Number of steady states of quantum evolutions

We prove sharp universal upper bounds on the number of linearly independent steady and asymptotic states of discrete- and continuous-time Markovian evolutions of open quantum systems. We show that the bounds depend only on the dimension of the system and not on the details of the dynamics. A comparison with similar bounds deriving from a recent spectral conjecture for Markovian evolutions is also provided.


Introduction
Spectral theory is still a hot topic in quantum mechanics.Indeed, quantum theory was developed at the beginning of the last century in order to explain the energy spectra of atoms [1].
In particular, the dynamics of a closed quantum system, namely isolated from its surroundings, is encoded in the eigenvalues (energy levels) of its Hamiltonian [2].Similarly, for an open quantum system under the Markovian approximation [3], studying the spectrum of the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) generator (the open-system analogue of the Hamiltonian) allows us to obtain information about the dynamics of the system [4].
In spite of this general interplay between spectrum and dynamics, a complete understanding of open-quantum-system evolutions still remains a formidable task.However, a more detailed analysis may be performed if we restrict our attention to the large-time dynamics of open systems.This amounts to study the steady and, more generally, asymptotic states towards which the evolution converges in the asymptotic limit.
Besides their theoretical importance, stationary states also play a central role in reservoir engineering [18][19][20], consisting of properly choosing the system-environment coupling for preparing a target quantum state, or in phase-locking and synchronization of quantum systems [21].
Moreover, GKLS generators with multiple steady states [22] may be used in order to drive a dissipative system into (degenerate) subspaces protected from noise [23] or decoherence [24], in which only a unitary evolution, related to purely imaginary eigenvalues of the generator [11], may be exploited for the realization of quantum gates [25][26][27].For this reason, the analysis of stationary states and, more generally, the study of the relaxation of an open quantum system towards the equilibrium is needed for applications in quantum information storage and processing [28][29][30][31].
The asymptotic properties of open quantum systems have also been deeply studied in quantum statistical mechanics.In particular, dissipative quantum phase transitions [32,33], as well as drivendissipative systems [34,35], require the study of the large-time dynamical behaviour of the system.More generally, determining the steady states of an open system sheds light on the transport properties of the system itself.In particular, the existence of discontinuities of the dimension of the steady-state manifold should correspond to a jump for the transport features of the system [36,37].
Finally, open-quantum-system asymptotics naturally emerges in quantum implementations of Hopfield-type attractor neural networks [38].Indeed, the stored memories of such type of network may be identified with the stationary states of its (non-unitary) evolution [39,40].
Despite the much effort devoted to the asymptotic dynamics of open quantum systems, general constraints for the number of steady and asymptotic states of quantum evolutions are still to be found, as far as we know.Besides the theoretical relevance of this problem, they may allow us to elucidate the potential of some of the above mentioned applications.
In this Article, we find sharp universal upper bounds on the number of linearly independent steady and asymptotic states of discrete-time and Markovian continuous-time quantum evolutions.Importantly, these bounds are only related to the dimension of the system and not on the properties of the dynamics.
The Article is organized as follows.After introducing some preliminary notions in Section 2, we will discuss our main results in Section 3, then we will provide explicit examples proving the sharpness of the bounds in Section 4. Subsequently, before proving the theorems in Section 6, our results will be compared with analogous bounds derived from a recent universal spectral conjecture proposed in [41] in Section 5. Finally, we will draw the conclusions of the work in Section 7.

Preliminaries
In the present Section we will recall some basic notions about evolutions of finite-dimensional open quantum systems, see also Section 6 for more details.
The state of an arbitrary d-level open quantum system is given by a density operator ρ, namely a positive semidefinite operator on a Hilbert space H (d = dim H) with Tr ρ = 1, whereas its dynamics in a given time interval [0, τ ] with τ > 0 is described by a quantum channel Φ, namely a completely positive trace-preserving map (a superoperator) on B(H), the space of linear operators on H [42].
If the system state at time t = 0 is ρ, its discrete-time evolution at time t = nτ , with n ∈ N, will be given by the action of the n-fold composition Φ n of the map Φ, namely, As the Hilbert space H is finite-dimensional, B(H) is isomorphic to the space of complex matrices of order d.We will indicate the space of d × d ′ matrices with complex entries by M d,d ′ (C) and, for the sake of simplicity, Let µ α , α = 0, . . ., n − 1, with n ⩽ d 2 be the distinct eigenvalues of Φ, namely with A α being an eigenoperator corresponding to µ α .The spectrum spect(Φ) is the set of eigenvalues of Φ.Let ℓ α be the algebraic multiplicity [43] of the eigenvalue µ α , so that n−1 α=0 ℓ α = d 2 .It is well known [12] that: i) the spectrum is contained in the unit disk, ii) 1 is always an eigenvalue, namely, iii) the spectrum is symmetric with respect to the real axis, i.e., iv) the unimodular or peripheral eigenvalues µ α ∈ ∂D, the boundary of D, are semisimple, i.e. their algebraic multiplicity ℓ α coincides with their geometric multiplicity.The eigenspace Fix(Φ) corresponding to µ 0 = 1, called the fixed-point space of Φ, is spanned by a set of ℓ 0 density operators, which are the steady (or stationary) states of the channel Φ.
Also, the space Attr(Φ) corresponding to the peripheral eigenvalues µ α ∈ ∂D is known as the asymptotic [44] or the attractor subspace [45,46] of the channel Φ, since the evolution Φ n (ρ) of any initial state ρ asymptotically moves towards this space for large times, i.e. as n → ∞, see Section 6 for more details.These limiting states may be called oscillating or asymptotic states, and it is always possible to construct a basis of such states for the subspace Attr(Φ), analogously to Fix(Φ).
Note that closed-system evolutions are described by a unitary channel and some unitary U .Importantly, a quantum channel is unitary if and only if spect(Φ) ⊆ ∂D, i.e. all its eigenvalues belong to the unit circle [12].The Markovian continuous-time evolution of an open quantum system is described by a quantum dynamical semigroup [8] ρ where the generator L takes the well-known GKLS form [47,48] L where the square (curly) brackets represent the (anti)commutator, H = H † is the system Hamiltonian, the noise operators A k are arbitrary, and the first and second terms L H and L D in Eq. ( 8) are called the Hamiltonian and dissipative parts of the generator, respectively.Notice that the GKLS form ( 8) is not unique and, in particular, so is the decomposition of L into Hamiltonian and dissipative contributions.L is called a Hamiltonian generator if L D = 0 for one (and hence all) GKLS representation (8).
If λ α , α = 0, . . ., m − 1, with m ⩽ d 2 , denote the distinct eigenvalues of L, from the GKLS form one obtains that λ 0 = 0 and, given an eigenoperator X 0 ⩾ 0 corresponding to this eigenvalue, then X 0 / Tr(X 0 ) is a steady state of Φ t = e tL [49].The kernel of L, i.e. the eigenspace corresponding to the zero eigenvalue, will be denoted by Ker(L).Moreover, with Γ α being the relaxation rates of L. These parameters, describing the relaxation properties of an open system [50], may be experimentally measured.A condition for the relaxation rates of a quantum dynamical semigroup, recently conjectured in [41] and which we will call Chruściński-Kimura-Kossakowski-Shishido (CKKS) bound, is recalled in Section 5 in order to investigate its relation with the main results of this work, stated in Section 3. Finally, note that the purely imaginary (peripheral) eigenvalues of L are semisimple and are related to the large-time dynamics of Φ t = e tL , as the space corresponding to such eigenvalues is the asymptotic manifold Attr(L) of the Markovian evolution, see Section 6 for details.Importantly, as for unitary channels, the generator L is Hamiltonian if and only if Γ α = 0 for all α = 0, . . ., m − 1, i.e. all its eigenvalues are peripheral.

Bounds on the dimensions of the asymptotic manifolds
In this Section we will present the main results of this work, whose proofs are postponed to Section 6.First, let us introduce the quantities involved in our findings.Remember that we denoted with µ α the α-th distinct eigenvalue of Φ and with ℓ α its algebraic multiplicity with α = 0, . . ., n − 1.In particular, ℓ 0 is the algebraic multiplicity of µ 0 = 1, and coincides with the dimension of its eigenspace, the steady-state manifold, i.e.
We define the peripheral multiplicity ℓ P of Φ as the sum of the multiplicities of all peripheral eigenvalues, which coincides with the dimension of the attractor subspace Attr(Φ), made up of asymptotic states.Namely, Physically, ℓ 0 and ℓ P are respectively the number of independent steady and asymptotic states of the evolution described by Φ. Analogously, denote with m α (α = 0, . . ., m − 1) the algebraic multiplicity of the α-th distinct eigenvalue λ α of the generator L of the continuous-time semigroup (7).In particular m 0 denotes the multiplicity of the zero eigenvalue λ 0 = 0, so that Moreover, the peripheral multiplicity m P of L is the sum of the multiplicities of its purely imaginary eigenvalues and measures the dimension of its attractor manifold: The integers m 0 and m P represent respectively the number of independent steady and asymptotic states of the Markovian evolution Φ t = e tL generated by L. Now we will provide sharp upper bounds on such multiplicities.Let us call a quantum channel non-trivial if it is different from the identity channel, Φ(ρ) = ρ.Theorem 1 (Unitary discrete-time evolution).Let Φ be a non-trivial unitary quantum channel on a d-dimensional system.Then the multiplicity ℓ 0 of the eigenvalue 1 and the peripheral multiplicity Theorem 2 (Non-unitary discrete-time evolution).Let Φ be a non-unitary quantum channel.Then the multiplicity ℓ 0 of the eigenvalue 1 and the peripheral multiplicity ℓ P of Φ obey The content of the latter result is schematically illustrated in Fig. 1.Now, it is possible to construct quantum channels with ℓ 0 and ℓ P attaining the equalities in Eqs. ( 14) and (15), namely all the upper bounds are sharp, see Section 4 for explicit examples.Obviously, for a trivial quantum channel ℓ 0 = ℓ P = d 2 , therefore the bounds ( 14) and ( 15) are not valid.
The above results, valid for discrete-time evolutions (1) are perfectly mirrored by the following results on Markovian continuous-time evolutions (7), with GKLS generators (8).
Theorem 3 (Hamiltonian generator).Let L be a non-zero Hamiltonian GKLS generator.Then the multiplicity m 0 of the zero eigenvalue and the peripheral multiplicity m P of L fulfill Theorem 4 (Non-Hamiltonian GKLS generator).Let L be a non-Hamiltonian GKLS generator.
Then the multiplicity m 0 of the zero eigenvalue and the peripheral multiplicity m P of L satisfy The bounds ( 16) and ( 17) are also sharp as the previous ones, see Section 4. Clearly, the two latter theorems do not apply to the zero operator because in such case m 0 = m P = d 2 .
Theorem 1 provides a tight universal upper bound on the number of linearly independent steady states of a (non-trivial) unitary quantum channel Φ, depending only on the dimension d of the system.Similarly, Theorem 2 shows that the number of linearly independent steady and asymptotic states of a non-unitary channel Φ is bounded from above by the same d-dependent quantity.Theorems 3 and 4 provide analogous constraints for non-zero Hamiltonian and non-Hamiltonian generators respectively and, indeed, Theorem 4 easily follows from Theorem 2, as shown in Section 6.
Interestingly, Theorem 4 implies that when we add to a Hamiltonian generator a dissipative part, no matter how small, the peripheral multiplicity m P undergoes a jump larger than the gap ∆ = 2(d − 1), (18) varying linearly with d.Consequently, we have ì forbidden values for m P .The same jump for the peripheral multiplicity ℓ P occurs when we pass from unitary channels to non-unitary ones according to the bound (15).

Sharpness of the bounds
In this Section we will prove the sharpness of the bounds stated in Theorems 1-4.Let us start with the proof of the sharpness of the bound ( 16) for non-zero Hamiltonian GKLS generators.If we take for some basis {|e i ⟩} d i=1 of H, then it is immediate to check that whence m 0 = (d − 1) 2 + 1 = d 2 − 2d + 2. Furthermore, if we require that the multiplicity ℓ 0 of the corresponding unitary channel Φ = e L attains the inequality in Eq. ( 14).Note that condition (22) guarantees that Φ is not trivial.
Let us now turn our attention to the sharpness of the bounds (17) for GKLS generators.Recall that the commutant S ′ of a system of operators S = {A k } M k=1 ⊂ B(H) is defined as Now consider the system S = {A k } N k=1 of diagonal operators with respect to the basis Here, λ 2 , are the eigenvalues of A k , and are the corresponding spectral projections, with I being the identity operator on H.Note that, by construction, the eigenvalues λ 2 have respectively multiplicities m 1 = 1, m 2 = d − 1 for all k = 1, . . ., N .Let us now take into account the generator for which L(I) = 0. We have Here, the second and fourth equalities follow respectively from Proposition 8 and Corollary 5.1 in Section 6, whereas the third one is a consequence of Eq. ( 24).Moreover, as L is non-Hamiltonian by construction, we necessarily have by Theorem 4. A quantum channel saturating the equalities in Eq. ( 15) is simply Φ = e L , with L given by Eq. ( 26).
In particular, we can construct a more physically transparent example of GKLS generator saturating the bounds ( 17) by taking S = {P 1 , P 2 }.The associated Markovian channel Φ acts as follows with respect to the basis with Therefore we realize that Φ is a phase-damping channel causing an exponential suppression of the coherences x 12 , . . ., x 1d ∈ X 12 , and we immediately see that it attains the equalities in the bounds (15), in line with the discussion above.

Relation with the Chruściński-Kimura-Kossakowski-Shishido bound
In this Section we will make a comparison between the bounds given in Theorems 1-4 and similar bounds arising from a recent spectral conjecture discussed in [41].As already noted in Section 1, the real parts of the eigenvalues λ α of a quantum dynamical semigroup Φ t = e tL with GKLS generator L are non-positive.However, it was recently conjectured in [41] that the relaxation rates Γ α = − Re(λ α ) are not arbitrary non-negative numbers, but they must obey the CKKS bound where m β is the algebraic multiplicity of λ β .This upper bound was not proved yet in general, but it holds for qubit systems, while for d ⩾ 3 it is valid for generators of unital semigroups, i.e. with L(I) = 0 , and for a class of generators obtained in the weak coupling limit [41] (see also [51] for further results).Also, it was experimentally demonstrated for two-level systems [52,53].
The CKKS bound (30) implies the following inequalities for non-Hamiltonian generators Indeed, summing Eq. ( 30) over the bulk, i.e. non-peripheral, eigenvalues of L yields where m B = Γα<0 m α is the number of the repeated eigenvalues in the bulk.If L is not Hamiltonian, viz.m B ̸ = 0 as noted in Section 2, this implies that namely the assertion.Interestingly, the CKKS bound (30) implies also the following bound on the real parts x α of the eigenvalues µ α of an arbitrary quantum channel Φ [41] n−1 β=0 where ℓ β is the algebraic multiplicity of µ β .Although Eq. ( 34) does not yield an upper bound similar to Eq. ( 31) for the peripheral multiplicity ℓ P of Φ, the multiplicity ℓ 0 of the eigenvalue µ 0 = 1, i.e. the number of steady states of Φ, satisfies if Φ is not trivial.The proof goes as follows: when ℓ 0 = d 2 we have the identity channel, so suppose Then from Eq. ( 34) one gets Now, the right-hand side of Eq. ( 36) is the arithmetic mean of the set therefore condition (36) is equivalent to require that all the elements of S exceed their arithmetic mean, which is true if and only if N = 0 and which concludes the proof of Eq. ( 35).
Furthermore, from Eq. ( 31) it follows that for non-unitary Markovian channels, viz. of the form Φ = e L with L non-Hamiltonian generator.Now let us compare the bounds (31), (35), and (39) arising from the CKKS conjecture (30) discussed in the present Section with the ones stated in Section 3. First, the upper bound (15) for ℓ P is also valid for non-Markovian channels, differently from the bound (39) and, in the Markovian case, it is stricter than Eq. ( 39) when d ̸ = 2. Analogously, the bound in Eq. ( 14) and the one for ℓ 0 in Eq. ( 15) boil down to Eq. ( 35) in the two-dimensional case, but they are stricter otherwise.
Similarly, the bounds (31) for m 0 and m P are not tight for all d ⩾ 3, whereas they are equivalent to condition (17) in the case d = 2. Consequently, the jump for m P is also predicted by the bound (31) but ∆ = d, which is loose for all d ̸ = 2.In conclusion, the bounds given in Theorems 1-4 imply the bounds (31), (35), and (39) deriving from the CKKS conjecture (30), in favor of the validity of the conjecture itself.

Proofs of Theorems 1-4
In this Section we will prove Theorems 1-4 stated in Section 3. To this purpose, let us recall several preliminary concepts, besides the ones introduced in Section 2.
First, given a quantum channel Φ, it always admits a Kraus representation [42], in terms of some operators {B k } N k=1 ⊂ B(H).Note that the second equation in (40) expresses the trace-preservation condition.
Note that Φ * has the same eigenvalues with the same algebraic multiplicities of Φ [54].In addition, the spectral projections P and P P onto Fix(Φ) and Attr(Φ) are quantum channels themselves [12].Finally, let M 0 ≡ Ker(L) be the kernel of L, given by Before discussing the proofs of Theorems 1-4, we need a few preparatory results.Consider A ∈ B(H) with spectrum spect(A) = {λ k } N k=1 .If m k , n k are the algebraic and geometric multiplicities of the eigenvalue λ k , let d j,k with j = 1, . . ., n k and k = 1, . . ., N indicate the dimension of the j-th Jordan block corresponding to the eigenvalue λ k of the Jordan normal form J of A [43]. where with i = 1, . . ., m k , k = 1, . . ., N and |I| being the cardinality of the set I.
In particular, the equality holds if and only if A is diagonalizable with spectrum spect(A) = {λ 1 , λ 2 } having algebraic multiplicities m 1 = 1 and m 2 = d − 1.

Proof. By definition {s
where the equality holds if and only if A is diagonalizable, viz.m k = n k for all k = 1, . . ., N .Now, by the fundamental theorem of algebra [43], where the equality holds if and only if A = cI , c ∈ C. If A is a non-scalar matrix, then the maximum value is attained when A is diagonalizable and has spectrum spect(A) = {λ 1 , λ 2 } with multiplicities m 1 = 1 and m 2 = d − 1, and it reads which concludes the proof.
Let us now recall several known facts about open-system asymptotics.Let us start with the following definition.
If Φ is not faithful, then we can define the faithful channel φ 00 as in Eq. ( 54), therefore we have as a consequence of Proposition 9 where d 0 = dim(H 0 ) ⩽ d − 1 and B 0 = {B 0,k , B † 0,k } N 0 k=1 is the system of Kraus operators of φ 00 .Let us now prove the analogous bound on the peripheral multiplicity ℓ P of Φ. Observe that the spectral projection P P of Φ onto Attr(Φ) satisfies because not all the eigenvalues of the non-unitary channel Φ are peripheral.Indeed, P P is a nonunitary channel as P P is non-invertible.Therefore, since the fixed-point space of P P is Attr(Φ), it is sufficient to apply the bound (58) to P P .□ Proof Theorem 3: Let L be a non-zero Hamiltonian GKLS generator.Then it is straightforward to show that [27] λ where Attr(Φ) is the attractor subspace of the non-unitary channel Φ = e L , the second inequality follows from Theorem 2. □ Notice that the universal bounds given in Theorems 1-4 may also be proved by using the structure theorems on the asymptotic evolution of quantum channels [12,14].

Conclusions and outlooks
We found dimension-dependent sharp upper bounds on the number of independent steady states of non-trivial unitary quantum channels and an analogous bound on the number of independent asymptotic states of non-unitary channels.Moreover, similar sharp upper bounds on the number of independent steady and asymptotic states of GKLS generators were also obtained.We further made a comparison of our bounds with similar ones obtained from the CKKS conjecture (30) and (34).
Interestingly, the upper bound on the peripheral multiplicity of GKLS generators reveals that adding a dissipative perturbation to an initially Hamiltonian generator causes a jump for the peripheral multiplicity across a gap linearly depending on the dimension, and an analogous remark may be made for the peripheral multiplicity ℓ P of quantum channels on the basis of condition (15).These findings may be framed in a series of works, addressing the general spectral properties of open quantum systems [12,41,51,[56][57][58][59], in particular Markovian ones, and can motivate further study of the spectral properties of channels and generators, far from being completely understood.In particular, the bounds found in this Article may be the consequence of a generalization of the CKKS bound (30) involving also the imaginary parts of the eigenvalues of a GKLS generator.Moreover, structure theorems on the asymptotic dynamics [12,14] may be employed in order to find further constraints for the quantities studied in this work.

Figure 1 :
Figure 1: Schematic representation of the content of Theorem 2. A system S coupled to a bath B evolves according to the non-unitary discrete-time evolution Φ n with n ⩾ 1.The asymptotic states ρ 1 , . . ., ρ ℓ P of S, spanning the attractor subspace Attr(Φ), are at most d 2 − 2d + 2, where d is the dimension of the system.
where h k , with k = 1, . . ., d, are the (repeated) real eigenvalues of the Hamiltonian H. Therefore this implies that the maximum value of the algebraic multiplicity m 0 of the zero eigenvalue of L is d 2 − 2d + 2, obtained by settingh 1 = h 2 = • • • = h d−1 ̸ = h d .The equality m P = d 2 follows immediately from Eq. (60).□ Proof Theorem 4: Let L be a non-Hamiltonian GKLS generator.The first inequality is trivial.Since m P = dim Attr(Φ) = dim Attr(L),