and ρ * is faithful, we can prove the thesis by invoking point (i) to say that there must exist U ω unitary such that (up to a proportionality constant) Bρ * = U ω ρ * . But this immediately implies that B must be proportional to U ω .
(vi) The thesis follows by noticing that if ω is a peripheral eigenvalue of M then point (i) says that its eigenvector can be written as A ω = U ω ρ * (up to a proportionality constant) with U ω satisfying (50). From this it immediately follows thatω is an Note that the converse is guaranteed by lemma 3: if U ω is an eigenvector of M † with eigenvalueω, then U ω ρ * is an eigenvector of M with eigenvalue ω.
The fact that the peripheral eigenvalues of an ergodic channel with faithful fixed point are Lth roots of the unit for some L ∈ {1, . . . , d 2 } was known from [12,30]. However, the condition in terms of Kraus operators in theorem 9 allows us to prove a slightly stronger result, namely that the peripheral eigenvalues of an ergodic random-unitary channel are dth roots of the unit: Using the fact that the peripheral eigenvalues are roots of unit we obtain a characterization of mixing channels with a faithful fixed point. The interesting feature of this characterization is that it connects the two properties of mixing and ergodicity: Proof. If M is mixing, then also M k must be mixing for every k, and, therefore, ergodic. Conversely, suppose that M k is ergodic for all k d 2 and assume by contradiction that M is not mixing, namely that M has a peripheral eigenvalue ω = 1 for some eigenvector A ω which is not a multiple of ρ * . By theorem 9 it follows that there exists an L d 2 such that ω L = 1. Since M L (A ω ) = ω L A ω = A ω , this means that M L has two distinct fixed points A ω and ρ * , namely, it is not ergodic, in contradiction with the hypothesis.

Ergodicity and mixing for unital channels
Here we focus our attention on a particular type of channels, unital channels, which includes a large number of physically interesting examples, and allows for an even more specific characterization of ergodicity and mixing.
We remind that a channel M ∈ C(H) is called unital if and only if it preserves the identity, that is, if and only if M(I) = I . The easiest example of unital channels is given by the class of random-unitary channels, of the form M(ρ) = i p i U i ρU † i , with U i unitary operator and p i 0 for every i. Note, however, that there are many examples of unital channels that are not of the random-unitary form [41,44,45]: as a matter of fact, as anticipated in section 3.2 the two sets coincides only for qubit systems.
A first useful observation is that, in the case of peripheral eigenvalues, the eigenvectors of a unital channel are also eigenvectors of its adjoint: Moreover, the eigenspace of M corresponding to ω is spanned by partial isometries.
Proof. The proof is provided in the appendix.

From ergodicity to mixing in the case of unital channels
We now give a sufficient condition for mixing, which has a nice algebraic form and connects mixing with ergodicity. Unfortunately, in general this is only a sufficient condition. However, the condition is also necessary for a particular class of channels, here called diagonalizable channels.
Since M † M must be mixing, it must also be ergodic.
Note that the ergodicity of the square modulus is a necessary condition for mixing only in the case of diagonalizable channels. For example, consider the (non-diagonalizable) channel Specializing to random-unitary channels, theorem 13 becomes: Proof. The proof can be found in the appendix.

Corollary 9 (Ergodicity of the square modulus for random-unitary channels). A random-
The above proposition shows that Streater's condition is sufficient for mixing, and also necessary in the case of diagonalizable channels.

Ergodic semigroups and ergodic channels
Conditions for the existence and for the uniqueness of a fixed point for dynamical semigroups were the subject of an intense analysis at the end of the 1970s, see e.g. [13] and references therein. Various characterizations of irreducibility of the dynamical semigroup in terms of the Lindblad decomposition were given. However, we will not review these results here. Instead, we will give an alternative characterization of ergodicity of a semigroup in terms of ergodicity of a suitable quantum channel associated to the generator (theorem 14). This new characterization is useful because it allows one to translate the convexity property of ergodic channels (theorem 4) into a convexity property of dynamical semi-groups.

Ergodic channels as generators for mixing completely positive and trace-preserving semigroups
It has recently been pointed out [38] that any CPTP channel M can be used to induce a continuous CPTP semigroup evolution on S(H), described by a Markovian master equatioṅ ρ(t) = L (M) (ρ(t)), ∀t 0.
where γ > 0 is a constant, which scales the unit of time 8 whose properties may be rather different from those of the maps M n . Specific instances of dynamical semigroups of the form (55) have been analysed in [46] (see also [13]). One can easily verify that if M is ergodic, then L (M) admits a unique eigenvector associated with the null eigenvalue (up to a multiplicative factor), i.e.
where c is a complex number and ρ * is the fixed point state of M. Indeed from (55) it follows that the eigenvalues of L must be of the form µ = γ (λ − 1) with λ being the eigenvalues of M. The condition µ = 0 hence implies λ = 1, which according to lemma 1 is only possible if the eigenvector is of the form described in (57). Accordingly [4,7,47,48] in the limit of implying that (for t > 0) each of the maps T (M) t is mixing. As an example consider the case of the ergodic (but not mixing) qubit channel defined in (14). In this case for n 1 integer we have with M D (ρ) := 0|ρ|0 |0 0| + 1|ρ|1 |1 1| being the fully depolarizing channel. Therefore,

Necessary and sufficient condition for the ergodicity of a Lindblad generator
A generalization of the result discussed in the previous section to arbitrary Lindblad generators can be obtained by reversing the connection M → L (M) of (55). Specifically we will show that the ergodicity of a generic Lindblad generator L (and hence the asymptotic mixing property of its integrated trajectory e Lt ) is equivalent to the ergodicity of a suitable quantum channel M L which can be associated to L. To see this we recall that any L can always be written as where H = H † is an Hamiltonian operator and A is a completely positive (not necessarily trace preserving) map. Equation (64) can be conveniently rewritten as a difference of two completely positive maps T L (ρ) := 1 where the second term is invertible with completely positive inverse (the invertibility of T L being a direct consequence of the invertibility of the operator I + G, the latter following from the fact that A † (I ) is non-negative). The definition of the quantum channel M L is now obtained by observing that

Convexity of ergodic semigroups
Consider now two dynamical semigroups, with Lindblad generators L and L and define the Lindblad generator Assuming that L generates an ergodic semigroup, we may ask whether L p also generates an ergodic semigroup. At the end of section 6.1 we have already seen that this is indeed the case when (55) hold for both generators. Answering to the question for the general case is difficult. Still it is possible to provide a relatively simple answer when, cast in the form (64), the two semigroups have the same Hamiltonian and the same positive operators A † (I ) and A † (I ), that is, (incidentally this case covers also the scenario addressed in section 6.1).

Theorem 15 (Convexity of ergodic semigroups). Suppose that the condition of (71) is
satisfied and that L generates an ergodic semigroup. Then, L p generates an ergodic semigroup for every p ∈ (0, 1]. Proof. Under the condition of (71), we have and therefore, M L p = pM L + (1 − p)S L . Using theorems 4 and 14 we then obtain that L p is ergodic.
Because of condition (71), this is a weaker theorem than the corresponding theorem 4 for discrete channels. It is easy to see that generally, ergodicity is not stable under convex combination. As a simple example, consider the qubit Lindbladians L ± (ρ) = ±i[X, ρ] + (Zρ Z − ρ). As they contain dephasing to the Z-axis combined with a rotation around the X-axis, they are easily seen to be ergodic, with the centre of the Bloch sphere as fixed point. However, their midpoint convex combination L = (L + + L − )/2 is simply a dephasing map, which leaves the whole Z-axis invariant.
Despite being weaker than theorem 4, we can still conclude that the set of non-ergodic dynamical semigroups have measure zero. Note that the condition (71) is equivalent to G = G . We decompose the set of all Lindblad superoperators into convex sets with G = G . Now, each of this set contains at least one ergodic Lindblad superoperator: simply choose A(ρ) = ρ * Tr[ρ(G + G † )/2], where ρ * is faithful. The channel M associated to this Lindbladian is of the form M(ρ) = ρ * Tr[Pρ] + Q(ρ), where P is a positive operator and Q is a quantum operation.
A channel of this form is necessarily ergodic, due to our characterization theorem 1, because M cannot have two distinct invariant subspaces. Therefore, the only non-ergodic maps can be at the boundary of the sets with G = G .

Conclusions
We have discussed structural properties of quantum channels in finite dimensions, focusing on criteria for ergodicity and mixing. Because these notions are relevant to many protocols in quantum information processing, our characterization paves the way to simpler proofs of quantum convergence in those applications. One of our main results, i.e. the convexity of ergodicity, has potential applications for toy models [21,49] in quantum statistical dynamics, where the ergodicity of a given model is usually hard to establish. Since thermal states provide a natural convex decomposition, implying a convex decomposition of the corresponding map, our result implies that it suffices to establish ergodicity at zero temperature only.

A.1 Other Proofs
Here we conclude the appendix by providing the details of the proofs that had been skipped in the main text.
Proof of lemma 1. Since A is a fixed point, also A † is a fixed point (see (8)) and so are the linear combinations X := (A + A † )/2 and Y := (A − A † )/2i. Let us denote by P + (P − ) the projectors on the eigenspaces of X with non-negative (negative) eigenvalues and write X as In order for the equality to hold, it is necessary to have P + M(X − ) = P − M(X + ) = 0. Hence, we have X + = P + M(X + ) = M(X + ), which also implies X − = M(X − ). Repeating the same reasoning for Y = Y + − Y − we obtain that also Y + and Y − are fixed points of M.
Alternative proof of corollary 2. From lemma 6 of [14], we know that |A| := √ A † A must be a fixed point state of the map. If M is ergodic with fixed point state ρ * then we must have |A| = A 1 ρ * . Consider first the case in which A is Hermitian, i.e. A = A † (the non-Hermitian case will be considered below). In this case we can then write ρ * = n |c n ||φ n φ n |/ m |c m |, with {|φ n } being the orthonormal eigenvectors of A and c n the corresponding eigenvalues. Furthermore for each α real and satisfying the inequality 0 < |α| 1/ m |c m |, we can also conclude that the operator ρ * = α A + ρ * αTr[A] + 1 = Let us diagonalize |A| as |A| = s k=1 α k P k , where α 1 > α 2 > · · · > α s 0 are the eigenvalues, S k is the eigenspace corresponding to α k , and P k is the projector on S k . Then, for every unit vector ϕ ∈ S 1 we have where the inequality comes from the Cauchy-Schwarz inequality. To saturate the inequality (A.6) we need to have Clearly, in order for M to be trace-preserving we must have λ = 1, as λ 2 Moreover, since |ϕ is a generic element of S 1 , (A.9) with λ = 1 is equivalent to (U † M i U )P 1 = ωM i P 1 , ∀i. (A.10) Similarly, the relation M i |ϕ ∈ S 1 , ∀|ϕ ∈ S 1 , which was needed to saturate the inequality (A.8), is equivalent to M i P 1 = P 1 M i P 1 , ∀i. (A.11) Recalling that the eigenvalue equation M † (A) =ω A is equivalent to M(A) = ω A (corollary 6), we can use the same reasoning as above to prove also Putting together the two relations (A.11) and (A.12) we then obtain M i P 1 = P 1 M i , ∀i. Finally, defining the partial isometry T 1 := U P 1 we obtain Hence, we proved that the partial isometry T 1 must satisfy (51) in the statement of the theorem. In particular, we then have M(T 1 ) = ωT 1 . This means that for every peripheral eigenvalue ω, the channel M must have at least one eigenvector that is a partial isometry and satisfies (51). Moreover, defining the operator A := A − α 1 T 1 we have M(A ) = ω A .
The polar decomposition of A is A = U (|A| − α 1 P 1 ) = U s k=2 α k P k , so that the eigenspace of |A | with maximum eigenvalue is S 2 . Iterating the above proof we obtain that the partial isometry T 2 := U P 2 is an eigenvector of M satisfying (51), and by further iteration we obtain