Construction of propagators for divisible dynamical maps

Divisible dynamical maps play an important role in characterizing Markovianity on the level of quantum evolution. Divisible maps provide important generalization of Markovian semigroups. Usually one analyzes either completely positive or just positive divisibility meaning that the corresponding propagators are defined in terms of completely positive or positive maps, respectively. For maps which are invertible at any moment of time the very existence of propagator is already guaranteed and hence the only issue is (complete) positivity and trace-preservation. However, for maps which are not invertible the problem is much more involved since even the existence of a propagator is not guaranteed. In this paper we propose a simple method to construct propagators of dynamical maps using the concept of generalized inverse. We analyze both time-continuous and time-discrete maps. Since the generalized inverse is not uniquely defined the same applies for the corresponding propagator. In simple examples of qubit evolution we analyze it turns out that additional requirement of complete positivity possibly makes the propagator unique.


I. INTRODUCTION
The evolution of open quantum systems [1,2] attracts a lot attention due to rapidly developing modern fields of science and quantum technologies like quantum communication or quantum computation [3]. Any quantum evolution is represented by a quantum dynamical map, that is, a continuous family of linear maps {Λ t } t≥0 where L(H) denotes the space of linear operators acting on the system's Hilbert space H. One requires that for any t ≥ 0 the map Λ t is completely positive and trace-preserving (CPTP) [4][5][6]. Moreover, it satisfies the standard initial condition Λ t=0 = id, where id denotes the identity map in L(H). Hence, for any t ≥ 0 the map Λ t represents a quantum channel, a basic object of quantum information [3,6]. Very often one also considers a different scenario: time-discrete dynamical map {Λ n } n≥0 , where n = 0, 1, . . ., and Λ n : L(H 0 ) → L(H n ), where H n is a Hilbert space at 'time' n. In principle for different n the corresponding Hilbert space may be different. Such scenario is important to deal with many processes like encodings, decodings, quantum measurement, and many others. In this paper we consider both time-continuous and time-discrete scenarios.
In the time-continuous case it well known that under appropriate Markovian approximation the evolution of an open quantum system can be described by quantum dynamical semigroup satisfying the following master equatioṅ where L : L(H) → L(H) has the celebrated Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) structure [7,8] with Hermitian H † = H ∈ L(H), arbitrary noise (Lindblad) operators L α ∈ L(H), and dissipation/decoherence rates γ α ≥ 0 (we keep = 1). Markovian approximation leading to (2) assumes weak coupling between system and its environment and separation of characteristic time scales. Such approximation works well in quantum optical systems where the coupling between atoms/molecules and electromagnetic field is weak. Recently, the notion of non-Markovian quantum evolution received considerable attention (see review papers [9][10][11][12][13]). There is no unique universal approach to deal with quantum (non)Markovianity. In this paper we concentrate on two very popular approaches based on divisibility of dynamical maps [14] and so called information flow [15]. The very concept of divisibility of quantum channels was initiated in [16,17] and recently reviewed in [18]. According to [14] the evolution is Markovian iff there exist a family of CPTP propagators V t,s (t ≥ s) such that Λ t = V t,s Λ s (see Section II for more details). One calls such evolution CP-divisible. Requiring that V t,s defines a positive (and not necessarily completely positive) map one calls {Λ t } t≥0 P-divisible (for the intricate relation between P-and CP-divisibility cf. [19]). Actually, one may introduce a whole hierarchy of so called k-divisibility, where k runs from k = 1 (P-divisibility) up to d = dim H (CP-divisibility) [20]. In a recent paper [21] so called KS-divisibility was analyzed based on so called Kadison-Schwarz property of the propagator V t,s which is stronger than P-but weaker than CP-divisibility. An interesting relation of divisibility of quantum dynamical maps and collision models was studied in [22]. Moreover k-divisibility was linked to discrimination of quantum channels [23]. For recent review of various properties of quantum open systems related to divisibility of dynamical maps see recent review [24]. On the other hand authors of [15] proposed the following approach: they call quantum evolution represented by {Λ t } t≥0 Markovian if for any pair of states ρ 1 and ρ 2 so called information flow where X 1 denotes the trace norm. Since Λ t (ρ 1 − ρ 2 ) 1 corresponds to distinguishability of time evolved states at time t inequality (4) states that distinguishability monotonically decreases in time. It is interpreted as a flow of information from the system to the environment. Violation of (4) is therefore interpreted as an information back-flow and provides clear evidence of memory effects and hence non-Markovianity [9][10][11][12]. These two approaches are not independent [25]. Actually, CP-divisibility implies (4). However, violation of (4) is immediately recognized as a clear sign of non-Markovianity. The approach based on information flow is very popular in the literature. Intriguing connections of divisibility, information back-flow and quantum correlations were reported recently in [26,27].We stress, however, that there are other approaches to quantum non-Markovianity (see detailed review in [12] and the recent papers [28,29] relating (non)Markovianity and quantum stochastic processes).
In this paper we analyze the problem of construction of propagators V t,s (or V j,i in the time-discrete case). We propose a very simple approach based on the concept of generalized inverse which reduces to the standard inverse if the map is invertible. Generalized inverse and hence the corresponding propagator is not uniquely defined. We provide several simple examples of qubit evolution for which the non-uniqueness of the propagator is removed when one requires that the propagator is completely positive and trace-preserving.

II. DIVISIBLE DYNAMICAL MAPS
A. Time-continuous maps Consider a dynamical map {Λ t } t≥0 acting on L(H). We assume that the family of maps {Λ t } t≥0 is differentiable (w.r.t. t), and dim H = d < ∞.
and V t,s : L(H) → L(H). {Λ t } t≥0 is called CP-divisible iff V t,s is CPTP, and P-divisible iff V t,s is PTP.
Clearly, any invertible map Λ t is always divisible since the propagator V t,s can be uniquely defined by V t,s = Λ t Λ −1 s . Moreover, the propagator V t,s satisfies local composition law for t ≥ u ≥ s. In this case one proves the following for any Hermitian X ∈ L(H). It is CP-divisible iff for any Hermitian X ∈ L(H ⊗ H).
Note that if X is traceless then X = a(ρ 1 − ρ 2 ), where a ∈ R, and hence (7) recovers BLP criterion (4). Interestingly, one proves for any pair of density operators ρ 1 , The essence of (9) is that one enlarges the dimension of the ancilla d → d + 1, but uses only traceless operators X = a(ρ 1 − ρ 2 ) like in the original approach to the information flow [15]. For non-invertible maps the divisibility is not guarantied. Actually, one proves for any t > s.
One has the following generalization of Theorem 1 for any Hermitian X ∈ L(H ⊗ H), then there exists completely positive propagator V t,s : L(H) → L(H) which is trace preserving on the image of Λ s .
Moreover, in the qubit case (d = 2) one has the following for any Hermitian X ∈ L(H ⊗ H).

B. Time-discrete maps
Consider now the family {Λ n } n≥0 of CPTP maps where H S , H 1 , . . . are finite dimensional Hilbert spaces of dimensions d S , d 1 , d 2 , . . ., respectively. Moreover, we assume that Λ 0 : L(H S ) → L(H S ) is an identity map. We call time discrete dynamical map {Λ n } n≥0 CP-divisible iff there exists a family of CPTP propagators such that Consider now an ensemble of states ρ x prepared with probability p x : E = {p x , ρ x } x . To distinguish between these states one defines guessing probability where the maximum is over all POVMs {P x } x defined on the Hilbert space H. Buscemi and Datta [33] introduced the following interesting concept enabling one to compare quantum channels Definition 2 Time discrete dynamical map {Λ n } n≥0 is information decreasing iff for any j > i one has where E n = {p x , Λ n (ρ x )} x . {Λ n } n≥0 is completely information decreasing iff id S ⊗ Λ n is information decreasing (id S stands for the identity map on L(H S )).
Note, that if E consists of two members E = {p 1 , ρ 1 ; p 2 , ρ 2 }, then [34] (see also [35] for the review) and hence the monotonicity condition (17) defines a necessary condition for CP-divisibility. (19) is satisfied for all j > i.
Proof: We show that (19) implies CP-divisibility. Note, that {Λ n } n≥0 is divisible, that is, there exists the family V j,i satisfying (14). We can define where , and we do not require it to exist here either.
We see that V j,i defined thus is a valid propagator: For all X ∈ L(H S ) for some δ ∈ Ker(Λ i ), and hence as which means that for any Y = [id S ⊗ Λ i ](X). Now, since Im(Λ i ) = L(H i ), V j,i is trace-preserving and id S ⊗ V j,i defines a contraction w.r.t. trace norm, and thus one finds that V j,i is CPTP (note that dim H S ≥ dim H n for n > 0). Clearly, Λ −1R i is not uniquely defined but it is trace-preserving. Note however that Proof: Suppose that there are two propagators V j,i andṼ j,i defined via Corollary 1 In particular, if H S = H n for n > 0, and Λ n is invertible, then the map {Λ n } n≥0 is CP-divisible iff (19) holds for all Hermitian X ∈ L(H S ⊗ H S ).

III. CONSTRUCTION OF PROPAGATORS
For invertible maps the propagator V t,s is uniquely defined via V t,s = Λ t Λ −1 s . Suppose now that the dynamical map {Λ t } t≥0 is divisible but not necessarily invertible. There is a natural construction of a propagator via the following formula where Λ − s : L(H) → L(H) denotes a generalized inverse of Λ s [36][37][38][39] (cf. Appendix for more details). A generalized inverse Λ − s coincides with the inverse Λ −1 s for invertible maps and it is defined by the following property Clearly, Λ − s is not uniquely defined. Suppose, that the image of Λ s is a proper subspace of L(H), and let C s be a complementary subspace such that for any s one has It should be stressed that C s is not uniquely defined. Actually, any linear subspace C s such that dim(C s ) + dim(Im(Λ s )) = d 2 , and C s ∩ Im(Λ s ) = {0} does the job. Now, for any Y ∈ L(H) we have the unique decomposition where, by the last statement, we mean that Y 1 ∈ Im(Λ s ) and Y 0 ∈ C s . The following linear map is a generalized inverse Λ − s : where Λ s (X 1 ) = Y 1 and B s : C s → Ker(Λ s ) (cf. [36,37] and the Appendix) is an arbitrary linear map. One can therefore define Hence such a propagator is characterized by the family of subspaces {C t } t≥0 transversal to images of {Λ t } t≥0 . Such propagators all agree on Im(Λ s ), due to the kernel non-decreasing property.
Proof: One has which ends the proof.
is trace-preserving iff the subspace C s contains only traceless operators.
Proof: One has for any t ≥ s.
Proof: Note, that the most general propagator satisfying (31) reads as follows where Clearly, the same techniques applies for the time-discrete dynamical maps {Λ n } n≥0 . One defines Example 1 Consider the following dynamical map where ω t is a time-dependent density operator, and f : Hence, for t ≥ t * the map Λ t (ρ) = ω t Tr ρ projects any density operator into ω t and clearly is not invertible. Now, the propagator is defined by fixing the family of subspaces C s transversal to the image of Λ s and hence Clearly, one can choose for s ≥ t * . Interestingly, Λ − s = Λ s and hence in this case Λ − s is reflexive (satisfies (A3)). Note, however, that neither (B3) nor (B4) is satisfied. Propagator (37) is completely positive and trace preserving. Moreover, it does satisfy the composition law. Indeed, one has for all X, Y ∈ L(H), and (X, Y ) := Tr(X † Y ) stands for the Hilbert-Schmidt inner product. One has and hence One finds Note, that formula (42) implies Λ s Λ − s = Λ − s , and hence (42) is also reflexive. However, neither (B3) nor (B4) is satisfied.
The second propagator is completely positive but not trace-preserving due to the fact that Ker(Λ * t ) does not contain traceless operators only. However, it does satisfy the composition law. Indeed, one has Clearly both propagators agree on the image of Λ s , that is, for Y = α ω s one has (Y, ω s ) (ω s , ω s ) = ω t TrY.

IV. PROPAGATOR FROM SPECTRAL PROPERTIES OF DYNAMICAL MAPS
Generalized inverse Λ − s is highly non unique. There is, however, a natural way to define a generalized inverse (and hence the propagator) using spectral properties of the dynamical map. In this section we analyze both diagonalizable and non-diagonalizable cases.

A. Diagonalizable dynamical maps
Let us assume that dynamical map {Λ t } is diagonalizable, that is, for any t ≥ 0 one has for some F α (t), G α (t) ∈ L(H), α = 0, 1, . . . , d 2 − 1. One has One has for the spectral resolution where are projectors (not necessarily Hermitian) satisfying Note, that trace-preservation condition implies λ 0 (t) = 1 and G 0 (t) = 1l. Suppose that the eigenvalues satisfy λ α (s) = 0 for α > n and are non-zero for α ≤ n. Divisibility implies that λ α (t) = 0 for α > n for all t > s. Let us define generalised inverse via The corresponding propagator reads where the 'correlator' C αβ (t, s) reads project to the image of Λ t .
The corresponding propagator Λ t Λ − s is evidently trace-preserving. Moreover, in this case the family of subspaces C t satisfy C t = Ker(Λ t ).
Example 2 Let us observe that dynamical map in Example 1 is diagonalizable. Indeed, one finds where F α is traceless for α = 1, . . . , d 2 − 1. The dual map reads and hence where G α satisfy Tr(G α ω t ) = 0. Hence the spectral resolution reads as follows Now, for s ≥ t * , it reduces to that is, there is only one non-vanishing eigenvalue λ 0 (t) = 1. Hence, the formula (48) leads to which reproduces (37).

B. Non-diagonalizable dynamical maps
Consider now a general case corresponding to the following spectral Jordan decomposition [40] Λ t = where N α (t) are nilpotent maps satisfying and P nα α (t) = 0, with 1 ≤ n α ≤ rank P α (t). Suppose that for t ≥ t * the eigenvalues satisfy λ α (t) = 0 for α > n. One has therefore Clearly nilpotent maps are represented by Jordan blocks.
Proposition 8 Let J k (λ) be a Jordan block of size k. One finds and and hence J T k (0) is a reflexive generalized inverse of J k (0).

V. BLOCH REPRESENTATION AND PROPAGATORS
For a qubit system one often uses well known Bloch representation where r = (r 1 , r 2 , r 3 ) T is the Bloch vector corresponding to ρ. Now, for a qubit map Φ one defines a real 4 × 4 matrix with σ 0 = 1l. T αβ has the following structure where x ∈ R 3 and ∆ is a 3 × 3 real matrix and the map ρ → Φ(ρ) in the Bloch representation is realized via the following affine transformation This representation may be generalized for arbitrary dimension d: let τ α (α = 0, 1, . . . , d 2 − 1) be Hermitian orthonormal basis in L(H) such that τ 0 = 1l/ √ d. Any CPTP map Φ : L(H) → L(H) gives rise to real d 2 × d 2 matrix Consider now the dynamical map {Λ t }. One finds T αβ (t) = Tr(τ α Λ t (τ β )) where x t ∈ R d 2 −1 , and ∆ t is a real (d 2 − 1) × (d 2 − 1) matrix. In general a generalised inverse T − (t) is not tracepreserving. There is, however, a natural class of trace-preserving generalised inverses corresponding to the following matrix representation where, the defining condition that is, y t + ∆ − t x t ∈ Ker(∆ t ). One finds for the matrix representation of propagator This propagator is by construction trace-preserving.

VI. PROPAGATORS FOR QUBIT DYNAMICAL MAPS
In the qubit case any quantum channel Λ : M 2 → M 2 can be represented as follows [3,6,41] where U and V are unitary channels, and Φ has the following Bloch representation T αβ = 1 2 Tr(σ α Φ(σ β )) where λ i 's are (up to a sign) the singular values of ∆ defined in (67). Now, since U and V are invertible, one has for the generalized inverse of Λ that is, the generalized inverse of Λ is completely determined by that of Φ. In [32] the following theorem was proved Theorem 6 There is no CPTP projector P : M 2 → M 2 projecting M 2 to the 3-dimensional subspace of M 2 .
Here we provide an independent proof based on the very concept of generalized inverse. Note, that any projector P such that Im(P) = Σ may be realized via where Λ is a map satisfying Im(P) = Σ (cf. Appendix). Using representation (70) one has Since the image of Λ is 3-dimensional let as assume that in the formula (71) one has λ 3 = 0 and let us look for the general inverse represented by the general formula (68) (75) Using defining property T T − T = T one finds together with The remaining parameters are completely free. Hence the Bloch representation of the corresponding projector T T − reads with β 1 = λ 1 α 13 and β 2 = λ 2 α 23 . One easily finds the corresponding map ΦΦ − : and ΦΦ − (σ 1 ) = σ 1 , ΦΦ − (σ 2 ) = σ 2 . To check complete positivity one has to analyze the spectrum of the corresponding Choi matrix with E ij := |i j|. Using Now, observe that 2 × 2 submatrix is not positive, and hence the projector ΦΦ − is not completely positive.
Consider now the structure of projectors onto 2-dimensional subspaces of M 2 , that is, let us assume that in the formula (71) one has λ 2 = λ 3 = 0. Now, using again the defining property T T − T = T one finds The remaining parameters are completely free. Hence the Bloch representation of the corresponding projector T T − reads with γ 2 = λ 1 α 12 and γ 3 = λ 1 α 13 . One easily finds the corresponding map ΦΦ − : and hence the corresponding Choi matrix reads Lemma 1 The Choi matrix (82) is positive semidefinite iff x 2 = x 3 = 0 and γ 1 = γ 2 = 0.
Finally, if the image is 1-dimensional then the corresponding projector reads where the Bloch vector x = (x 1 , x 2 , x 3 ) satisfies |x| ≤ 1.

VII. EXAMPLES
In this Section we illustrate the construction of qubit propagators for divisible dynamical maps using Bloch representation.
Example 3 A well known example of a commutative diagonalizable qubit dynamical map is generated by the following generator (it was already analyzed in [31]) where L k (ρ) = 1 2 (σ k ρσ k − ρ). The corresponding dynamical map reads with σ 0 = 1l, and has the following representation with where {i, j, k} is a permutation of {1, 2, 3}, and Γ k (t) = t 0 γ k (τ )dτ . The map Λ t is invertible if all Γ k (t) are finite for finite times. Now, if for example one has Γ 1 (t * ) = ∞, then λ 2 (t * ) = λ 3 (t * )= 0 which means that the image of Λ t * is 2-dimensional and of course it is orthogonal to the 2-dimensional kernel. Now, for any s > t * (assuming that the image of Λ s is 2-dimensional) Moreover, V t,t defines CPTP projector.
Example 4 Consider the qubit dynamical map given by: where ω is a density matrix, and f : R ≥0 → [0, 1] is a monotonic function with f (0) = 0 and f (t) = 1 for all t ≥ t * . It is direct to see that, {Λ t } t≥0 is CP-divisible. In the Pauli basis, with X = (x 0 x 1 x 2 x 3 ) T , and ω = (1 ω 1 ω 2 ω 3 ) T the map can be written as: For s < t * , the map Λ s is invertible, and and hence, we have the unique propagator Evidently, this is a channel for 0 ≤ s ≤ t * , and in particular, for 0 ≤ s ≤ t * ≤ t, it is a CPTP projection operator onto the 1 dimensional subspace spanned by ω. Now, for 0 ≤ t * ≤ s ≤ t, we have A direct computation shows that the following matrix is a generalised inverse of Λ s giving rise to the same propagator V t,s through Λ t Λ − s : (97) One sees that such a matrix is non-diagonalizable, has rank 2, singular values 1, 1, 0, 0, eigenvalues 1, 0, 0, 0, and Kraus rank 2 with the following Kraus operators Now, we define the dynamical map as follows where f (t) ∈ [0, 1]. Suppose, again that f (t) = 1 for t ≥ t * . Clearly, the map is divisible. Now, for s < t * one finds for the propagator Simple analysis of the corresponding Choi matrix shows that in general V t,s is not completely positive and hence Λ t is not CP-divisible. Now, for t * ≤ s one has Λ t = Λ s = Ψ, and the most general (and trace-preserving) generalised inverse reads where all the α ij 's are completely arbitrary, and the corresponding propagator reads: One finds for the Choi matrix with the corresponding eigenvalues 1 ± α 2 32 + α 2 33 + 1. Clearly, C is positive definite if and only if α 32 = α 33 = 0. Hence, the requirement of complete positivity makes V t,s unique. We stress in this example the image is not complimentary to the kernel, and hence one cannot make the choice C s = Ker(Λ s ).
Example 6 Consider now a phase covariant evolution governed by the following time-local master equatioṅ with the following time-local generator [42][43][44] where with σ ± = (σ x ± iσ y )/2. In general it defines a non-commutative family of maps, that is, L t L s = L s L t . The corresponding dynamical map Λ t = T e t 0 Lτ dτ is given by where One finds the corresponding Bloch representation which already has the Ruskai representation [41]. Now, if Γ(t) = +∞ = Γ 3 (t), the map is invertible, and hence divisible. For divisibility while non-invertible, we need the following: rank first decreases to 2 from 4 at t = t 1 , and Γ 3 (t) = +∞ for t ≥ t 1 . Hence, for s ≥ t 1 , we have (108) The most general trace-preserving generalized inverse of this map can be written as follows where all the y i 's and α ij 's are arbitrary. Hence, the corresponding propagator, for t ≥ s ≥ t 1 > 0 can be written as: where ξ(t, s) = (1 − e −Γ(t) (2G(t) + 1)) − e −(Γ(t)−Γ(s)) (1 − e −Γ(s) (2G(s) + 1)) .
One finds for the projection The Choi matrix of this map reveals that it is CP iff α 31 = α 32 = 0, which is, when it is just the depolarizing channel. If the rank decreases once more at t = t 2 ≥ t 1 , that is, if Γ(t) = +∞ for all t ≥ t 2 , then for t ≥ t 2 which is a CPTP projector onto the space spanned by vacuum state |0 0|.

VIII. CONCLUSION
CP-divisibility is a recent topic of growing interest in the field of open quantum systems. Though extensive characterizations of necessary and (some) sufficient conditions for CP-divisibility do exist in the literature [25,[30][31][32][33], so far, to the best of our knowledge, before us no work has been done in the direction of explicit construction of propagators for divisible but non-invertible dynamical maps. We have successfully proved that if a propagator has range coinciding with the image of the map itself, we can always obtain it through a suitable generalized inverse. For all our examples, this is possible through a reflexive generalized inverse even. Several (non-unique) generalized inverses give the same propagator. Moreover, for such propagators, for all our examples, we see that the CPTP propagator is unique, though the map is non-invertible, which is an interesting observation perhaps not predictable through earlier works in this direction. It turns out that the generalized inverse is a very powerful tool for attacking such problems. For example, it was a well-established fact [32] that there is no qubit channel which is also a projection onto a 3-dimensional subspace, but it was relatively cumbersome to prove. Through the generalized inverse, we have in effect exhaustively characterized all trace-preserving projections for qubit channels, CP and otherwise. And now, with a concrete structure for the propagator at hand, we hope to make some concrete statement about CP-divisibility or P-divisibility in general, perhaps even solve the long-standing problem of whether CP-divisibility is equivalent to monotonicity of the trace norm beyond qubit dynamics. Future work is intended along these directions. Also, it should be mentioned that in our work, the generalized inverses considered are not "dynamical", that is, they have not been interpreted to represent some physical process. They are not necessarily CP, even when the corresponding propagators are CPTP. It is perhaps possible to view generalized inverses too as "time reversals" in some sense, possibly along the lines of [45]. Perhaps this could be done through Kraus operators, as done in [45]. If possible, this would give a strong "physical" flavour to our work on obtaining propagators through generalized inverses. Possible connections between their work and ours, are under consideration as well.
In this case A − A and AA − are orthogonal projectors. (B3) Actually, there is a unique generalized inverse satisfying (A3), (B3) and (B4) and it is called Moore-Penrose generalized inverse.
Proposition 12 Let V be a finite dimensional vector space, P be any projection onto some subspace S ⊂ V , and A be any linear map in L(V ) with Im(A) = S. Then, there exists a generalized inverse A − of A such that AA − = P.
Proof: Let dim(V ) = n, and dim(S) = r < n. To begin with let S 0 be the span of the first r of the n vectors in the canonical basis. Let P 0 be an arbitrary projection onto this subspace. As a matrix resolved in the canonical basis P 0 can be represented as follows where X is an arbitrary r × (n − r) matrix. If A = U ΣV † stands for SVD for A, then S 0 can be transformed to S via U , and P = U P 0 U † . Now, we choose P 0 = U † P U . This fixes the matrix X. Now, Σ = D 0 0 0 , where D is an invertible diagonal matrix of dimension r × r. A valid generalized inverse of Σ is, Σ − = D −1 D −1 X 0 0 , and we immediately observe that Hence, we have P = AA − , where A − = V Σ − U † is a generalized inverse of A.