Quantum and classical dynamical semigroups of superchannels and semicausal channels

Quantum devices are subject to natural decay. We propose to study these decay processes as the Markovian evolution of quantum channels, which leads us to dynamical semigroups of superchannels. A superchannel is a linear map that maps quantum channels to quantum channels, while satisfying suitable consistency relations. If the input and output quantum channels act on the same space, then we can consider dynamical semigroups of superchannels. No useful constructive characterization of the generators of such semigroups is known. We characterize these generators in two ways: First, we give an efficiently checkable criterion for whether a given map generates a dynamical semigroup of superchannels. Second, we identify a normal form for the generators of semigroups of quantum superchannels, analogous to the GKLS form in the case of quantum channels. To derive the normal form, we exploit the relation between superchannels and semicausal completely positive maps, reducing the problem to finding a normal form for the generators of semigroups of semicausal completely positive maps. We derive a normal for these generators using a novel technique, which applies also to infinite-dimensional systems. Our work paves the way to a thorough investigation of semigroups of superchannels: Numerical studies become feasible because admissible generators can now be explicitly generated and checked. And analytic properties of the corresponding evolution equations are now accessible via our normal form.


I. INTRODUCTION AND MOTIVATION
Anybody who has ever owned an electronic device knows: These devices have a finite lifespan after which they stop working properly. At least from a consumer perspective, a long lifespan is a desirable property for such devices. Thus, it is important for an engineer to know which kind of decay processes can affect a device, in order to suppress them by an appropriate design. Certainly, these considerations will also become important for the design of quantum devices. We therefore propose to study systematically the decay processes that quantum devices can be subject to.
In this work, we take a first step in this direction by deriving the general form of linear time-homogeneous master equations that govern how quantum channels behave when inserted into a circuit board at different points in time. This leads to the study of dynamical semigroups of superchannels. Here, superchannels are linear transformations between quantum channels [1]. FIG Let us consider a concrete example, see Fig. 1. Suppose we are trying to estimate the optical transmissivity of some material (M), which we assume to depend on the polarization of the incident light. A simple approach is to send photons from a light source (S) through the material and to count how many photons arrive a the detector (D). We model the material by a quantum channel T M , acting on the states of photons described as three-level systems, with the levels corresponding to vacuum, horizontal, and vertical polarization. In an idealized world, with a perfect vacuum in the regions between the source, the material, and the detector, we can infer the transmissivity from the measurement statistics of the state T M (σ ), where σ is the state of the photon emitted from the source. However, in a more realistic scenario, even though we might have created an (almost) perfect vacuum between the devices at construction time, some particles are leaked into that region over time. Then, interactions between the photons and these particles might occur, causing absorption or a change in polarization. Hence, the situation is no longer described accurately by T M alone, but also requires a description of the particle-filled regions.
To find such a description, we us argue that the effect of particles in some region (here, either between S and M; or M and D) can be modeled by a quantum dynamical semigroup, parametrized by the particle density δ . If the particle density is reasonably low and Q δ is the quantum channel describing the effect of the particles on the incident light at a given δ , then, as explained in Fig. 2, Q δ satisfies the semigroup property Q δ 1 +δ 2 = Q δ 1 • Q δ 2 . Furthermore, if there are no particles then there should be no FIG. 2. If the particle density is low, then the incident photon interacts with the particles in the region sequentially and independently. The effect of a single interaction can be described by a channel ∆Q. Hence, the state after the first interaction is ∆Q(σ ), the state after the second interaction is ∆Q(∆Q(σ )), and so forth. The number of interactions is given by the product of the particle density δ and the volume V . Hence, the effect of an region with fixed volume is described by the channel Q δ = (∆Q) δV . It follows that if δ = δ 1 + δ 2 , then Q δ 1 +δ 2 = (∆Q) δ 1 V (∆Q) δ 2 V = Q δ 1 • Q δ 2 . The semigroup property for real δ can then be obtained in the continuum limit. effect. Hence, Q 0 = id. After adding continuity in the parameter δ as a further natural assumption, the family {Q δ } δ ≥0 forms a quantum dynamical semigroup. That is, we can write Q δ = e Lδ , for some generator L in GKLS-form.
If we assume in our example that particles of type A are leaked into the region between S and M at a rate γ A and that particles of type B are leaked into the region between M and D at a rate γ B , then the overall channel describing the transformation that emitted photons undergo at time t is given byŜ We note that at any fixed time,Ŝ t interpreted as a map on quantum channels is a superchannel written in 'circuit'-form. This means, thatŜ t describes a transformation of quantum channels implemented via pre-and post-processing. Furthermore,Ŝ t (T M ) can be determined by solving the time-homogenous master equation whereL(T ) = γ A L A • T + γ B T • L B , with initial condition T (0) = T M . In other words, we havê S t = eL t and thus the family {Ŝ t } t≥0 forms a dynamical semigroup of superchannels. By inductive reasoning, we thus arrive at our central physical hypothesis: Decay-processes of quantum devices with some sort of influx are well described by dynamical semigroups of superchannels. It follows that such decay-processes can be understood by characterizing dynamical semigroups of superchannels. Such a characterization is the main goal of our work.
In particular, we aim to understand dynamical semigroups of superchannels in terms of their generators. We characterize these generators fully by providing two results: First, we give an efficiently checkable criterion for whether a given map generates a dynamical semigroup of superchannels. Second, we identify a normal form for the generators of semigroups of quantum superchannels, analogous to the GKLS form in the case of quantum channels. Interestingly, we find that the most general form of dynamical semigroups of superchannels goes beyond the simple introductory example above.
We arrive at these results through a path (see Fig. 3) that also illuminates the connection to the classical case. We start by studying dynamical semigroups of classical superchannels, which (analogously to quantum superchannels being transformations between quantum channels) are transformations between stochastic matrices. We do so by establishing a one-to one correspondence between classical superchannels and certain classical semicausal channels, that is, stochastic matrices on a bipartite system (AB) that do not allow for communication from B to A (see Definition IV.2). We can then obtain a full characterization of the generators of semigroups of classical superchannels by characterizing generators of semigroups of classical semicausal maps first and then translating the results back to the level of superchannels. The study of (dynamical semigroups of) classical superchannels and classical semicausal channels is the content of Section IV. Armed with the intuition obtained from the classical case, we then go on to study the quantum case. We start by characterizing the generators of semigroups of semicausal [2] completely positive maps (CP-maps) -our main technical result, and one of independent interest. This characterization can be obtained from the classical case by a 'quantization'-procedure that allows us to see exactly which features of semigroups of semicausal CP-maps are "fully quantum." Dynamical semigroups of semicausal CP-maps are discussed Section V B. Finally, in Section V C, we use the one-to one correspondence (via the quantum Choi-Jamiołkowski isomorphism) between certain semicausal CP-maps and quantum superchannels to obtain a full characterization of the generators of semigroups of quantum superchannels. While the classical section (IV) and the quantum section (V) are heuristically related, they are logically independent and can be read independently.
This work is structured as follows. In the remainder of this section, we discuss results related to ours. Section II contains an overview over our main results. In Section III, we recall relevant notions from functional analysis and quantum information, as well as some notation. The (logically) independent sections IV and V comprise the main body of our paper, containing complete statements and proofs of our results on dynamical semigroups of superchannels and semicausal channels. We study the classical case in Section IV and the quantum case in Section V. Finally, we conclude with a summary and an outlook to future research in Section VI.

A. Related work
The study of quantum superchannels goes back to [1] and has since evolved to the study of higher-order quantum maps [3][4][5]. A peculiar feature of higher-order quantum theory is that it allows for indefinite causal order [6,7]. However, it was recently discovered that the causal order is preserved under (certain) continuous evolutions [8,9]. It therefore seems interesting to study continuous evolutions of higher-order quantum maps systematically. Our work can be seen as an initial step into his direction.
The study of (semi-)causal and (semi-)localizable quantum channels goes back to [2]. By proving the equivalence of semicausality and semilocalizability for quantum channels, [10] resolved a conjecture raised in [2] (and attributed to DiVincenzo). Later, [11] provided an alternative proof for this equivalence, and further investigated causal and local quantum operations.

II. RESULTS
We give an overview over our answers to the questions identified in the previous section. In our first result, we identify a set of constraints that a linear map satisfies if and only if it generates a semigroup of quantum superchannels. Therefore, we can efficiently check whether a given linear map is a valid generator of a semigroup of quantum superchannels. We can even solve optimization problems over such generators in terms of semidefinite programs. Thereby, this first characterization of generators of semigroups of quantum superchannels facilitates working with them computationally.
As our second result, we determine a normal form for generators of semigroups of quantum superchannels. Similar to the GKLS-form, we decompose the generator into a "dissipative part" and a "Hamiltonian part," where the latter generates a semigroup of invertible superchannels such that the inverse is a superchannel as well.
with local Hamiltonians H B and H A , and where the "dissipative part" is of the formD(T )(ρ) = tr E D (T )(ρ) , wherê with unitary U and arbitrary A and B.
The "dissipative part" consists of three terms: Term (1a) itself generates a semigroup of superchannels (for B = 0), with the interpretation that the transformed channel (Ŝ t (T )) arises due to the stochastic application of T → tr E U(T ⊗ id E )(A(ρ ⊗ σ )A † )U † at different points in time (Dyson series expansion). Term (1b) itself generates a semigroup of superchannels (for A = 0) of the formŜ t (T ) = e L B t • T , where L B is a generator of a quantum dynamical semigroup (and hence in GKLS-form). Term (1c) is a "superposition" term, which is harder to interpret. It will become apparent from the path taken via the 'quantization' of semicausal semigroups that this term is a pure quantum feature with no classical analogue. Therefore, the presence of (1c) can be regarded as one of our main findings. It is also worth noting that the normal form in Result 1.2 is more general than the form of of the generator we found in our introductory example. Hence, nature allows for more general decay-processes than the simple ones with an independent influx of particles before and after the target object. We also complement this structural result by an algorithm that determines the operators U, A, B, H A and H B , if the conditions in Result 1.1 are met.
The proof of these results relies on the relation (via the Choi-Jamiołkowski isomorphism) between superchannels and semicausal CP-maps. Our next findings -and from a technical standpoint our main contributions -are the corresponding results for semigroups of semicausal CP-maps. Based on this insight, we can efficiently check whether a given linear map is a valid generator of a semigroup of semicausal CP-maps.
Since semigroups of semicausal CP-maps are in particular semigroups of CP-maps, our normal form for generators giving rise to semigroups of semicausal CP-maps is a refining of the the GKLS-form.
, and the K in the non-CP part is of the form with a self-adjoint H B and an arbitrary K A .
This characterization has both computational and analytical implications: On the one hand, it provides a recipe for describing semicausal GKLS generators in numerical implementations. On the other hand, the constructive characterization of semicausal GKLS generators makes a more detailed analysis of their (e.g., spectral) properties tractable. It is also worth noting that in Result 2.2 we can allow for (separable) infinite-dimensional spaces. In the finite-dimensional case, we also provide an algorithm to compute the operators U, A, B, K A and H B , if the conditions of Result 2.1 are met.
Let us now turn to the corresponding results in the classical case. Here, instead of looking at (semigroups of) CP-maps and quantum channels, we look at (entry-wise) non-negative matrices and row-stochastic matrices (see Section III and Section IV for details) that we assume to act on R X , for (finite) alphabets X ∈ {A, B, E}.
The following result is the classical analogue of Result 2.2.
generates a semigroup of (Heisenberg) B → A semicausal non-negative matrices if and only if it can be written as with a row-stochastic matrix U ∈ B(R B ; R E ⊗ R B ), a non-negative matrix A ∈ B(R A ⊗ R E ; R A ), a diagonal matrix K A and maps B (i) ∈ B(R B ) that generate semigroups of row-stochastic matrices.
We will discuss in detail how Result 2.2 arises as the 'quantization' of Result 3 in the paragraph following the proof of Lemma V.5. Here, we highlight that in both the quantum and the classical case, the generators of semicausal semigroups are constructed from two basic building blocks. In the quantum case, these are a B → A semicausal CP-map Φ sc , with Φ sc (X) = V † sc (X ⊗ 1 E )V sc and V sc = (1 A ⊗U)(A ⊗ 1 B ); and a GKLS generator of the form id A ⊗B. And in the classical case, they are a B → A semicausal non-negative map Φ sc = (1 A ⊗ U)(A ⊗ 1 B ); and operators of the form |a i a i | ⊗ B (i) , where B (i) generates a semigroup of row-stochastic maps. The difference between the quantum case and the classical case then lies in the way the general form is constructed from the building blocks. While we simply take convex combinations of the building-blocks in the classical case, we have to take superpositions of the building-blocks, by which we mean that we need to combine the corresponding Strinespring operators, in the quantum case.
As our last result, we present the normal form for generators of semigroups of classical superchannels.
generates a semigroup of classical superchannels if and only if it can be written asQ , a diagonal matrix K A , and a collection of generators of semigroups of column-stochastic matrices B (i) ∈ B(R B ).
As in the quantum case, we have two kinds of evolutions: a stochastic application of M → U(M ⊗ 1 E )A at different points in time; and a conditioned post-processing evolution of the form ∑ i e B (i) t M|a i a i |. Note that there are no "superposition" terms, like (1c).

III. NOTATION AND PRELIMINARIES
In this section, we review basic notions from Functional Analysis, Quantum Information Theory, and the theory of dynamical semigroups. We also fix our notation for these settings as well as for a classical counterpart of the quantum setting.

A. Functional analysis
Throughout the paper, H (with some subscript) denotes a (in general infinite-dimensional) separable complex Hilbert space. Whenever H is assumed to be finite-dimensional, we explicitly state this assumption. We denote the Banach space of bounded linear operators with domain H A and codomain H B , equipped with the operator norm, by B(H A ; H B ) and write B(H) for B(H; H). For X ∈ B(H A ; H B ), the adjoint X † ∈ B(H B ; H A ) of X is the unique linear operator such that ψ B |Xψ A = X † ψ B |ψ A for all |ψ A ∈ H A and all |ψ B ∈ H B . Here, and throughout the paper, we use the standard Dirac notation. An Besides the norm topology, we will use the strong operator topology and the ultraweak topology. The strong operator topology is the smallest topology on B( The partial trace w.r.t. the space H A is the unique linear map tr A : and all X ∈ B(H C ; H B ). If the spaces involved have subscripts, the partial trace will always be denoted with the corresponding subscript. The partial trace with respect to ρ ∈ S 1 (H A ) is the unique linear map tr ρ : Furthermore, the Stinespring dilation can be chosen to be minimal, that is, the pair (V, H E ) can be chosen such that span{( . A trace-preserving CP-map is called a (quantum) channel. The facts in this section are contained or follow directly from results in [14,15].
, for all elements of the orthonormal basis.
The (quantum) Choi-Jamiołkowski isomorphism [16,17], defined with respect to an orthonormal basis , and its inverse is given by

D. Non-negative matrices and duality
As we provide characterizations for both the quantum and the classical case, we now also introduce the notation and definitions required for the latter. With a classical system A, we associate a finite alphabet A = {a 1 , a 2 , . . . , a |A| } and a 'state-space' R A , with orthonormal basis {|a i } |A| i=1 . We define by |1 A := ∑ i |a i the all-one-vector. A vector |x ∈ R A is called non-negative if a|x ≥ 0, for all a ∈ A. A linear operator M ∈ B(R A ; R B ) is called non-negative if M|x is non-negative, whenever |x is non-negative (equivalently, all matrix elements are non-negative). A non-negative M is called column-stochastic if 1 B |M = 1 A |; column-sub-stochastic if there exists a non-negative P, such that M + P is column-stochastic; row-stochastic, if M|1 A = |1 B ; and row-sub-stochastic if there exists a non-negative P, such that M + P is row-stochastic. Given |x or x|, we denote by diag(|x ) = diag( x|) the diagonal matrix with the components of x on the diagonal. Finally, we will use the 'classical Choi-Jamiołkowski isomorphism' (also known as vectorization), which is a convenient notation to make the connection to the quantum case more transparent. The classical Choi-Jamiołkowski isomorphism, defined w.
We will sometimes refer to elements of the range of C C A;B as Choi vectors.

E. Dynamical semigroups
Let X be a Banach space. A family of operators {T t } t≥0 , with T t ∈ B(X ) for all t ≥ 0, is called a norm-continuous oneparameter semigroup on X , or short, dynamical semigroup, if T 0 = 1, T s+t = T s T t for all t, s ≥ 0 and the map R ≥0 t → T t is norm-continuous. Norm-continuous dynamical semigroups are automatically differentiable and have bounded generators, that is, there exists L ∈ B(X ) such that T t = e tL for all t ≥ 0 and L = d dt t=0 + T t [18, Thm. I.3.7]. Lindblad [19] proved that T t ∈ CP σ (H) for all t ≥ 0 if and only if there exist Φ ∈ CP σ (H) and K ∈ B(H) such that T t = e tL , with L(X) = Φ(X) − K † X − XK. In this case, we refer to {T t } t≥0 as a CP semigroup. We call the corresponding form of the generator L the GKLS form [19,20] and Φ its CP part. If H is finite-dimensional, then T t = e tL ∈ CP σ (H) for all t ≥ 0 if and only if the operator L := C A;B = (id ⊗ L)(|Ω Ω|) is self-adjoint and P ⊥ LP ⊥ ≥ 0, where |Ω = ∑ i |a i ⊗ |a i , for some orthonormal basis {a i } of H and P ⊥ ∈ B(H ⊗ H) is the orthogonal projection onto the orthogonal complement of {|Ω } [21,22]. The corresponding classical result is as follows: is a dynamical semigroup of non-negative linear maps if and only if there exists a non-negative linear map Φ ∈ B(R A ) and a diagonal map K ∈ B(R A ) (w.r.t. the basis orthogonal basis {|a i } i ) such that the generator L has the form Φ − K [23].

IV. THE CLASSICAL CASE
Before studying the quantum scenario, we consider the classical version of our main question. I.e., we study continuous semigroups of classical superchannels and their generators. On the one hand, this allows us to develop an intuition that we can build upon for the quantum case. On the other hand, a comparison between the classical and the quantum case elucidates which features of the latter are actually quantum. For the purpose of this section, A, B and E denote finite alphabets as in Subsection III D.
A classical superchannel is a map that maps classical channels, i.e., stochastic matrices, to classical channels while preserving the probabilistic structure of the classical theory. To achieve the latter requirement, we require that a classical superchannel is a linear map and that probabilistic transformations, i.e., sub-stochastic matrices, are mapped to probabilistic transformations. Expressed more formally, we have A related concept is that of a classical semicausal channel, which is a stochastic matrix on a bipartite space A × B such that no communication from B to A is allowed. We formalize this as follows: To emphasize the analogy to the quantum case, we will often refer to a column B → A semicausal map as a Schrödinger B → A semicausal map and to a row B → A semicausal map as a Heisenberg B → A semicausal map. In both cases, the maps M A and N A will be called the reduced maps.
The structure of this section is as follows: We start by establishing the connection between classical superchannels and classical non-negative semicausal maps, followed by a characterization of classical non-negative semicausal maps as a composition of known objects; such a characterization is known in the quantum case as the equivalence between semicausality and semilocalizability. We then turn to the study of the generators of semigroups of semicausal and non-negative maps and finally use the correspondence between superchannels and semicausal channels to obtain the corresponding results for the generators of semigroups of superchannels.

A. Correspondence between classical superchannels and semicausal nonnegative linear maps
We first show, with a proof inspired by the one given in [1] for the analogous correspondence in the quantum case, that we can understand classical superchannels in terms of classical semicausal channels. To concisely state this correspondence, we use the classical version of the Choi-Jamiołkowski isomorphism. Let us mention here one again that we assume all alphabets (A, B, . . . ) to be finite for our treatment of the classical case.
Then, S is a classical superchannel if and only if S is non-negative and (Schrödinger B → A) semicausal such that the reduced map S A satisfies S A |1 A = |1 A . In this case, S A is automatically non-negative.
Proof. We first show the "if"-direction, i.e., that if S is non-negative and ( Suppose M is a non-negative matrix. ThenŜ(M) is non-negative, since C C A;B maps non-negative matrices to non-negative vectors, S maps non-negative vectors to non-negative vectors and (C C A;B ) −1 maps non-negative vectors to non-negative matrices.
Furthermore, if M is column stochastic, then In the preceding calculation, we used that S is semicausal in the third line, that M is stochastic in the fifth line, and that This proves thatŜ is a superchannel. The claim about the non-negativity of S A now follows directly from the semicausality condition. For the converse, supposeŜ is a superchannel. Since for all a ∈ A and all b ∈ B, the matrix |b a| is sub-stochastic, it follows by linearity ofŜ thatŜ(M) is non-negative whenever M is non-negative. Thus, since (C C A;B ) −1 maps non-negative vectors to nonnegative matrices,Ŝ maps non-negative matrices to non-negative matrices and C C A;B maps non-negative matrices to non-negative vectors, it follows that S is non-negative.
Next, we want to show that S is Schrödinger B → A semicausal. SinceŜ is a superchannel, S maps Choi vectors of stochastic matrices to Choi vectors of stochastic matrices, that is, As a tool, we define the set of scaled differences of Choi vectors of stochastic matrices by We claim that To see this, first note that C 0 ⊆ C 0 follows directly from the definition. For the other inclusion, C 0 ⊇ C 0 , we decompose |x ∈ C 0 as |x = |p − |n , for two non-negative vectors |p , |n ∈ R A ⊗ R B . It follows that Furthermore, for ε > 0 small enough, we have that |y := |1 A −ε(1 A ⊗ 1 B |)|p is non-negative. But then, for any non-negative unit |v ∈ R B , with 1 B |v = 1, the vectors |p := ε|p + |y ⊗ |v and |n := ε|n + |y ⊗ |v are Choi vectors of stochastic where we used in the second line that C 0 is invariant under S, a fact that follows directly from (2). This calculation exactly shows that S is Schödinger A → B semicausal.
It remains to show that S A |1 A = |1 A . This follows easily, since where we used that 1 |B| |1 B 1 A | is stochastic and that thusŜ( 1 |B| |1 B 1 A |) is stochastic. In summary, Theorem IV.3 tells us that, via the classical Choi-Jamiołkowski isomorphism, we can view classical superchannels equivalently also as suitably normalized semicausal non-negative maps.

B. Relation between classical semicausality and semilocalizability
The goal of this section is to get a better understanding of the structure of semicausal maps. For non-negative semicausal maps, we have the following structure theorem: In that case, we can choose |E| = |A| 2 .
Borrowing the terminology from the quantum case [2,10], the preceding theorem tells us that non-negative semicausal maps are semilocalizable. We formally define the latter notion for the classical case as follows: The requirement that U is stochastic and A is non-negative in the decomposition above is essential. In fact, if one drops these requirements, then a decomposition M = (1 A ⊗U)(A ⊗ 1 B ) can be found for any matrix M ∈ B(R A ⊗ R B ).
Due to Theorem IV.4, a non-negative Schrödinger B → A semicausal and column-stochastic map M admits an operational interpretation. First, note that if M is not only semicausal, but also stochastic, then also the matrix A in Eq. (3) is stochastic. Thus, the interpretation of the decomposition is: First, Alice applies some probabilistic operation (A) to the composite system A × E. Then she transmits the E-part to Bob, who now applies a stochastic operation (U) to his part of the system. Given this interpretation, the idea behind the construction in the proof of Theorem IV.4 is that Alice first looks the input of system A and generates the output of system A according to the distribution given by the matrix N A . Then she copies the input as well as her generated output and sends this information to Bob, who is then able to complete the operation by generating an output conditional on his input and the information he got from Alice. Given that this construction requires copying, it might be considered surprising that a quantum analogue is true nevertheless [10].
U := ∑ m,n,r,s a n |N A a m =0 a n ⊗ b r |N a m ⊗ b s a n |N A a m |a m ⊗ a n ⊗ b r b s | + For the last step, observe that the second sum vanishes and that one can drop the constraint that a j |N A a k = 0 in the first sum (after cancellation), because a j ⊗ b r |N a k ⊗ b s = 0, if a j |N A a k = 0. To see this last claim, note that, since N is non-negative and semicausal, we have It is clear, that A and U are non-negative, since N and thus also N A are non-negative by assumption. It remains to show that U is row-stochastic. We have U|1 B = ∑ m,n,r,s a n |N A a m =0 a n ⊗ b r |N a m ⊗ b s a n |N A a m |a m ⊗ a n ⊗ b r + ∑ m,n,s a n |N A a m =0 |a m ⊗ a n ⊗ b s = ∑ m,n,r a n |N A a m =0 a n ⊗ b r |N a m ⊗ 1 B a n |N A a m |a m ⊗ a n ⊗ b r + ∑ m,n,s a n |N A a m =0 |a m ⊗ a n ⊗ b s = ∑ m,n,r a n |N A a m =0 |a m ⊗ a n ⊗ b r + ∑ m,n,s a n |N A a m =0 |a m ⊗ a n ⊗ b s where we used the condition that N is semicausal to obtain the third line. This finishes the proof.
Remark IV.6. Theorem IV.4 can be extended to weak- * continuous non-negative maps on the Banach space of bounded real sequences, but this requires extra care and does not yield additional insight beyond the previous proof.

C. Generators of semigroups of classical semicausal non-negative maps
The main goal of this section is to establish a structure theorem for the generators of semigroups of non-negative semicausal maps. First, recall that a (norm)-continuous semigroup A classical result states that N t is non-negative for all t ≥ 0 if and only if the generator Q can be written in the form Q = Φ − K, where Φ is non-negative and K is a diagonal matrix w.r.t. the canonical basis [24]. A second, crucial observation is that N t is Heisenberg B → A semicausal for all t ≥ 0 if and only if Q is Heisenberg B → A semicausal. To see this, let us first show that the reduced maps N A t t≥0 also form a norm-continuous semigroup of non-negative maps. Since non-negativity is clear, we derive the semigroup properties (N A 0 = 1 A , N A t+s = N A t N A s and continuity) from the corresponding ones of {N t } t≥0 : Thus, we conclude that N A t = e tQ A for some generator Q A ∈ B(R A ). We further have Therefore, our task reduces to characterizing semicausal maps of the form Q = Φ − K. Let us first remark that it is straightforward to check (numerically) whether a given map satisfies these two conditions: We just need to check for non-negativity of the off-diagonal elements and whether (1 A ⊗ b|)Q|a i ⊗ 1 B = 0, for all a i ∈ A and all b ∈ {|1 B } ⊥ . I.e., semicausality can be checked in terms of |A| (|B| − 1) linear equations and |A| |B| (|A| |B| − 1) linear inequalities. Thus, a desirable result would be a normal form for all Heisenberg B → A semicausal generators Q, which allows for generating such maps, rather than checking whether a given maps is of the desired form. The main result of this section is exactly such a normal form.
To understand our normal form below, note that there are two natural ways of constructing a generator (remember that the matrix elements are interpreted as transition rates) that does not transmit information from system B to system A. First, we can leave system A unchanged and have transitions only on system B. The most basic form of such a map is |a i a i | ⊗ B (i) , for some 1 ≤ i ≤ |A| and for some B (i) ∈ B(R B ) that is itself a valid generator of a semigroup of row-stochastic maps. That means that . Second, if we want to act non-trivially on system A, we can make both of the two parts of a generator Q = Φ − K, the non-negative part Φ ∈ B(R A ⊗ R B ) and the diagonal part The fact that (convex) combinations of these basic building blocks already give rise to the most general form of semicausal generators for semigroups of non-negative bounded linear maps is the content of our next theorem, which establishes the desired normal form.
In that case, Φ sc can be chosen 'block-off-diagonal', i.e., Φ sc = ∑ i = j |a i a j | ⊗ Φ (i j) sc , for some collection of (non-negative) maps Φ Proof. It is straight-forward to check that a generator Q of the given form has non-negative off-diagonal entries w.r.t. the standard basis and is Heisenberg B → A semicausal. By the above discussion, this means that such a generator indeed gives rise to a semigroup of semicausal non-negative maps. We prove the converse. Suppose Q is the generator of a semigroup of non-negative linear maps. Then we can expand it as Q = ∑ |A| i, j=1 |a i a j | ⊗ Q (i j) , where the operators Q (i j) ∈ B(R B ) are non-negative for i = j and of the form of a generator of a non-negative semigroup (i.e., non-negative minus diagonal) for i = j. This decomposition, together with semicausality, implies that for all 1 ≤ i, j ≤ |A|, In other words, |1 B is an eigenvector of every Q (i j) , with corresponding eigenvalue λ (i j) = a i |Q A |a j . Hence, if we define then B i generates a semigroup of non-negative maps (since Q (i j) does and λ (ii) 1 B is diagonal) and satisfies (by construction), B (i) |1 B = 0. So B (i) generates a semigroup of row-stochastic maps. With this notation, we can rewrite Q as Note that Φ sc is semicausal, since it can be written as the linear combination of the three semicausal maps Q, K A ⊗ 1 B and ∑ i |a i a i | ⊗ B (i) . Thus, we have reached the claimed form.
By applying Theorem IV.4, we can further expand the Φ part: In that case, we can choose |E| = |A| 2 .
One should also note that with the notation of Corollary IV.8, the reduced map is given by Q A = (A(1 A ⊗ |1 B )) − K A . So, the reduced dynamics only depends on the operators A and K A . Further note that if we require the semigroup to consist of nonnegative semicausal maps that are also row-stochastic, then we obtain the additional requirement that K A |1 A = A|1 AE , which completely determines K A . For completeness and later use, we write down the form of the generators non-negative semigroups that are Schrödinger B → A semicausal.
In that case, we can choose |E| = |A| 2 .
Similar to the row-stochastic case, B (i) generates a semigroup of column-stochastic maps if and only if

D. Generators of semigroups of classical superchannels
We finally turn to semigroups of classical superchannels, that is, a collection of classical superchannels Ŝ t t≥0 , such that S 0 = id,Ŝ t+s =Ŝ tŜs and the map t →Ŝ t is continuous Furthermore,Q generates a semigroup of classical superchannels if and only ifQ generates a semigroup of preselecting supermaps and a i |K A a i = 1 AE |Aa i , for all 1 ≤ i ≤ |A|. In this case,Q is given bŷ Proof. The main idea is to relate the generators of superchannels to those of semicausal maps. This relation is given by definition for preselecting supermaps and by Theorem IV.3 for superchannels. For a generatorQ of a semigroup of preselecting supermaps ThusQ generates a semigroup of preselecting supermaps if and only ifQ can be written asQ = (C C A;B ) −1 • Q • C C A;B , for some generator Q of a semigroup of non-negative Schrödinger B → A semicausal maps. Thus to prove the first part of our Theorem, we simply take the normal form in Corollary IV.9 and compute the similarity transformation above.
can be proven by a direct calculation. Simmilarly, it is easy to show that for X A ∈ B(R A ; R A ⊗ R E ), the slightly more general identity (X A ⊗ 1 A )|Ω = (1 A ⊗ F A;E X T A A )|Ω holds, where F A;E is the flip operator that exchanges systems A and E. We use these two identities in the following calculations.
And finally, for an operatorB (i) ∈ B(R B ) and for any 1 ≤ i ≤ |A|, we have, for any M ∈ B(R A ; R B ), Applying the results of these calculations term by term to the normal form in Corollary IV.9 yields the first claim, where we defined A =Ã T A , U =ŨF B;E , K A =K T A and B (i) =B (i) . If the semigroup {Ŝ t } t≥0 consists of superchannels, that is, preselecting maps s.t. (by Theorem IV.3) the reduced maps S A t of the semigroup of semicausal maps S t :

then differentiating this relation yields
We conclude thatQ generates a semigroup of superchannels if and only if Q generates a semigroup of semicausal maps and Q A |1 A = 0. We obtain directly from Corollary IV.9 that where we used thatK A = K A is diagonal in the last step. This is the condition claimed in the theorem. Finally, (6) is obtained by combining this condition with (5).

V. THE QUANTUM CASE
We now turn to the quantum case. As introduced and described in more detail in [1], a quantum superchannel is a map that maps quantum channels to quantum channels while preserving the probabilistic structure of the theory. To achieve the latter, it is usually required that a quantum superchannel is a linear map and that probabilistic transformations, i.e., trace nonincreasing CP-maps, should be mapped to probabilistic transformations, even if we add an innocent bystander. When dealing with superchannels, we will restrict ourselves to the finite-dimensional case, and leave the infinite-dimensional case [25] for future work. We follow [1] and define superchannels as follows: is called a superchannel if for all n ∈ N the mapŜ n = id B(S 1 (C n )) ⊗Ŝ satisfies thatŜ n (T ) is a probabilistic transformation whenever T ∈ B(S 1 (C n ⊗ H A ); S 1 (C n ⊗ H B )) is a probabilistic transformation and thatŜ n (T ) is a quantum channel whenever T ∈ B(S 1 (C n ⊗ H A ); S 1 (C n ⊗ H B )) is a quantum channel.
A related concept is that of a semicausal quantum channel, which is a quantum channel on a bipartite space H A ⊗ H B such that no communication from B to A is allowed. Following [2,10], we formalize this as follows: The map L * is Schrödinger B → A semicausal if and only if the dual map L := L * * is normal and Heisenberg B → A semicausal. We will often omit the Schrödinger or Heisenberg attribute if it is clear from the context. This section is structured analogously to the section about the classical case. Namely, we will start by reminding the reader of the connection between semicausal maps and superchannels as well as the characterization of semicausal CP-maps in terms of semilocalizable maps, as schematically shown in Fig. 4. We then turn to the study of the generators of semigroups of semicausal CP-maps and finally use the correspondence between superchannels and semicausal channels to obtain the corresponding results of the generators of semigroups of superchannels.  4. Visualization of the relation between the notions of superchannels, semicausal maps and semilocalizable maps. Superchannels and semicausal maps are related via a similarity transform with the Choi-Jamiołkowski isomorphism. Schrödinger B → A semicausal maps are those maps whose output, after tracing out system 4, does not depend on input 2 (ρ orρ). Semicausal maps are precisely those maps that allow for one-way communication only. This is called semilocalizability.

A. Superchannels, semicausal channels, and semilocalizable channels
We first state the characterization of superchannels in terms of semicausal maps, obtained in [1]: The next result is due to Eggeling, Schlingemann, and Werner [10], who proved it in the finite-dimensional setting. The following form, which is a generalization of [10] to the infinite-dimensional case, and which has previously been shown in [26,Theorem 4], can be obtained from our main result (Theorem V.6) by setting K = 0:

semicausal if and only if there exists a (separable) Hilbert space H E , a unitary U ∈ U(H E ⊗ H B ; H B ⊗ H E ) and arbitrary operator A ∈ B(H
We call a normal CP-map Φ ∈ CP σ (H A ⊗ H B ) semilocalizable if its Stinespring dilation can be written in the form of Eq. (8). With that nomenclature, the above Theorem is exactly the quantum analogue of Theorem IV.4.

B. Generators of semigroups of semicausal CP maps
The main goal of this section is to establish a structure theorem for the generators of semigroups of semicausal CP-maps, the proof-structure of which is highlighted in Fig. 5. This is our main technical contribution. To get started, recall that a generator . We first observe that semicausality of the CP semigroup is equivalent to semicausality of the corresponding GKLS generator L. The insight is then that we can construct a CP-map Φ 0 , that is closely related to the CP-part of L and that is semicausal (Lemma V.13). From the semilocalizable form of Φ 0 , we then obtain an explicit form for the CP-part of L. This, together with the observation that a semicausal non-CP part has to have a local form, yields the desired normal form.
As in the classical case, we continue by showing that T t is Heisenberg B → A semicausal, for all t ≥ 0, if and only if L is Heisenberg B → A semicausal. We start by showing that the family of reduced maps {T A t } t≥0 also forms a norm-continuous semigroup of normal CP-maps. That T A t is normal and CP follows, since for any density operator ρ B ∈ S 1 (H B ), we have So T A t is a normal CP-map as composition of normal CP-maps. It remains to check the semigroup properties (T A 0 = id A , T A t+s = T A t T A s and norm-continuity). We have Thus, we conclude that T A t = e tL A , for some generator L A ∈ B(B(H A )) of normal CP-maps. We further have Thus, L is semicausal if T t is semicausal for all t ≥ 0. Conversely, if L is semicausal, then T t is semicausal for all t ≥ 0, since Therefore, our task reduces to characterizing semicausal maps in GKLS-form, i.e., we want to determine the corresponding Φ and K. Our main result (Theorem V.6) is a normal form which allows us to list all semicausal generators L. • L is self-adjoint and P ⊥ LP ⊥ ≥ 0, and The generated semigroup is unital (i.e., T t (1 AB ) = 1 AB , for t ≥ 0) if and only if tr Furthermore, a linear map L :

generator of a semigroup of Schrödinger B → A semicausal CP-maps if and only if
• L is self-adjoint and P ⊥ LP ⊥ ≥ 0, and The generated semigroup is trace-preserving (i.e., tr [T t (ρ)] = tr [ρ], for ρ ∈ B(H A ⊗ H B ) and t ≥ 0) if and only if tr A 2 L A = 0.
Thus, checking whether a map L is the generator of a semigroup of semicausal CP-maps reduces to checking several semidefinite constraints. In particular, the problem to optimize over all semicausal generators is a semidefinite program.
Proof. It is known (see, e.g., the appendix in [21]) that L generates a semigroup of CP-maps if and only if L is self-adjoint and P ⊥ LP ⊥ ≥ 0. This criterion goes by the name of conditional complete positivity [22]. Thus, it remains to translate the other criteria to the level of Choi-Jamiołkowski operators. If L is Heisenberg B → A semicausal, then Finally, it is known that a semigroup of CP-maps is unital if and only if L(1 A 2 B 2 ) = 0. But this is equivalent to our criterion, since a simple calculation shows that This finishes the proof for the Heisenberg picture case. The Schrödinger case can be proven along similar lines, or be obtained directly from the Heisenberg case via the identity Let us now return to the main goal of this section: finding a normal form for semicausal generators in GKLS-form. We motivate (and interpret) our normal form as the 'quantization' of the normal form for generators of classical semicausal semigroups (Theorem IV.7). In the classical case, the normal form had two building blocks: an operator of the form where Φ sc is non-negative and semicausal and an operator of the form where Φ sc ∈ CP σ (H A ⊗ H B ), given in Stinespring form by Φ sc (X) = V † sc (X ⊗ 1 E )V sc , is semicausal. One readily verifies that L 1 defines a semicausal generator. To 'quantize' the second building block, note that Q 2 does not induce any change on system A. Indeed, since the generated semigroup looks like the identity on system A. In the quantum case, semigroups that do not induce any change on system A are more restricted, since any information-gain about system A inevitably disturbes system A -so there can be no conditioning as in the classical case. Indeed, if one requires that T t ∈ CP σ (H A ⊗ H B ) satisfies the quantum analogue of Eq. (11), namely for all X A ∈ B(H A ), then T t = id A ⊗ Θ t for some unital map Θ t ∈ CP σ (H B ), see Appendix B for a proof. Differentiation of T t = id A ⊗ Θ t at t = 0 now implies that the generator of a semigroup of CP-maps that satisfy (12) are of the form id A ⊗B, wherê B generates a semigroup of unital CP-maps (i.e.,B(1 B ) = 0). To conclude, the two building blocks are operators of the form of L 1 in Eq. (10) and maps L 2 of the form In the classical case, we obtained the normal form (Theorem IV.7) by taking a convex combination of the basic building blocks. This corresponds to probabilistically choosing one or the other. In quantum theory, there is is a more general concept: superposition. To account for this, we construct our normal form not as a convex combination of the maps L 1 and L 2 , but by taking a linear combination (superposition) of the Stinespring operators V sc and 1 A ⊗ B as the Stinespring operator of the CP-part of the GKLS-form (note here that the coefficients can be absorbed into V sc and 1 A ⊗ B, respectively). This means that if L is given by Eq. (9) with Φ(X) = V † (X ⊗ 1 E )V , then we take V = V sc + 1 A ⊗ B. It turns out that K can then be chosen such that L becomes semicausal. Also note that we can further decompose V sc = (1 A ⊗U)(A ⊗ 1 B ), as in Theorem V.4.
Our main technical result is that the heuristics employed in the 'quantization' procedure above is sound, i.e., that the generators constructed in that way are the only semicausal generators in GKLS-form.
Remark V.7. Note that the characterization in Theorem V.6 is for generators of Heisenberg B → A semicausal dynamical semigroups. There are two special cases of interest: First, if we want the dynamical semigroup to be unital, then we need to further impose L(1 A ⊗ 1 B ) = 0 in the normal form above, which is equivalent to A † A = K A + K † A -a constraint that also appears in the usual Linblad form. Second, if the dynamical semigroup corresponds (in the sense of Theorem V.3) to a semigroup of superchannels, then we additionally require that the reduced generator satisfies L A * (1 A ) = 0. We will use this in the "translation step" in Theorem V.18.
Remark V.8. In the finite-dimensional case the proof of Theorem V.6 is constructive. In Appendix C we discuss in detail how to obtain the operators A, U, K A , B and H B starting from the conditions in Lemma V.5.
The remainder of this section is devoted to the proof of Theorem V.6, whose structure is highlighted in Fig. 5. We begin with a technical observation about certain Haar integrals.
Lemma V.9. Let H n be an n-dimensional subspace of H A with orthogonal projection P n ∈ B(H A ) and let V ∈ B(H A ⊗ H B ; H A ⊗ H C ). Then where the integration is w.r.t. the Haar measure on U P (H n ). It follows that P n ⊗ 1 n tr P n [V ] ≤ V . Furthermore if H is separable infinite-dimensional, with orthonormal basis {|e i } i∈N and H n = span{|e 1 , |e 2 , . . . , |e n }, then there exists B ∈ B(H B ; H C ) and an ultraweakly convergent subsequence of P n ⊗ 1 n tr P n [V ] n∈N with limit 1 A ⊗ B. Proof. To calculate the integral, we employ the Weingarten formula [27][28][29], which for the relevant case reads: . . , | f n } of H n . A basis expansion then yields For the second claim, we note that a standard estimate of the integral yields 1 n tr P n [V ] = P n ⊗ 1 n tr P n [V ] ≤ V . Thus the sequence 1 n tr P n [V ] n∈N is bounded and hence, by Banach-Alaoglu, has an ultraweakly convergent subsequence, whose limit we call B. The claim then follows by observing that, under the separability assumption, (P n ) n∈N converges ultraweakly to 1 A and that the tensor product of two ultraweakly convergent sequences converges ultraweakly.
As a first step towards our main result, we provide a characterization of those semicausal Lindblad generators that can be written with vanishing CP part.
Let H n be an n-dimensional subspace of H A and U ∈ U P (H n ). Then where P n ∈ B(H A ) is the orthogonal projection onto H n . We integrate both sides w.r.t. the Haar measure on U P (H n ). Lemma V.9 and some rearrangement and taking the conjugate yields for some operator L A n ∈ B(H A ). If H A is finite-dimensional, we can take H n = H A , so that P n = 1 A . Hence If H A is separable infinite-dimensional, we obtain the same result via a limiting procedure n → ∞ as follows: Let {|e i } i∈N be an orthonormal basis of H A and set H n = span{|e 1 , |e 2 , . . . , |e n }. Then, the second part of Lemma V.9 allows us to pass to a subsequence of P n ⊗ 1 n tr P n K † n∈N that converges ultraweakly to a limit 1 A ⊗ B. The corresponding subsequence of ((P n ⊗ 1 B )K) n∈N converges ultraweakly to K, and hence that subsequence of L A n ⊗ 1 B n∈N converges ultraweakly to a limitK A ⊗ 1 B . I.e., we get which can only be true for all If we had restricted our attention to Hamiltonian generators and unitary groups in finite dimensions, an analog of this Lemma would have already followed from the fact that semicausal unitaries are tensor products, which was proved in [2] (and reproved in [11]).
As another technical ingredient, the following lemma establishes a closedness property of the set of semicausal maps.
, since the trace-class operators are an ideal in the bounded operators. Hence, by definition of the ultraweak topology, Since tr ρ Φ A m,n (X A ) ⊗ 1 B converges as n → ∞ for every ρ ∈ S 1 (H A ⊗ H B ), the sequence Φ A m,n (X A ) ⊗ 1 B n∈N converges ultraweakly [30]. We call the limit Φ A m (X A ) ⊗ 1 B . It is then easy to see that Φ A m (X A ), viewed as a map on B(H A ) is linear and continuous. This tells us that the map Φ m : Repeating the argument above then shows that Φ is semicausal.
As a final preparatory step we observe that, given a semicausal Lindblad generator, we can use its CP part to define a family of semicausal CP-maps.
is Heisenberg B → A semicausal for every Y, Z ∈ B(H A ).

Proof. For every M ∈ B(H
This map has already been used, for a different purpose, in Lindblad's original work [19,Eq. 5.1]. It follows from the semicausality of L that, if we choose M = M A ⊗ 1 B , for some M A ∈ B(H A ), then Ψ M is semicausal. Furthermore, a calculation shows that By choosing N = Z ⊗ 1 B and M = Y ⊗ 1 B , it follows that S Y,Z is the linear combination of four semicausal maps, and hence is itself semicausal.
We now combine this Lemma with an integration over the Haar measure to obtain the key Lemma in our proof. Proof. Let H n and H m be n and m dimensional subspaces of H A with respective orthogonal projections P n ∈ B(H A ) and P m ∈ B(H A ). Since for every U ∈ U P (H n ) and W ∈ U P (H m ), the map S U,W , defined in Lemma V.12 is semicausal, also the map is semicausal. Writing out the definition of S U,W yields where the last line was obtained by using Lemma V.9. If H A is finite-dimensional, we can choose H n = H m = H A , so that P n = P m = 1 A , and obtain the desired result immediately. If H A is separable infinite-dimensional and {|e i } i∈N is an orthonormal basis and H k := span{|e 1 , |e 2 , . . . , |e k }, then, by Lemma V.9, the sequence P k ⊗ 1 k tr P k [V ] k∈N has an ultraweakly convergent subsequence with a limit 1 A ⊗ B, where B ∈ B(H B ; H B ⊗ H E ). Furthermore, since (P k ) k∈N converges ultraweakly to 1 A , we have that the sequence V (P k ⊗ 1 B ) − P k ⊗ 1 k tr P k [V ] k∈N has a subsequence that converges ultraweakly to V − 1 A ⊗ B. Hence, by passing to subsequences, we can apply Lemma V.11, which yields that S is semicausal.
Remark V.14. The previous two lemmas are at the heart of our result. They illustrate a (to the best of our knowledge) novel technique that allows to characterize GKLS generators with a certain constraint, if this constraint is well understood for completely positive maps. It seems useful to develop this method more generally, but this is beyond the scope of the present work.
With these tools at hand, we can now prove our main result.
Proof. (Theorem V.6) A straightforward calculation shows that L, defined via (22a) and (22b) is semicausal. To prove the converse, note that by the Stinespring dilation theorem, there exist a separable Hilbert spaceH E andṼ ∈ B( It follows that the map defined by X → −(K − κ) † X − X(K − κ) is semicausal. Thus, Lemma V.10 implies that there exist What we have achieved so far is thatṼ = V sc + 1 ⊗B and , then we are basically done. But this decomposition is given (up to details) by the equivalence between semicausal and semilocalizable channels [10]. Since the conclusion in [10] was in the finite-dimensional setting, we will repeat the argument here, showing that it goes through also for infinite-dimensional spaces, while paying special attention to the dimensions of the spaces involved.
The last condition is called the minimality condition. We then get As a first consequence, we obtain the analogous theorem for semigroups of Schrödinger B → A semicausal CP-maps: . Then L is Schrödinger B → A semicausal, if and only if, K, V and H E can be chosen as in (22a) and (22b).
As a further corollary, we translate the results above to the familiar representation in terms of jump-operators (by going from Stinespring to Kraus).
Proof. A simple calculation by defining the Kraus operators as (1 AB ⊗ |e i )V , with {|e j } j an orthonormal basis of H E and V given by Theorem V.6.
We conclude this section about semicausal semigroups with an example that uses our normal form in full generality.
FIG. 6. Systems A and B describe 2-level systems, respectively. The allowed interactions are infinitesimally described as follows: If A is in its excited state, it can emit a photon. Through parametric down-conversion, the photon is converted into two photons (of lower energy). One of those two photons, k 1 , is sent to a detector D 1 . The other, k 2 , is sent to B. If B is in its ground state, it absorbs k 2 . If B is in its excited state, it cannot absorb k 2 , so k 2 passes through B and travels to a detector D 2 . Additionally, in this case, B can emit a photon, indistinguishable from k 1 , to D 1 .
Example. We consider the scenario of two 2-level atoms that can interact according to the processes specified in Fig. 6. We can describe this process either via a dilation (as in Theorem V.6) or via the Kraus operators (as in Corollary V.16). In the dilation picture, we introduce an auxiliary Hilbert space H E := H 1 ⊗ H 2 , where H i is for the i th photon. Then, the process is described The crucial feature of this example is that the CP-part of the generator (tr E V ·V † ) cannot be written as a convex combination of the two building blocks (Φ sc and id A ⊗B). As mentioned also in the quantization procedure before, this is a pure quantum feature and stems from the fact that it cannot be determined if a photon arriving at the detector D 1 came from B or A. Hence, the system remains in a superposition state. We can also look at the usual representation via jump operators. This can be achieved by switching from dilations to Kraus operators. We obtain the two jump-operators where L e = |0 1| and L a = L † e describe emission and absorption of a photon, respectively. Thus, the usual Lindblad equation reads: It is also possible and instructive to consider the reduced dynamics on system A, which can also be described by a Lindblad equation, since B does not communicate to A (this is not true otherwise): where ρ A (t) = tr B [ρ(t)]. Not surprisingly (given our model), this describes an atom emitting photons.

C. Generators of semigroups of quantum superchannels
We finally turn to semigroups of quantum superchannels (on finite-dimensional spaces), that is, a collection of quantum superchannels {Ŝ t } t≥0 ⊆ B(B(B(H A ); B(H B ))), such thatŜ 0 = id,Ŝ t+s =Ŝ tŜs and the map t →Ŝ t is continuous (w.r.t. any and thus all of the equivalent norms on the finite-dimensional space B (B(B(H A ); B(H B )))). To formulate a technically slightly stronger result, we call a mapŜ ∈ B (B(B(H A ); B(H B ))) a preselecting supermap if C A;B •Ŝ • C −1 A;B is a Schrödinger B → A semicausal CP-map. Theorem V.3 then tells us that a superchannel is a special preselecting supermap. Again, as for semicausal CP-maps, we characterize the generators of semigroups of preselecting supermaps and superchannels in two ways: First, we answer how to determine if a given mapL ∈ B (B(B(H A ); B(H B ))) is such a generator. Second, we provide a normal form for all generators.
The answer to the first question is really a corollary of Lemma V.5 together with Theorem V.3. To this end, defineL := and C AB;AB is defined w.r.t. the product of the two bases. Furthermore, we introduced the spaces to be the orthogonal projection onto the orthogonal complement of {|Ω }, where |Ω = ∑ i, j |a i ⊗ |b j ⊗ |a i ⊗ |b j . We then have •L is self-adjoint and P ⊥L P ⊥ ≥ 0, L is preselecting if and only if the first two conditions hold.
Proof. Theorem V.3 tells us that {Ŝ t } t≥0 forming a semigroup of superchannels is eqiuvalent to S t = C A;B •Ŝ t • C −1 A;B forming a semigroup of Schrödinger B → A semicausal CP-maps and that the reduced map S A t satisfies S A t (1 A ) = 1 A . By Lemma V.5 the semicausal semigroup property is equivalent to the first two conditions in the statement. This proves the claim about preselectinĝ L. By differentiation, it follows that S A t (1 A ) = 1 A is satisfied if and only if L A , the generator of {S A t } t≥0 , satisfies L A (1 A ) = 0. But since tr A 1 L A = L A (1 A ), the claim follows.
We finally turn to a normal form for generators of semigroups of preselecting supermaps and superchannels: In that case, we can splitL into a dissipative partD and a 'Hamiltonian' partĤ, i.e., a part which generates a (semi-)group of invertible superchannels whose inverses are superchannels as well. We haveL(T ) =D(T ) +Ĥ(T ), withD where H A is the imaginary part of K A , wherê and where [·, ·] and {·, ·} denote the commutator and anticommutator, respectively.
Remark V. 19. Similarly to Theorem V.6, the proof of Theorem V.18 is constructive. In Appendix D we discuss in detail how to obtain the operators A, U, K A , B, H A and H B starting from the conditions in Theorem V.17.
As in the classical case, the proof strategy is to use the relation between superchannels and semicausal channels and Theorem V.6. As this translation process is more involved than in the classical case, we need two auxilliary lemmas. with Here, the partial transpose on H A is taken w.r.t. the basis used to define the Choi-Jamiołkowski isomorphism.
Proof. The proof is a direct calculation. We present it in detail in Appendix A.
Proof. The proof is a direct calculation. We present it in detail in Appendix A.
We are finally ready to prove Theorem V.18 Proof. (Theorem V.18) The idea is to relate the generators of superchannels to semicausal maps. This relation is given by definition for preselecting superamps and by Theorem V.3 for superchannels. For a generatorL of a semigroup of preselecting supermaps {Ŝ t } t≥0 , we haveL ThusL generates a semigroup of preselecting supermaps if and only ifL can be written asL = C −1 A;B • L • C A;B for some generator L of a semigroup of Schrödinger B → A semicausal CP-maps. Thus to prove the first part of our theorem, we can take the normal form in Corollary V.15 and compute the similarity transformation above. We now execute this in detail. To start with, Corollary V.15 tells us that which is an expression suitable for a term by term application of Lemma V.20. Doing so yieldŝ where we defined U :=ŨF B;E , B :=BΞ † B F B;E , A :=Ã T A Ξ † A and σ := |ξ ξ |. This proves Equation (17). Similarly, upon defining κ L (ρ) := Kρ we can write [32] and apply Lemma V.20 term by term, which yieldŝ ) and also S A t (1 A ) = 1 A holds. Differentiating that expression at t = 0 yields the equivalent condition L A (1 A ) = 0. So, our goal is to incorporate the last condition into the form of (22). To do so, we determine L A by calculating tr B [L(ρ)], where L is in the form of (22). We obtain tr B A . Thus, the condition L A (1) = 0 holds if and only if tr E ÃÃ † =K A +K † A . Transposing both sides of this equation and using that the definition of A implies thatÃ = A T A Ξ A , yields tr E A T A (1 A ⊗ σ )(A † ) T A T = K A + K † A . But the left hand side is, by Lemma V.21, equal to tr σ A † A . This proves the claim thatL generates a semigroup of superchannels if and only ifL is hyper-preselecting and tr σ A † A = K A + K † A . Finally, defining H A := 1 2i (K A − K † A ) and a few rearrangements lead to (19).

VI. CONCLUSION
a. Summary The underlying question of this work was: How can we mathematically characterize the processes that describe the aging of quantum devices? We have argued that, under a Markovianity assumption, such processes can be modeled by continuous semigroups of quantum superchannels. Therefore, the goal of this work was to provide a full characterization of such semigroups of superchannels.
We have derived such a general characterization in terms of the generators of these semigroups. Crucially, we have exploited that superchannels correspond to certain semicausal maps, and that therefore it suffices to characterize generators of semigroups of semicausal maps. We have demonstrated both an efficient procedure for checking whether a given generator is indeed a valid semicausal GKLS generator and a complete characterization of such valid semicausal GKLS generators. The latter is constructive in the sense that it can be used to describe parametrizations of these generators. Aside from the theoretical relevance of these results, they will be valuable in studying properties of these generators numerically. Finally, we have translated these results back to the level of superchannels, thus answering our initial question.
We have also posed and answered the classical counterpart of the above question. I.e., we have characterized the generators semigroups of classical superchannels and of semicausal non-negative maps. These results for the classical case might be of independent interest. From the perspective of quantum information theory, they provide a comparison helpful to understand and interpret the characterizations in the quantum case.
b. Outlook and open questions We conclude by presenting some open questions raised by our work. First, in our proof of the characterization of semicausal GKLS generators, we have described a procedure for constructing a semicausal CP-map associated to such a generator. We believe that this method can be applied to a wide range of problems. Determining the exact scope of this method is currently work in progress.
Second, there is a wealth of results on the spectral properties of quantum channels and, in particular, semigroups of quantum channels. With the explicit form of generators of semigroups of superchannels now known, we can conduct analogous studies for semigroups of quantum superchannels. Understanding such spectral properties, and potentially how they differ from the properties in the scenario of quantum channels, would in particular lead to a better understanding of the asymptotic behavior of semigroups of superchannels, e.g., w.r.t. entropy production [33,34], the thermodynamics of quantum channels [35] or entanglement-breaking properties [36].
A further natural question would be a quantum superchannel analogue of the Markovianity problem: When can a quantum superchannelŜ be written as eL for someL that generates a semigroup of superchannels? Several works have investigated the Markovianity problem for quantum channels [21,[37][38][39] and a divisibility variant of this question, both for quantum channels and for stochastic matrices [40][41][42]. It would be interesting to see how these results translate to quantum or classical superchannels. Similarly, we can now ask questions of reachability along Markovian paths. Yet another question aiming at understanding Markovianity: If we consider master equations arising from a Markovianity assumption on the underlying process formalized not via semigroups of channels, but instead via semigroups of superchannels, what are the associated classes of (time-dependent) generators and corresponding CPTP evolutions?
Two related directions, both of which will lead to a better understanding of Markovian structures in higher order quantum operations, are: Support our mathematical characterization of the generators of semigroups of superchannels by a physical interpretation, similar to the Monte Carlo wave function interpretation of Lindblad generators of quantum channels. And extend our characterization from superchannels to general higher order maps.
This work has focused on generators of general semigroups of superchannels, without further restrictions. For quantum channels and their Lindblad generators, there exists a well developed theory of locality, at the center of which are Lieb-Robinson bounds [43]. If we put locality restrictions on generators of superchannels, how do these translate to the generated superchannels?
Finally, an important conceptual direction for future work is to identify further applications of our theory of dynamical semigroups of superchannels. In the introduction we gave a physical meaning to semigroups of superchannels by relating them to the decay process of quantum devices. This, however, is only one possible interpretation. For example, semigroups of superchannels might also describe a manufacturing process, where a quantum device is created layer-by-layer. We hope that other use-cases will be found in the future. network pictures, can be executed as follows: This proves the claim.