Quantum capacity analysis of multi-level amplitude damping channels

Evaluating capacities of quantum channels is the first purpose of quantum Shannon theory, but in most cases the task proves to be very hard. Here, we introduce the set of Multi-level Amplitude Damping quantum channels as a generalization of the standard qubit Amplitude Damping Channel to quantum systems of finite dimension d. In the special case of d = 3, by exploiting degradability, data-processing inequalities, and channel isomorphism, we compute the associated quantum and private classical capacities for a rather wide class of maps, extending the set of models whose capacity can be computed known so far. We proceed then to the evaluation of the entanglement assisted quantum and classical capacities. Estimating the capacity of quantum channels is relevant to understand at which rate quantum information can be exchanged. Here, the authors introduce multi-level amplitude damping channels that generalize the qubit amplitude damping noise model, analyse their quantum and private capacities, and exactly derive the capacities of quantum channels that are not anti-degradable.


T
he main goal of quantum information and communication theory is to understand how can we store, process, and transfer information in a reliable way and, from the physical point of view, to individuate realistic platforms by means of which performing these tasks.All this is done by exploiting the characteristic features of quantum mechanics.Focusing on quantum communication, every communication protocol can be seen as a physical system (the encoded message) undergoing some physical transformation that translates it in space or time.Any real-world application though suffers from some kind of noise, each of which can be in turn described as a quantum process or equivalently as a quantum channel.Following the work of Shannon 1 and the later quantum generalizations, the ability of a quantum channel to preserve the encoded classical or quantum information is described by its capacities 2,3 .In the classical case, we can only transfer classical information, hence we only need to deal with the classical capacity.In the quantum framework, we can also transfer quantum states and consequently, in addition to the classical capacity, we count also the quantum capacity.Moreover, the family of capacities associated with a quantum channel can be enlarged assuming the communicating parties to be able to perform specific tasks or to share further resources such as, for instance, entanglement 2,[4][5][6][7][8] .
One among the simplest models for quantum noise is given by the amplitude damping channel (ADC).While the ADC has been thoroughly studied and characterized, in terms of capacities in various settings, for the qubit framework [9][10][11][12] , a general treatise for qudit (d-dimensional) systems is still missing and likely not possible to attain.Because of these reasons ADC for d > 2 has to be approached case by case, and the literature regarding capacities of fixed finite dimensions ADC is still remarkably short [12][13][14] .In recent years though higher dimensional systems have attracted the attention of a growing number of researchers, since they have been shown to provide potential advantages both in terms of computation (see e.g.6][17][18][19][20] ) and communication or error correction (see e.g.2][23][24] ) together with the fact that more experimental implementations have been progressively made available (see e.g.6][27][28][29][30][31][32] ).Both near-term and long-term applications of quantum information, whether in a computational (e.g.distributed quantum computing) or communication (e.g.long-distance quantum communication) framework, will necessitate high-fidelity quantum state transmission to achieve reliable and advantageous purposes.This drives the need of an extensive characterization of communications performances in all available noise regimes and most general noise models.In addition to these considerations, new results on the quantum capacity of finite dimensional channels can also be applied to higher dimensional maps via the partially coherent direct sum (PCDS) channels approach 33 , placing in a wider context the efforts dedicated to the analysis of non-qubit channels.Among non-qubit systems, threedimensional systems (qutrit) have received particular attention because of their relative accessibility both theoretically and experimentally (see e.g.5][36][37][38][39][40][41][42][43]. Considering this, in this paper we focus on the model for quantum noise given by the ADC in the multi-level setting, that we will denote as multi-level amplitude damping (MAD) channel.In particular, we perform a systematic analysis of the MAD on the qutrit space: while we will not approach the issue of the classical capacity of the channel, we will focus on the quantum capacity, private classical capacity, and entanglement-assisted capacities, trying to understand in which conditions these quantities can be known.We find that the quantum and private classical capacities are exactly computable in large regions of the damping parameters space, even when the qutrit MAD are not degradable.When an exact value is missing, we are still able to provide upper bounds exploiting composition rules and data-processing inequalities.Finally we compute the quantum and classical entanglement-assisted capacities.

Results and discussion
MAD channels and composition rules.The transformations we focus on in the present work are special instances of the multilevel versions of the qubit ADC 9 , hereafter indicated as MAD channels in brief, which effectively describe the decaying of energy levels of a d-dimensional quantum system A. In its most general form, given i j i f g i¼0;ÁÁÁ;dÀ1 an orthonormal basis of the Hilbert space H A associated with A (hereafter dubbed the computational basis of the problem), a MAD channel D is a completely positive trace preserving (CPTP) mapping 2,4-8 acting on the set LðH A Þ of linear operators of the system, defined by the following set of with γ ji real quantities describing the decay rate from the j-th to the i-th level that fulfill the conditions 0 ≤ γ ji ≤ 1; 8i; j s:t: 0 ≤ i<j ≤ d À 1; Accordingly, given ρ 2 SðH A Þ a generic density matrix of the system A, the MAD channel D will transform it into the output state defined as By construction, D always admits the ground state 0 j i as a fixed point, i.e.Dð 0 j i 0 h jÞ ¼ 0 j i 0 h j, even though, depending on the specific values of the coefficients γ ji , other input states may fulfill the same property as well.Limit cases are γ ji = 0 ∀ i, j, where all levels are untouched and D reduces to the noiseless identity channel Id which preserves all the input states of A. On the opposite extreme are those examples in which for some j we have ξ j = 1, corresponding to the scenario where the j-th level becomes totally depopulated at the end of the transformation.The maps in Eq. ( 3) provide also a natural playground to describe PCDS channels 33 .Last but not the least, an important and easy to verify property of the maps in Eq. ( 3) is that they are covariant under the group formed by the unitary transformations Û which are diagonal in the computational basis i j i f g i¼0;ÁÁÁ;dÀ1 , i.e.
For what concerns the present work, we shall restrict our analysis to the special set of MAD channels as in Eq. ( 3) associated with a qutrit system (d = 3) whose decay processes, pictured in the top panel of Fig. 1, are fully characterized by only three rate parameters γ ji that for the ease of notation we rename with the cartesian components of a 3D vector γ !ðγ 1 ; γ 2 ; γ 3 Þ.
Accordingly, expressed in terms of the matrix representation induced by the computational basis 0 j i; 1 j i; 2 j i f g , the Kraus operators in Eq. ( 1) write explicitly as with CPTP conditions from Eq. ( 2) given by 0 ≤ γ j ≤ 1; 8j ¼ 1; 2; 3; which produce the volume visualized in the bottom panel of Fig. 1.
The resulting mapping as in Eq. ( 3) for the channel D ðγ 1 ;γ 2 ;γ 3 Þ reduces, hence, to the following expression while the associated complementary CPTP transformation 2,4-6 computed as in Eq. (67) of "Methods", for generic choices of the system parameters, transforms A into a four-dimensional state via the mapping where for i, j ∈ 0, 1, 2, ρ ij i h jρ j j i are the matrix entries of the input density operator ρ 2 SðH A Þ.
Quantum and private classical capacities for qutrit MAD.The quantum capacity Q(Φ) of a quantum channel Φ is a measure of how faithfully quantum states can be transmitted from the input to the output of the associated CPTP map by exploiting proper encoding and decoding procedures that act on multiple transmission stages 2,[4][5][6][7][8] .The private classical capacity C p instead quantifies the amount of classical information transmittable per channel use under the extra requirement that the entire signaling process allows the communicating parties to be protected by eavesdropping by an adversary agent that is controlling the communication line.The explicit evaluation of these important functionals is one of the most elusive task of quantum information theory, as testified by the limited number of examples which allow for an explicit solution.A closed expression for the quantum capacity is provided by the formula [44][45][46] QðΦÞ ¼ lim where the maximization in Eq. ( 10) is performed over the set of density matrices of n channel uses, and J is the coherent information JðΦ n ; ρðnÞ Þ SðΦ n ðρ ðnÞ ÞÞ À Sð Φn ðρ ðnÞ ÞÞ; ð11Þ with SðρÞ ÀTr½ρ log 2 ρ the von Neumann entropy of the state ρ, and Φ the complementary channel of Φ, see "Methods"-Complementary channels and degradability.For the private classical capacity instead, we have 46,47 : where the maximization is now performed over all quantum ensembles E n p i ; ρðnÞ i n o of n channel uses, and where is the Holevo information functional.The difficulties related to the evaluation of the above formulas are well known and ultimately the reason underlying our efforts here.An exception to this predicament is given by degradable 48 and antidegradable 49 channels.Degradable channels are those for which exists a CPTP map N s.t.Φ ¼ N Φ, while antidegradable channels are those for which exists a CPTP map M s.t.Φ ¼ M Φ; for more details, see "Methods"-Complementary channels and degradability.For degradable channels, Q and C p result to be additive, so the regularization over n in Eq. ( 9) is not needed, leading to the following single-letter formula 50 C p ðΦÞ ¼ QðΦÞ ¼ Q ð1Þ ðΦÞ: ð15Þ For antidegradable channels instead, due to a no-cloning argument Q = 0 while, from expression in Eq. ( 13), positivity of private classical capacities and data processing, we have C p = 0.So no maximizations are needed.
Building up from these premises, here we present a thoughtful characterization of the quantum capacity QðD γ !Þ and the private defined in Eq. ( 7).We stress that while failing to provide the explicit solution for all rate vectors γ ! in the allowed domain defined by Eq. ( 6), in what follows we manage to deliver the exact values of QðD γ !Þ and C p ðD γ !Þ for a quite a large class of qutrit MAD channels by making use of degradability properties 48 , dataprocessing (or bottleneck) inequalities 51,52 , and channel isomorphism.In particular, we anticipate here that, for those D γ ! which are provably degradable, we shall exploit the covariance property in Eq. ( 4) to further simplify the single-letter formula in Eq. ( 15) as where the maximization is performed on input states of A which are diagonal in the computational basis of the problem, i.e. the density matrices of the form ρdiag ¼ P 2 i¼0 p i i j i i h j with p 0 , p 1 , p 2 ∈ [0, 1] being usually called "populations" and fulfilling the normalization constraint p 0 + p 1 + p 2 = 1, see "Methods", Eq. ( 82), and below for details.Notably, when applicable, Eq. ( 16) relies on an optimization of a functional of only d − 1 real variables in the case of a qudit MAD and consequently just two real variables in the case of a qutrit MAD (namely the populations p 0 and p 1 ), which can be easily carried out (at least numerically).
To begin with, observe that, as anticipated in Eq. ( 8), the complementary map D γ ! of a generic qutrit MAD channel D γ !sends the input states of A into a four-dimensional "environment state".In the end, this is a consequence of the fact that the (minimal) number of Kraus operators we need to express Eq. ( 7) is 4. Unfortunately, this number also ensures us that the channel is not degradable: it has been indeed shown 53 that a necessary condition for any CPTP map with output dimension 3 to be degradable is that its associated Choi rank, and consequently the minimal number of Kraus operators we need to express such transformation, is at most 3.This brings us to consider some simplification in the problem, e.g. by fixing some of the values of the damping parameters.One approach is represented by the selective suppression of one (or two) of the decaying channels, i.e. imposing one (or two) of the parameters γ i equal to 0 or to their maximum allowed value, choices that as we shall see, will effectively allow us to reduce the number of degrees of freedom of the problem.
Single-decay qutrit MAD channels.We consider here instances of the qutrit MAD channel in which only one of the three damping parameters γ i is explicitly different from zero, i.e. the maps D ðγ 1 ;0;0Þ , D ð0;γ 2 ;0Þ , and D ð0;0;γ 3 Þ associated, respectively, with the edges DA, DF, and DE of Fig. 1.It is easy to verify that these three sets of transformations can be mapped into each other via unitary conjugations that simply permute the energy levels of the system: for instance D ð0;0;γ 3 ¼γÞ can be transformed into D ðγ 1 ¼γ;0;0Þ by simply swapping levels 1 j i and 2 j i. Accordingly, the capacities of these three sets must coincide, since each channel can be obtained from the other, i.e.
(similarly for C p ).By virtue of this fact, without loss of generality, in the following we report the analysis only for D ðγ 1 ;0;0Þ , being the results trivially extendable to the remaining two.For this purpose, we observe that from Eq. ( 5) it follows that D ðγ 1 ;0;0Þ possesses only two non-zero Kraus operators, i.e.
Transformation in Eq. ( 7) is then given by D ðγ 1 ;0;0Þ ðρÞ ¼ and the complementary channel Dðγ 1 ;0;0Þ that can be expressed as a mapping that connects the system A to a two-dimensional environmental system E, i.e.
By the study of degradability and the techniques discussed in "Methods" and in Supplementary Note 1, we are able to evaluate Q and C p for every γ.The results are summarized in the plot in Fig. 2.
Fig. 2 Quantum capacity for the single-decay.a Profile of the quantum and the private classical capacity for the channel D ðγ 1 ;0;0Þ w.r.t. the damping parameter γ 1 .For γ 1 ≤ 1/2, the channel is degradable and the reported value follows from the numerical maximization.For γ > 1/2, instead, the channel is neither degradable nor antidegradable: here the associated capacity value is equal to 1. Notice that the reported values respect the monotonicity property given by Eq. (48).b Populations p 0 , p 1 , and p 2 of those states that maximize the quantum capacity formula for the channel D ðγ 1 ;0;0Þ w.r.t. the damping parameter γ 1 .
Complete damping of the first excited state (γ 1 = 1).Assume next that our qutrit MAD channel of Eq. ( 7) is characterized by the maximum value of γ 1 allowed by CPTP constraint of Eq. ( 6), i.e. γ 1 = 1, region represented by the ABC triangle of Fig. 1.This map corresponds to the case where the initial population of the first excited level 1 j i gets completely lost in favor of the ground state 0 j i of the model so that Eqs. ( 7) and ( 8) rewrite as The above expressions make it explicit that, at variance with the case discussed in the previous section and in agreement with the conclusions of ref. 53 , the map D ð1;γ 2 ;γ 3 Þ is not degradable.Indeed we notice that while Dð1;γ 2 ;γ 3 Þ ðρÞ preserves information about the components ρ 11 , ρ 01 , ρ 10 , ρ 12 , ρ 21 of the input state ρ, no trace of those terms is left in D ð1;γ 2 ;γ 3 Þ ðρÞ: accordingly it is technically impossible to identify a linear (not mentioning CPTP) map N which applied to D ð1;γ 2 ;γ 3 Þ ðρÞ would reproduce Dð1;γ 2 ;γ 3 Þ ðρÞ for all ρ.Despite this fact, it turns out that also for D ð1;γ 2 ;γ 3 Þ , the capacity can still be expressed as the single letter expression in Eq. ( 16).For the technical details, we refer the reader to Supplementary Note 2, where we apply techniques expressed in "Methods".We report the results in Fig. 3.
exchange of γ 1 and γ 3 .Indeed, indicating with V the unitary gate that swaps levels 2 j i and 3 j i we have that which by data-processing inequality implies with an analogous identity applying in the case of the private classical capacity.As reported in Supplementary Note 3, applying techniques in "Methods", we produce results showed in Figs. 4  and 5.
The qutrit MAD channel on the γ 2 + γ 3 = 1 plane.Let us now consider the regime with γ 2 + γ 3 = 1 where rate vectors γ !belong to the rectangular area BEFC of Fig. 1.Under this condition, the map in Eq. ( 7) still admits four Kraus operators and becomes We notice that the level 2 j i gets completely depopulated and that the channel can be expressed as where D γ 1 is a standard qubit ADC channel connecting level 1 j i to level 0 j i with damping rate γ 1 , while now C is a CPTP transformation sending the qutrit A to the qubit system spanned by vectors 0 j i; 1 j i and completely erasing the level 2 j i, moving its population in part to 1 j i and in part to 0 j i, i.e.
Accordingly, the quantum capacity of D γ 1 computed in ref. 9 is an explicit upper bound for QðD ðγ 1 ;γ 2 ;1Àγ 2 Þ Þ and C p ðD ðγ 1 ;γ 2 ;1Àγ 2 Þ Þ (remember that for the qubit ADC Q and C p coincide).On the other hand, QðD γ 1 Þ is also a lower bound for QðD ðγ 1 ;γ 2 ;1Àγ 2 Þ Þ and C p ðD ðγ 1 ;γ 2 ;1Àγ 2 Þ Þ as its rate can be achieved by simply using input states of A that live on the subspace 0 j i; 1 j i f g.Consequently, we can conclude that the following identity holds true as shown in Fig. 6.
Double-decay qutrit MAD channel with γ 1 = 0.Here we consider the triangular surface DEF of Fig. 1.From Eq. ( 1), we have that   71), that implies that the quantum capacity Q = 0, bottleneck inequality that implies QðM N Þ QðN Þ; QðMÞ and composition rules, as shown in Supplementary Note 3, all points included in the green region of the plot have zero quantum (and private classical) capacity.
the Kraus operators for the MAD channel D ð0;γ 2 ;γ 3 Þ are three: The actions of D ð0;γ 2 ;γ 3 Þ and its complementary counterpart Dð0;γ 2 ;γ 3 Þ on a generic density matrix ρ can hence be described as (notice that in this case, differently of what happens with D ðγ 1 ;0;γ 3 Þ , the complementary channel is not an element of the MAD set).By close inspection of Eq. ( 33), and as intuitively suggested by Fig. 1, also these channels exhibit a symmetry analogous to the one reported in Eq. ( 26), but this time with V being the swap operation exchanging levels 0 j i and 1 j i, which gives us and an analogous identity for the private classical capacity.Furthermore, as in the case of the single-decay qutrit MAD channel D ð0;γ 2 ;0Þ , we notice that D ð0;γ 2 ;γ 3 Þ has a noiseless subspace, given here by 0 j i; 1 j i f g, and we can establish the following lower bound: In particular, this tells us that D ð0;γ 2 ;γ 3 Þ cannot be antidegradable (the same conclusion can be obtained by noticing that 53 the map Dðγ 2 ;0;γ 3 Þ has a kernel that cannot be included into the kernel set of D ðγ 2 ;0;γ 3 Þ -e.g. the former contains 0 j i 1 h j while the latter does not).
Via numerical inspection, we are also able to evaluate the magnitude of Q on the border of the degradability region, designated by showing that here it equals the lower bound in Eq. ( 36).This, in addition to the monotonicity in Eq. ( 52), allows us to conclude that Q assumes the value 1 over all the region above the degradability borderline (red curve of Fig. 7), i.e.
A straightforward approach is to exploit the right-hand-side of Eq. ( 16) and run them also outside the degradability region, in synthesis evaluating the maximum of the coherent information of D ðγ 1 ;γ 2 ;0Þ on the diagonal sources.Notice that since the map is not degradable, the coherent information is not necessarily concave and the restriction to diagonal sources does not even guarantee that the computed expression corresponds to the true Q ð1Þ ðD ðγ 1 ;γ 2 ;0Þ Þ functional.Clearly the task can be refined as much as needed, e.g. by choosing less specific families of states or by computing Q ðiÞ ðD ðγ 1 ;γ 2 ;0Þ Þ for i > 1, but these aspects are beyond the focus of this work and will be considered in future research.The results we obtain are reported in Fig. 8.
Entanglement-assisted quantum capacity of qutrit MAD channels.For the sake of completeness, the present section is devoted to studying the entanglement-assisted quantum capacity Q E ðDÞ of MAD CPTP maps which quantifies the amount of quantum information transmittable per channel use assuming the communicating parties to share an arbitrary amount of entanglement.A closed expression for it has been provided in ref. 54,55 and results in an expression which, in contrast to the quantum capacity formula, does not need a regularization w.r.t. to the number of channel uses, i.e.
IðΦ; ρÞ; ð41Þ where now IðΦ; ρÞ SðρÞ þ JðΦ; ρÞ is the quantum mutual information functional.The discussion in "Methods" about covariance of the channel and the concavity-in this case of the quantum mutual information-apply also here, and we can reduce the maximization in Eq. (41) to where ρdiag are input density matrices which are diagonal in the computational basis of the system, see Supplementary Note 5.The evaluations of Q E of the for the single and double-decay qutrit MAD channels are reported in Figs. 9 and 10.Notice that also the three-rate qutrit MAD channels Q E can be computed but not easily visualized, hence it is not reported.

Conclusion.
We introduce a finite dimensional generalization of the qubit ADC model which represents one of the most studied examples of quantum noise in quantum information theory.In this context, the quantum (and classical private) capacity of a large class of quantum channels (namely the qutrit MAD channels) has been explicitly computed, vastly extending the set of models whose capacity is known: this effort in particular includes some non-trivial examples of quantum maps which are explicitly non-degradable (neither antidegradable)-see e.g. the results of "Methods"-(Double-decay qutrit MAD channel with γ 3 = 0).Having also shown the covariance w.r.t.diagonalizing unitaries of the MAD, follows that, when degradable, the computational complexity associated with the quantum capacity evaluation grows only linearly with the dimension.Besides allowing generalizations to higher dimensional systems (see e.g.ref. 33 ), the analysis here presented naturally spawns further research, e.g.extending it to include other capacity measures, such as the classical capacity or the two-way quantum capacity 52,56 .We finally conclude by noticing that the MAD channel scheme discussed in the present paper can be also easily adapted to include generalizations of the (qubit) generalized ADC scheme 52 , by allowing reverse damping processes which promote excitations from lower to higher levels that could mimic, e.g., thermalization events.Notice that reported plot does not fulfill the monotonicity constraint in Eq. ( 52), hence explicitly proving that the function we present is certainly not the real capacity of the system.
Fig. 9 Entanglement-assisted quantum capacity for the single-decay channel.Profile of the entanglement-assisted quantum capacity Q E (blue) of the channel D ðγ 1 ;0;0Þ w.r.t. the damping parameter γ 1 (results should be compared with those of Fig. 2 where we present the quantum capacity QðD ðγ 1 ;0;0Þ Þ (dashed gray)).Notice that also in this case the expression fulfills the monotonicity constraint in Eq. (48).

Methods
Composition rules, data-processing, bottleneck inequalities.It is relatively easy to verify that the set of qutrit MAD channels in Eq. ( 7) is closed under concatenation.Specifically we notice that given D γ 0 !and D γ 00 ! with γ 00 !¼ ðγ 00 1 ; γ 00 2 ; γ 00 3 Þ and γ 0 !¼ ðγ 0 1 ; γ 0 2 ; γ 0 3 Þ two rate vectors fulfilling the conditions in Eq. ( 6), we have with γ !¼ ðγ 1 ; γ 2 ; γ 3 Þ a new rate vector of components which also satisfies Eq. ( 6).The importance of Eq. ( 44) for the problem we are facing stems from channel data-processing inequalities (or bottleneck) inequalities 7,51,52 , according to which, any information capacity functional Γ (ref. 2 ) such as the quantum capacity Q, the classical capacity C, the private classical capacity C p , the entanglement-assisted classical capacity C E etc., computed for a CPTP map Φ ¼ Φ 0 Φ 00 obtained by concatenating channel Φ 0 with channel Φ″, must fulfill the following relation Applied to Eq. ( 44), the above inequality can be used to predict monotonic behaviors for the capacity ΓðD γ !Þ as a function of the rate vector γ !, that allows us to provide useful lower and upper bounds which in some case permit to extend the capacity formula to domain where other techniques (e.g.degradability analysis) fail.In particular, we notice that for single-decay MAD channels where only one component of the rate vector is different from zero (say γ 1 ), we get D ðγ 0 1 ;0;0Þ D ðγ 00 1 ;0;0Þ ¼ D ðγ 00 1 ;0;0Þ D ðγ 0 1 ;0;0Þ ¼ D ðγ 1 ;0;0Þ ; ð47Þ with γ 1 as in the first identity of Eq. ( 45).Accordingly, we can conclude that all the capacities ΓðD ðγ 1 ;0;0Þ Þ should be non-increasing functionals of the parameter γ 1 , i.e.
Composing two single-decay MAD channels characterized by rate vectors pointing along different cartesian axis in general can create maps with a resulting vector rate with a component in the third direction.Specifically from Eq. ( 44) it follows that, for an arbitrary choice of the rate vector γ !¼ ðγ 1 ; γ 2 ; γ 3 Þ in the allowed CPTP domain, the MAD channel D ðγ 1 ;γ 2 ;γ 3 Þ can be expressed as ¼ D ð0; γ 2 ;0Þ D ð0;0;γ 3 Þ D ðγ 1 ;0;0Þ ; ð50Þ with which because of the constraint in Eq. ( 6) are properly defined rates.As a direct consequence of Eqs. ( 46) and ( 47), it then follows that the capacities ΓðD ðγ 1 ;γ 2 ;γ 3 Þ Þ must be non-increasing functionals of all the cartesian components of rate vector γ !, i.e.
ΓðD ðγ 1 ;γ and must be restricted by the upper bound As a further refinement notice that, setting γ 2 = 0 in Eqs. ( 49) and ( 50), we get which replaced back into Eq.( 50) gives us which allows us to replace Eq. ( 53) with the stronger requirement Similarly by setting γ 1 = 0, we get that yields Finally setting γ 3 = 0 in Eq. ( 49), we get D ðγ 1 ;γ 2 ;0Þ ¼ D ð0;γ 2 ;0Þ D ðγ 1 ;0;0Þ ; that leads to D ðγ 1 ;γ 2 ;γ 3 Þ ¼ D ð0;0; γ 3 Þ D ðγ 1 ;γ 2 ;0Þ ; ð61Þ and ΓðD ðγ 1 ;γ Complementary channels and degradability.A CPTP map Φ : LðH A Þ !LðH B Þ can be seen as the evolution induced by an isometry V : H A ! H B H E involving an environment E, called Stinespring dilation 57,58 .Specifically for all input states ρ A 2 S A we can write If instead we trace out the degrees of freedom in B, we obtain the complementary (or conjugate) channel Φ : Being Mk the Kraus operators generating Φ and k j i E a basis for the environment, the operator V can be written as: and being it is straightforward to verify that Eq. ( 64) can be equivalently expressed as A fact that it is worth mentioning, as it will play a fundamental role in our analysis, is that 59 for a channel Φ that is covariant under a unitary representation of some group G, i.e.
then also the complementary channel Φ is covariant under the same transformations, i.e.
where for X = A, B, E, Û X g is the unitary operator that represents the element g of the group G in the output space X.
We finally recall the definition of degradable and anti-degradable channels 48 .A quantum channel Φ is said degradable if a CPTP map N : while it is said antidegradable if it exists a CPTP map M : (the symbol "∘" representing channel concatenation).Notice that in case Φ is mathematically invertible, a simple direct way to determine whether it is degradable or not is to formally invert Eq. ( 70) constructing the super-operator Φ Φ À1 and check whether such object is CPTP (e.g. by studying the positivity of its Choi matrix) 60,61 , i.e. explicitly Concretely this can be done by using the fact that since quantum channels are linear maps connecting vector spaces of linear operators, they can in turn being represented as matrices acting on vector spaces.This through the following vectorization isomorphism: Following Eq. (70), we have, hence, that for a degradable channel the following identity must apply with MN the matrix representation of the CPTP connecting channel N , implying that the super-operator Φ Φ À1 is now represented by matrix M Φ MÀ1 Φ .
Covariance of the channel.Besides allowing for the single-letter simplification in Eq. ( 15), another important consequence of the degradability property of Eq. ( 70) is the fact that, for channels fulfilling such condition, the coherent information in Eq. ( 11) is known to be concave 62 with respect to the input state ρ, i.e.
J Φ; for all statistical ensemble of input states p k ; ρk È É .This last inequality allows for some further drastic simplification in particular when the channel Φ is covariant under a group of unitary transformations as in Eq. (68).Indeed, thanks to ref. 59 and the invariance of the von Neumann entropy under unitary operations, we can now observe that JðΦ; for all input states and for all elements g of the group.Given then a generic input state ρ of the system, construct the following ensemble of density matrices fdμðgÞ; ρg g with dμ(g) some properly defined probability distribution on G and with ρg Û A g ρ Û A y g .Defining then the average state of fdμðgÞ; ρg g we notice that if Φ is degradable the following inequality holds true: where in the last passage we used the invariance in Eq. ( 77).Accordingly, we can now restrict the maximization in Eq. ( 9) to only those input states ρG which result from the averaging operation of Eq. (78), i.e.
For the special case of the MAD channels D introduced in "Methods"-(MAD channels and composition rules), thanks to Eq. ( 4) we can identify the group G with the set of unitary operations which are diagonal in the computational basis i j i f g i¼0;ÁÁÁ;dÀ1 .Taking dμ g a flat measure, Eq. (78) allows us to identify Λ G ½ρ with the density matrices of A which are diagonal as well, i.e.
and therefore to derive from Eq. (80) the following compact expression: which for d C = 3 reduces to Eq. ( 16) of the main text.For completeness, we report also an alternative, possibly more explicit way to derive Eq. ( 82).This is obtained by observing that a special instance of the unitaries which are diagonal in the computational basis of a MAD channel and hence fulfill the identity in Eq.
from which Eq. ( 82) can once more be derived as a consequence of Eq. ( 80) for all degradable D.

Fig. 1
Fig. 1 Qutrit MAD and parameters region.a Schematic representation of the action of the Multi-level Amplitude Damping (MAD) channel D γ ! on a three-level system: arrows indicate the damping processes connecting different energy levels (black lines), γ ij are the associated damping parameters.b The admitted region of the damping parameters space: the transformation is Completely Positive and Trace Preserving (CPTP) if and only if the rate vector γ! belongs to the yellow region defined in Eq. (6).

Fig. 5
Fig. 5 Parameter values corresponding to zero quantum capacity.From antidegradability, Eq. (71), that implies that the quantum capacity Q = 0, bottleneck inequality that implies QðM N Þ QðN Þ; QðMÞ and composition rules, as shown in Supplementary Note 3, all points included in the green region of the plot have zero quantum (and private classical) capacity.

Fig. 8
Fig.8Lower bound for the quantum and private classical capacities of the channel D ðγ 1 ;γ 2 ;0Þ .Numerical evaluation of a lower bound for the quantum capacity Q and the private classical capacity C p , here they coincide, w.r.t. the damping parameters γ 1 , γ 2 .It is obtained by maximizing the single-use coherent information of the channel over all possible diagonal inputs.The parameters region (γ 1 , γ 2 , 0) corresponds to the CADF square of Fig.1.Notice that reported plot does not fulfill the monotonicity constraint in Eq. (52), hence explicitly proving that the function we present is certainly not the real capacity of the system.

( 4 )
, is provided by the subgroup O D ðdÞ formed by the operators represented by the diagonal d × d matrices for which all the non-zero (and diagonal) elements are ±1.Clearly the identity operator 1 is an element of O D ðdÞ and the group is finite with 2 d elements.Given then an arbitrary input state ρ of A, construct then the ensemble p k ; ρk È É formed by the density matrices ρk Ôk ρ Ôy k , with Ôk being the k-th element of O D ðdÞ, and by a flat probability set p k = 1/2 d .It can be shown 63 that the average state of p k ; ρk È É is diagonal in the computational basis, i.e.
k ¼ diag ðρÞ ; now MΦ is a d 2 B d 2 A matrix connecting H 2 A and H 2 B (d A and d B being, respectively, the dimensions of H A and H B ), which given a Kraus set Mk È É k for Φ it can be explicitly expressed as kMk MÃk :