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https://doi.org/10.1038/s42005-021-00524-4 OPEN Quantum capacity analysis of multi-level amplitude damping channels ✉ Stefano Chessa 1 & Vittorio Giovannetti1 1234567890():,; 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.

1 NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, Pisa, Italy. ✉email: [email protected]

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he main goal of quantum information and communication inequalities. Finally we compute the quantum and classical Ttheory is to understand how can we store, process, and entanglement-assisted capacities. transfer information in a reliable way and, from the phy- sical point of view, to individuate realistic platforms by means of Results and discussion which performing these tasks. All this is done by exploiting the MAD channels and composition rules. The transformations we characteristic features of quantum mechanics. Focusing on focus on in the present work are special instances of the multi- quantum communication, every communication protocol can be level versions of the qubit ADC9, hereafter indicated as MAD seen as a physical system (the encoded message) undergoing channels in brief, which effectively describe the decaying of some physical transformation that translates it in space or time. energy levels of a d-dimensional quantum system A. In its most Any real-world application though suffers from some kind of general form, given fgjii i¼0;ÁÁÁ;dÀ1 an orthonormal basis of the noise, each of which can be in turn described as a quantum Hilbert space HA associated with A (hereafter dubbed the com- process or equivalently as a quantum channel. Following the work putational basis of the problem), a MAD channel D is a com- 1 of Shannon and the later quantum generalizations, the ability of pletely positive trace preserving (CPTP) mapping2,4–8 acting on a quantum channel to preserve the encoded classical or quantum the set LðH Þ of linear operators of the system, defined by the 2,3 A information is described by its capacities . In the classical case, following set of d(d − 1)/2 + 1 Kraus operators we can only transfer classical information, hence we only need to pffiffiffiffiffi deal with the classical capacity. In the quantum framework, we ^ γ ; ; : : ≤ ≤ ; Kij  jijii hjj 8i j s t 0 i d À 1 can also transfer quantum states and consequently, in addition to X qffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ the classical capacity, we count also the quantum capacity. K^  ji0 hjþ0 1 À ξ jij hjj ; Moreover, the family of capacities associated with a quantum 0 j 1 ≤ j ≤ dÀ1 channel can be enlarged assuming the communicating parties to fi be able to perform speci c tasks or to share further resources such γ 2,4–8 with ji real quantities describing the decay rate from the j-th to as, for instance, entanglement . fi One among the simplest models for quantum noise is given by the i-th level that ful ll the conditions the amplitude damping channel (ADC). While the ADC has been ( ≤ γ ≤ ; ; : : ≤ ≤ ; thoroughly studied and characterized, in terms of capacities in 0 ji 1 8i j s t 0 i 2 has to ^ρ S be approached case by case, and the literature regarding capacities Accordingly, given 2 ðHAÞ a generic density matrix of the of fixed finite dimensions ADC is still remarkably short12–14.In system A, the MAD channel D will transform it into the output fi recent years though higher dimensional systems have attracted state de ned as the attention of a growing number of researchers, since they have X y y been shown to provide potential advantages both in terms of Dð^ρÞ¼ K^ ^ρK^ þ K^ ^ρK^ ; 15–20 0 0 ij ij computation (see e.g. refs. ) and communication or error 0 ≤ iquantum computing) or communication (e.g. By construction, D always admits the ground stateji 0 as a fixed long-distance quantum communication) framework, will neces- point, i.e. Dðji0 hjÞ0 ¼ ji0 hj0 , even though, depending on the fi fi fi γ fi sitate high- delity quantum state transmission to achieve reliable speci c values of the coef cients ji, other input states may ful ll γ = ∀ and advantageous purposes. This drives the need of an extensive the same property as well. Limit cases are ji 0 i, j, where all characterization of communications performances in all available levels are untouched and D reduces to the noiseless identity noise regimes and most general noise models. In addition to these channel Id which preserves all the input states of A. On the considerations, new results on the quantum capacity of finite opposite extreme are those examples in which for some j we have ξ = dimensional channels can also be applied to higher dimensional j 1, corresponding to the scenario where the j-th level becomes maps via the partially coherent direct sum (PCDS) channels totally depopulated at the end of the transformation. The maps in approach33, placing in a wider context the efforts dedicated to the Eq. (3) provide also a natural playground to describe PCDS analysis of non-qubit channels. Among non-qubit systems, three- channels33. Last but not the least, an important and easy to verify dimensional systems (qutrit) have received particular attention property of the maps in Eq. (3) is that they are covariant under because of their relative accessibility both theoretically and the group formed by the unitary transformations U^ which are experimentally (see e.g. refs. 34–43). diagonal in the computational basis fgjii i¼0;ÁÁÁ;dÀ1, i.e. Considering this, in this paper we focus on the model for quantum noise given by the ADC in the multi-level setting, that y y ^^ρ ^ ^ ^ρ ^ ; ð4Þ we will denote as multi-level amplitude damping (MAD) channel. DðU U Þ¼UDð ÞU In particular, we perform a systematic analysis of the MAD on the qutrit space: while we will not approach the issue of the classical for all inputs ^ρ. capacity of the channel, we will focus on the quantum capacity, For what concerns the present work, we shall restrict our private classical capacity, and entanglement-assisted capacities, analysis to the special set of MAD channels as in Eq. (3) trying to understand in which conditions these quantities can be associated with a qutrit system (d = 3) whose decay processes, known. We find that the quantum and private classical capacities pictured in the top panel of Fig. 1, are fully characterized by only γ are exactly computable in large regions of the damping para- three rate parameters ji that for the ease of notation we rename !γ γ ; γ ; γ meters space, even when the qutrit MAD are not degradable. with the cartesian components of a 3D vector ð 1 2 3Þ. When an exact value is missing, we are still able to provide upper Accordingly, expressed in terms of the matrix representation bounds exploiting composition rules and data-processing induced by the computational basisfgji 0 ; ji1 ; ji2 , the Kraus

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∈ ρ ^ρ where for i, j 0, 1, 2, ij  hji jij are the matrix entries of the ^ρ S input density operator 2 ðHAÞ.

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 trans- 2,4–8 mission stages . The private classical capacity Cp 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 informa- tion theory, as testified by the limited number of examples which allow for an explicit solution. A closed expression for the quan- tum capacity is provided by the formula44–46 QðΦÞ¼ lim QðnÞðΦÞ=n; ð9Þ n!1

ðnÞ Φ Φ n; ^ρðnÞ ; Q ð Þ max Jð Þ ð10Þ ^ρðnÞ2SðH nÞ where the maximization in Eq. (10) is performed over the set of density matrices of n channel uses, and J is the coherent infor- mation n Fig. 1 Qutrit MAD and parameters region. a Schematic representation of JðΦ n; ^ρðnÞÞSðΦ nð^ρðnÞÞÞ À SðΦ~ ð^ρðnÞÞÞ; ð11Þ the action of the Multi-level Amplitude Damping (MAD) channel D!γ on a three-level system: arrows indicate the damping processes connecting with Sð^ρÞÀTr½^ρ log ^ρŠ the von Neumann entropy of the state ~ 2 different energy levels (black lines), γij are the associated damping ^ρ, and Φ the complementary channel of Φ, see “Methods”— parameters. b The admitted region of the damping parameters space: the Complementary channels and degradability. For the private transformation is Completely Positive and Trace Preserving (CPTP) if and 46,47 ! classical capacity instead, we have : only if the rate vector γ belongs to the yellow region defined in Eq. (6). C ðΦÞ¼ lim CðnÞðΦÞ=n; ð12Þ p n!1 p operators in Eq. (1) write explicitly as 0 1 0 ffiffiffiffiffi 1 ðnÞ n ~ n pγ C ðΦÞmaxðχðΦ ; E ÞÀχðΦ ; E ÞÞ; 10 0 0 1 0 p n n ð13Þ B pffiffiffiffiffiffiffiffiffiffiffiffiffi C B C En K^ ¼ @ 0 1 À γ 0 A; K^ ¼ @ 000A; 0 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 01 where the maximization is now performed over all quantum 00 1 À γ À γ 000 no 0 1 2 3 0 ffiffiffiffiffi 1 ð5Þ ; ^ρðnÞ pγ ensembles En  pi i of n channel uses, and where 00 0 00 3 !! B pffiffiffiffiffi C B C  K^ ¼ @ 00 γ A; K^ ¼ @ 00 0A; X X 12 2 03 χðΦ n; E ÞS Φ n p ^ρðnÞ À p S Φ n ^ρðnÞ 00 0 00 0 n i i i i i i with CPTP conditions from Eq. (2) given by  ð14Þ 0 ≤ γ ≤ 1; 8j ¼ 1; 2; 3; j is the Holevo information functional. The difficulties related to γ γ ≤ ; ð6Þ 2 þ 3 1 the evaluation of the above formulas are well known and ulti- mately the reason underlying our efforts here. An exception to which produce the volume visualized in the bottom panel of 48 49 Fig. 1. this predicament is given by degradable and antidegradable channels. Degradable channels are those for which exists a CPTP The resulting mapping as in Eq. (3) for the channel Dðγ ;γ ;γ Þ 1 2 3 Φ~ Φ reduces, hence, to the following expression map N s.t. ¼N , while antidegradable channels are those 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Φ Φ~ ρ þ γ ρ þ γ ρ 1 À γ ρ 1 À γ À γ ρ for which exists a CPTP map M s.t. ¼M ; for more B 00 pffiffiffiffiffiffiffiffiffiffiffiffiffi1 11 3 22 1 01 pffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 3 02 C B C “ ”— D!ð^ρÞ¼ 1 À γ ρà ð1 À γ Þρ þ γ ρ 1 À γ 1 À γ À γ ρ ; details, see Methods Complementary channels and degrad- γ @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 01 pffiffiffiffiffiffiffiffiffiffiffiffiffip1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 2 22 1 2 3 12 A γ γ ρà γ γ γ ρà γ γ ρ ability. For degradable channels, Q and Cp result to be additive, so 1 À 2 À 3 02 1 À 1 1 À 2 À 3 12 ð1 À 2 À 3Þ 22 the regularization over n in Eq. (9) is not needed, leading to the ð7Þ following single-letter formula50 2,4–6 while the associated complementary CPTP transformation C ðΦÞ¼QðΦÞ¼Qð1ÞðΦÞ: ð15Þ computed as in Eq. (67)of“Methods”, for generic choices of the p system parameters, transforms A into a four-dimensional state For antidegradable channels instead, due to a no-cloning via the mapping argument Q = 0 while, from expression in Eq. (13), positivity of 0 1 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi pffiffiffiffiffi = ρ þð1 À γ Þρ þð1 À γ À γ Þρ γ ρ 1 À γ γ ρ γ ρ private classical capacities and data processing, we have Cp 0. B 00 1 11ffiffiffiffiffi 2 3 22 1 01 1 2 12 ffiffiffiffiffi 3ffiffiffiffiffi02 C B pγ ρà γ ρ pγ pγ ρ C ~ B 1 01 1 11 0 1 3 12 C So no maximizations are needed. D!ð^ρÞ¼B pffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi C ; γ γ pγ ρà γ ρ @ 1 À 1 2 12 0 2 22 0 A Building up from these premises, here we present a thoughtful pffiffiffiffiffi pffiffiffiffiffipffiffiffiffiffi γ ρà γ γ ρà 0 γ ρ 3 02 1 3 12 3 22 characterization of the quantum capacity QðD!γ Þ and the private ð8Þ classical capacity CpðD!γ Þ of the qutrit MAD channel D!γ

COMMUNICATIONS PHYSICS | (2021) 4:22 | https://doi.org/10.1038/s42005-021-00524-4 | www.nature.com/commsphys 3 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00524-4 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 CpðD!γ Þ for a quite a large class of qutrit MAD channels by making use of degradability properties48, data- processing (or bottleneck) inequalities51,52, and channel iso- morphism. 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  ~ QðD Þ¼C ðD Þ¼max SðD ð^ρ ÞÞ À SðD ð^ρ ÞÞ ; !γ p !γ ^ρ !γ diag !γ diag diag ð16Þ where the maximization is performed on input states of A which are diagonal in the computational basisP of the problem, i.e. the ^ρ 2 ∈ density matrices of the form diag ¼ i¼0 pijii hji with p0, p1, p2 [0, 1] being usually called “populations” and fulfilling the + + = “ ” normalization constraint p0 p1 p2 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 p0 and p1), 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 Fig. 2 Quantum capacity for the single-decay. a Profile of the quantum and (minimal) number of Kraus operators we need to express Eq. (7) the private classical capacity for the channel Dðγ ;0;0Þ w.r.t. the damping is 4. Unfortunately, this number also ensures us that the channel 1 parameter γ . For γ ≤ 1/2, the channel is degradable and the reported value is not degradable: it has been indeed shown53 that a necessary 1 1 follows from the numerical maximization. For γ > 1/2, instead, the channel is condition for any CPTP map with output dimension 3 to be neither degradable nor antidegradable: here the associated capacity value is degradable is that its associated Choi rank, and consequently the equal to 1. Notice that the reported values respect the monotonicity minimal number of Kraus operators we need to express such property given by Eq. (48). b Populations p , p , and p of those states that transformation, is at most 3. This brings us to consider some 0 1 2 maximize the quantum capacity formula for the channel Dðγ ;0;0Þ w.r.t. the simplification in the problem, e.g. by fixing some of the values of 1 damping parameter γ . the damping parameters. One approach is represented by the 1 selective suppression of one (or two) of the decaying channels, i.e. γ two non-zero Kraus operators, i.e. imposing one (or two) of the parameters i equal to 0 or to their 0 1 0 pffiffiffiffiffi 1 maximum allowed value, choices that as we shall see, will 100 0 γ 0 B pffiffiffiffiffiffiffiffiffiffiffiffiffi C B 1 C effectively allow us to reduce the number of degrees of freedom of K^ ¼ @ 0 1 À γ 0 A K^ ¼ @ 000A: ð18Þ the problem. 0 1 01 001 000 Single-decay qutrit MAD channels. We consider here instances of Transformation in Eq. (7) is then given by the qutrit MAD channel in which only one of the three damping 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ρ γ ρ γ ρ ρ parameters γ is explicitly different from zero, i.e. the maps 00 þ 1 11 1 À 1 01 02 i B pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi C D γ ; ; , D ;γ ; , and D ; ;γ associated, respectively, with the ^ρ @ γ ρà γ ρ γ ρ A; ð 1 0 0Þ ð0 2 0Þ ð0 0 3Þ Dðγ ; ; Þð Þ¼ 1 À ð1 À Þ 1 À 1 0 0 1 01 pffiffiffiffiffiffiffiffiffiffiffiffiffi1 11 1 12 edges DA, DF, and DE of Fig. 1. It is easy to verify that these three ρà 1 À γ ρà ρ sets of transformations can be mapped into each other via unitary 02 1 12 22 conjugations that simply permute the energy levels of the system: ð19Þ for instance D ; ;γ γ can be transformed into D γ γ; ; by ð0 0 3¼ Þ ð 1¼ 0 0Þ ~ and the complementary channel Dðγ ;0;0Þ that can be expressed as simply swapping levelsji 1 andji 2 . Accordingly, the capacities of 1 a mapping that connects the system A to a two-dimensional these three sets must coincide, since each channel can be obtained environmental system E, i.e. from the other, i.e. ! pffiffiffiffiffi 1 À γ ρ γ ρ ~ ^ρ 1 11 1 01 : QðD γ; ; Þ¼QðD ;γ; Þ¼QðD ; γ Þ; 8γ 2½0; 1Š; D γ ; ; ð Þ¼ pffiffiffiffiffi ð20Þ ð 0 0Þ ð0 0Þ ð0 0 Þ ð17Þ ð 1 0 0Þ γ ρà γ ρ 1 01 1 11

(similarly for Cp). By virtue of this fact, without loss of generality, By the study of degradability and the techniques discussed in in the following we report the analysis only for Dðγ ;0;0Þ, being the “Methods” and in Supplementary Note 1, we are able to evaluate 1 γ results trivially extendable to the remaining two. For this purpose, Q and Cp for every . The results are summarized in the plot in we observe that from Eq. (5) it follows that D γ ; ; possesses only Fig. 2. ð 1 0 0Þ

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fi γ = Complete damping of the rst excited state ( 1 1). Assume next that our qutrit MAD channel of Eq. (7) is characterized by the γ maximum value of 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 levelji 1 gets completely lost in favor of the ground state ji0 of the model so that Eqs. (7) and (8) rewrite as 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 Àð1 À γ Þρ 0 1 À γ À γ ρ B 3 22 2 3 02 C D ;γ ;γ ð^ρÞ¼@ 0 γ ρ 0 A; ð1 2 3Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 22 γ γ ρà γ γ ρ 1 À 2 À 3 02 0 ð1 À 2 À 3Þ 22 ð21Þ 0 ffiffiffiffiffi 1 ρ γ γ ρ ρ pγ ρ 00 þð1 À 2 À 3Þ 22 01 0 3 02 B pffiffiffiffiffi C B ρà ρ 0 γ ρ C ~ ^ρ B 01 11 3 12 C; Dð1;γ ;γ Þð Þ¼ 2 3 @ 00γ ρ 0 A ffiffiffiffiffi ffiffiffiffiffi 2 22 pγ ρà pγ ρà γ ρ 3 02 3 12 0 3 22 ð22Þ

for γ2, γ3 ∈ [0, 1] such that γ2 + γ3 ≤ 1. 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 ;γ ;γ is not degradable. Indeed we notice that while ð1 2 3Þ ~ ^ρ ρ ρ Dð1;γ ;γ Þð Þ preserves information about the components 11, 01, ρ ρ2 3 ρ ^ρ 10, 12, 21 of the input state , no trace of those terms is left in D ;γ ;γ ð^ρÞ: accordingly it is technically impossible to identify a ð1 2 3Þ linear (not mentioning CPTP) map N which applied to ~ D ;γ ;γ ð^ρÞ would reproduce D ;γ ;γ ð^ρÞ for all ^ρ. Despite this ð1 2 3Þ ð1 2 3Þ fact, it turns out that also for D ;γ ;γ , the capacity can still be ð1 2 3Þ 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.

Double-decay qutrit MAD channel with γ2 = 0. Here we consider the value of the capacity for !γ belonging to the square surface fi γ = ABED of Fig. 1, identi ed by the condition 2 0. From Eq. (5), we have that the Kraus operators for the MAD channel D γ ; ;γ ð 1 0 3Þ are three: 0 1 0 pffiffiffiffiffi 1 10 0 0 γ 0 B pffiffiffiffiffiffiffiffiffiffiffiffiffi C B 1 C ^ @ 0 1 À γ 0 A; ^ @ A; K0 ¼ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi K01 ¼ 000 γ 00 1 À 3 000 0 pffiffiffiffiffi 1 00 γ B 3 C ^ @ A K03 ¼ 00 0 00 0

Fig. 3 Quantum and private classical capacity of the channel Dð ;γ ;γ Þ. 1 2 3 ð23Þ a Values for quantum (Q) and private classical (C ) capacities (here p while Eqs. (7) and (8) become they are identical) of the Multi-level Amplitude Damping channel Dð ;γ ;γ Þ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 3 ρ þ γ ρ þ γ ρ 1 À γ ρ 1 À γ ρ w.r.t. γ2 and γ3—the associated parameter region corresponds to the ABC B 00 pffiffiffiffiffiffiffiffiffiffiffiffiffi1 11 3 22 1 01 pffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi3 02 C ^ρ B γ ρà γ ρ γ γ ρ C; Dðγ ; ;γ Þð Þ¼@ 1 À ð1 À Þ 1 À 1 À A triangle of Fig. 1. The gray region represent points where Dð1;γ ;γ Þ is not 1 0 3 pffiffiffiffiffiffiffiffiffiffiffiffiffi1 01 pffiffiffiffiffiffiffiffiffiffiffiffiffip1ffiffiffiffiffiffiffiffiffiffiffiffiffi11 1 3 12 2 3 γ ρà γ γ ρà γ ρ Completely Positive and Trace Preserving (CPTP); the points above the red 1 À 3 02 1 À 1 1 À 3 12 ð1 À 3Þ 22 line (γ3 = (1 − γ2)/2) have zero capacity, QðDð ;γ ;γ ÞÞ¼0. For γ2 = 0, the 1 2 3 ð24Þ value of QðDð ;γ ;γ ÞÞ and C ðDð ;γ ;γ ÞÞ coincides with the quantum capacity 1 2 3 p 1 2 3 0 ffiffiffiffiffi ffiffiffiffiffi 1 γ 9 γ ρ γ ρ pγ ρ pγ ρ of a qubit Amplitude Damping channel of transmissivity 3 (ref. ) (plot b): 1 À 1 11 À 3 22 1 01 3 02 B pffiffiffiffiffi pffiffiffiffiffipffiffiffiffiffi C this should be compared with the value of QðDð ;γ ;γ ÞÞ and the private ~ ^ρ @ γ ρà γ ρ γ γ ρ A: 1 2 3 Dðγ ; ;γ Þð Þ¼ 1 01 1 11 1 3 12 1 0 3 ffiffiffiffiffi ffiffiffiffiffi ffiffiffiffiffi classical capacity C ðDð ;γ ;γ ÞÞ on the other border (i.e. γ3 = 0), which we p à p p à p 1 2 3 γ ρ γ γ ρ γ ρ report in c. Notice finally that the reported values respect the monotonicity 3 02 1 3 12 3 22 requirement of Eq. (52). ð25Þ As evident from Fig. 2 and from the formal structure of γ = Eq. (24), for 2 0 the model exhibits a symmetry under the

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D Fig. 4 Quantum and private classical capacities of the channel ðγ ;0;γ Þ. 1 3 D Fig. 6 Quantum capacity of the channel ðγ ;γ ;1Àγ Þ. Evaluation of the Quantum (Q) and private classical (Cp) capacities (here they coincide) for 1 2 2 quantum capacity QðDðγ ;γ ; Àγ ÞÞ w.r.t. the damping parameter γ1, which is γ γ 1 2 1 2 the channel Dðγ ;0;γ Þ w.r.t. the damping parameters 1 and 3. The damping 1 3 equal to the qubit ADC quantum capacity with damping parameter γ parameters region (γ ,0,γ ) coincides with the square surface ABED of 1 1 3 (ref. 9). The parameters region (γ , γ ,1− γ ) corresponds to the Fig. 1. Notice that in the region γ , γ ≤ 1/2 the channel is degradable as in 1 2 2 1 3 rectangular region BEFC of Fig. 1): as shown in Eq. (31), the capacity exhibits Eq. (70), while for γ , γ > 1/2 the channel is antidegradable as in Eq. (71), in 1 3 no dependence upon the damping parameter γ in this case. the remaining regions are neither degradable nor antidegradable. 2

γ + γ = The qutrit MAD channel on the 2 3 1 plane. Let us now γ + γ = !γ consider the regime with 2 3 1 where rate vectors belong to the rectangular area BEFC of Fig. 1. Under this condition, the map in Eq. (7) still admits four Kraus operators and becomes 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ρ þ γ ρ þð1 À γ Þρ 1 À γ ρ 0 B 00 p1 ffiffiffiffiffiffiffiffiffiffiffiffiffi11 2 22 1 01 C D γ ;γ ; γ ð^ρÞ¼@ γ ρà γ ρ γ ρ A: ð 1 2 1À 2Þ 1 À 1 01 ð1 À 1Þ 11 þ 2 22 0 000 ð28Þ

We notice that the levelji 2 gets completely depopulated and that the channel can be expressed as

D γ ;γ ; γ ¼CDγ ; ð 1 2 1À 2Þ 1 ð29Þ

where Dγ is a standard qubit ADC channel connecting levelji 1 1 γ to levelji 0 with damping rate 1, while now C is a CPTP transformation sending the qutrit A to the qubit system spanned by vectorsji 0 ; ji1 and completely erasing the levelji 2 , moving its Fig. 5 Parameter values corresponding to zero quantum capacity. From population in part toji 1 and in part toji 0 , i.e. antidegradability, Eq. (71), that implies that the quantum capacity Q = 0,  bottleneck inequality that implies QðM  N ÞQðN Þ; QðMÞ and ρ þð1 À γ Þρ ρ Cð^ρÞ¼ 00 2 22 01 : ð Þ composition rules, as shown in Supplementary Note 3, all points included in ρ ρ γ ρ 30 10 11 þ 2 22 the green region of the plot have zero quantum (and private classical) 9 capacity. Accordingly, the quantum capacity of Dγ computed in ref. is an 1 explicit upper bound for QðDðγ ;γ ;1Àγ ÞÞ and CpðDðγ ;γ ;1Àγ ÞÞ exchange of γ and γ . Indeed, indicating with V^ the unitary gate 1 2 2 1 2 2 1 3 (remember that for the qubit ADC Q and Cp coincide). On the that swaps levelsji 2 andji 3 we have that other hand, QðDγ Þ is also a lower bound for QðD γ ;γ ; γ Þ and 1 ð 1 2 1À 2Þ y y C ðD γ ;γ ; γ Þ as its rate can be achieved by simply using input ^ρ ^ ^^ρ^ ^ ; p ð 1 2 1À 2Þ Dðγ ; ;γ Þð Þ¼VDðγ ; ;γ ÞðV V ÞV ð26Þ 3 0 1 1 0 3 states of A that live on the subspacefgji 0 ; ji1 . Consequently, we which by data-processing inequality implies can conclude that the following identity holds true ; QðD γ ; ;γ Þ¼QðD γ ; ;γ Þ; QðDðγ ;γ ;1Àγ ÞÞ¼CpðDðγ ;γ ;1Àγ ÞÞ¼QðDγ Þ ð31Þ ð 1 0 3Þ ð 3 0 1Þ ð27Þ 1 2 2 1 2 2 1 with an analogous identity applying in the case of the private as shown in Fig. 6. classical capacity. As reported in Supplementary Note 3, applying “ ” γ = techniques in Methods , we produce results showed in Figs. 4 Double-decay qutrit MAD channel with 1 0. Here we consider and 5. the triangular surface DEF of Fig. 1. From Eq. (1), we have that

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given here byfgji 0 ; ji1 , and we can establish the following lower bound:

C ðD ;γ ;γ Þ ≥ QðD ;γ ;γ Þ ≥ log ð2Þ¼1: p ð0 2 3Þ ð0 2 3Þ 2 ð36Þ

In particular, this tells us that D ;γ ;γ cannot be antidegradable ð0 2 3Þ (the same conclusion can be obtained by noticing that53 the map ~ D γ ; ;γ has a kernel that cannot be included into the kernel set ð 2 0 3Þ of D γ ; ;γ —e.g. the former containsji 0 hj1 while the latter ð 2 0 3Þ does not). Following the usual approach—see Supplementary Note 4—we ~ find that D ;γ ;γ is invertible for γ + γ < 1, and that D ;γ ;γ  ð0 2 3Þ 2 3 ð0 2 3Þ À1 1 D ;γ ;γ is CPTP for γ þ γ ≤ , which defines hence the ð0 2 3Þ 2 3 2 degradability region for the map. So, invoking Eq. (16) and the procedures described in “Methods”, we compute there the quantum capacity. Via numerical inspection, we are also able to evaluate the D Fig. 7 Quantum and private classical capacity of the channel ð0;γ ;γ Þ. 2 3 magnitude of Q on the border of the degradability region, Q C Evaluation of the quantum ( ) and private classical ( p) capacities, here γ γ 1 γ γ designated by 2 þ 3 ¼ 2, showing that here it equals the they coincide, for Dð0;γ ;γ Þ w.r.t. the damping parameters 2 and 3. The 2 3 lower bound in Eq. (36). This, in addition to the monotonicity in parameters region (0, γ2, γ3) corresponds to the triangular surface DEF of γ þ γ ¼ 1 Eq. (52), allows us to conclude that Q assumes the value 1 over Fig. 1. The DEG (degradable) zone below the red curve, 2 3 2, is the Qð Þ all the region above the degradability borderline (red curve degradability region for the channel: here we compute Dð0;γ ;γ Þ solving 2 3 of Fig. 7), i.e. numerically the maximization of Supplementary Eq. (40). Above the red curve, the channel capacity assumes constant value from Eq. (36). Notice QðD ;γ ;γ Þ¼ C ðD ;γ ;γ Þ¼1; ð0 2 3Þ p ð0 2 3Þ that the quantum capacity exhibits the symmetry as in Eq. (35) and the ð37Þ 8γ þ γ ≥ 1=2: monotonicity conditions from Eq. (52). The gray zone indicates the non- 2 3 accessible region, since it violates constraints in Eq. (6). γ = Double-decay qutrit MAD channel with 3 0. Here we consider the square region CADF of Fig. 1 identified by γ3 = 0. From Eq. (1), we have that the Kraus operators for D γ ;γ ; are three: the Kraus operators for the MAD channel Dð ;γ ;γ Þ are three: ð 1 2 0Þ 0 1 0 02 3 1 0 1 0 pffiffiffiffiffi 1 10 0 10 0 0 γ 0 00 0 B pffiffiffiffiffiffiffiffiffiffiffiffiffi C B 1 C ^ B C ^ B pffiffiffiffiffi C ^ γ ^ @ 00 0 A @ γ A K ¼ @ 0 1 À 1 0 A K ¼ @ 000A K0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K12 ¼ 00 2 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi 01 γ γ 00 1 À γ 000 00 1 À 2 À 3 00 0 2 0 1 0 pffiffiffiffiffi 1 00 γ 00 0 B 3 C B pffiffiffiffiffi C ^ @ A: K^ ¼ @ 00 γ A; K03 ¼ 00 0 02 2 00 0 00 0 ð32Þ ð38Þ ~ The actions of Dð0;γ ;γ Þ and its complementary counterpart while the actions of Dðγ ;γ ;0Þ and Dðγ ;γ ;0Þ on a generic density ~ 2 3 ^ρ ^ρ 1 2 1 2 Dð0;γ ;γ Þ on a generic density matrix can hence be described as matrix are: 2 3 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 0 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ρ γ ρ ρ γ γ ρ ρ γ ρ γ ρ γ ρ 00 þ 3 22 01 1 À 2 À 3 02 00 þ 1 11 1 À 1 01 1 À 2 02 B pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C B pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi C ^ρ @ ρà ρ γ ρ γ γ ρ A; ^ρ γ ρà γ ρ γ ρ γ γ ρ ; Dð0;γ ;γ Þð Þ¼ þ 1 À À D γ ;γ ; ð Þ¼@ 1 À ð1 À Þ þ 1 À 1 À A 2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi01 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 2 22 2 3 12 ð 1 2 0Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi1 01 pffiffiffiffiffiffiffiffiffiffiffiffiffi1 p11 ffiffiffiffiffiffiffiffiffiffiffiffiffi2 22 1 2 12 γ γ ρà γ γ ρà γ γ ρ γ ρà γ γ ρà γ ρ 1 À 2 À 3 02 1 À 2 À 3 12 ð1 À 2 À 3Þ 22 1 À 2 02 1 À 1 1 À 2 12 ð1 À 2Þ 22 ð33Þ ð39Þ 0 pffiffiffiffiffi pffiffiffiffiffi 1 0 pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi 1 1 Àðγ þ γ Þρ γ ρ γ ρ 1 À γ ρ À γ ρ γ ρ 1 À γ γ ρ 2ffiffiffiffiffi 3 22 2 12 3 02 B 1 ffiffiffiffiffi11 2 22 1 01 1 2 02 C ~ B p à C ~ p à ^ρ @ γ ρ γ ρ A; D γ ;γ ; ð^ρÞ¼@ γ ρ γ ρ 0 A: Dð ;γ ;γ Þð Þ¼ 2 12 2 22 0 ð34Þ ð 1 2 0Þ 1 01 1 11 0 2 3 ffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffi pγ ρà γ ρ 1 À γ γ ρà 0 γ ρ 3 02 0 3 22 1 2 02 2 22 (notice that in this case, differently of what happens with ð40Þ Dðγ ;0;γ Þ, the complementary channel is not an element of the At variance with the previous sections, we have that while D γ ;γ ; 1 3 ð 1 2 0Þ MAD set). By close inspection of Eq. (33), and as intuitively is invertible for γ1, γ2 < 1, for no range of these values the appli- suggested by Fig. 1, also these channels exhibit a symmetry ~ À1 cation D γ ;γ ; Dγ ;γ ; produces a CPTP map. We can hence ^ ð 1 2 0Þ ð 1 2 0Þ analogous to the one reported in Eq. (26), but this time with V conclude that the map is never degradable. About antidegrad- being the swap operation exchanging levelsji 0 andji 1 , which ~ ability, here also we have that kerfDðγ ;γ ; Þg⊈ kerfDðγ ;γ ; Þg,so gives us 1 2 0 1 2 0 53 Dðγ ;γ ; Þ is also not antidegradable . As a matter of fact, the only ; 1 2 0 QðDð0;γ ;γ ÞÞ¼QðDð0;γ ;γ ÞÞ ð35Þ 2 3 3 2 cases for which we can produce explicit values of QðDðγ ;γ ;0ÞÞ are γ γ 1 2 and an analogous identity for the private classical capacity. the limiting cases where either 1 or 2 equals 0 (in these cases, the Furthermore, as in the case of the single-decay qutrit MAD map is a single-rate MAD channel discussed in “Results”—(Single- channel D ;γ ; , we notice that D ;γ ;γ has a noiseless subspace, decay qutrit MAD channels), or 1 where instead findings of ð0 2 0Þ ð0 2 3Þ

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ð1Þ Q ðD γ ;γ ; Þ functional. Clearly the task can be refined as ð 1 2 0Þ much as needed, e.g. by choosing less specific families of states or ðiÞ by computing Q ðD γ ;γ ; Þ for i > 1, but these aspects are ð 1 2 0Þ 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 fi QEðDÞ of MAD CPTP maps which quanti es the amount of quantum information transmittable per channel use assuming the communicating parties to share an arbitrary amount of entan- glement. 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. Fig. 8 Lower bound for the quantum and private classical capacities of 1 the channel Dðγ ;γ ; Þ. Numerical evaluation of a lower bound for the Φ Φ; ^ρ ; 1 2 0 QEð Þ¼ max Ið Þ ð41Þ 2 ^ρ2SðHÞ quantum capacity Q and the private classical capacity Cp, here they coincide, w.r.t. the damping parameters γ1, γ2. It is obtained by maximizing the single-use coherent information of the channel over all possible where now diagonal inputs. The parameters region (γ , γ , 0) corresponds to the CADF 1 2 Φ; ^ρ ^ρ Φ; ^ρ square of Fig. 1. Notice that reported plot does not fulfill the monotonicity Ið Þ Sð ÞþJð Þ ð42Þ constraint in Eq. (52), hence explicitly proving that the function we present ¼ Sð^ρÞþSðΦð^ρÞÞ À SðΦ~ð^ρÞÞ; is certainly not the real capacity of the system. 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 no 1 Q ðDÞ ¼ max Sð^ρ ÞþSðDð^ρ ÞÞ À SðDð~ ^ρ ÞÞ ; E ^ρ diag diag diag ð43Þ 2 diag

^ρ where diag are input density matrices which are diagonal in the computational basis of the system, see Supplementary Note 5. The evaluations of QE 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 QE 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 Fig. 9 Entanglement-assisted quantum capacity for the single-decay large class of quantum channels (namely the qutrit MAD chan- channel. Profile of the entanglement-assisted quantum capacity QE (blue) nels) has been explicitly computed, vastly extending the set of of the channel Dðγ ; ; Þ w.r.t. the damping parameter γ1 (results should be models whose capacity is known: this effort in particular includes 1 0 0 compared with those of Fig. 2 where we present the quantum capacity some non-trivial examples of quantum maps which are explicitly Qð Þ — Dðγ ;0;0Þ (dashed gray)). Notice that also in this case the expression non-degradable (neither antidegradable) see e.g. the results of 1 “ ”— γ = fulfills the monotonicity constraint in Eq. (48). Methods (Double-decay qutrit MAD channel with 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 “ ”— fi γ = Results (Complete damping of the rst excited state ( 1 1)) grows only linearly with the dimension. Besides allowing gen- — γ + γ = or Results (The qutrit MAD channel on the 2 3 1 plane) eralizations to higher dimensional systems (see e.g. ref. 33), the can be applied. For the remaining cases, we resort in presenting a analysis here presented naturally spawns further research, e.g. lower bound for QðD γ ;γ ; Þ and C ðD γ ;γ ; Þ. ð 1 2 0Þ p ð 1 2 0Þ extending it to include other capacity measures, such as the A straightforward approach is to exploit the right-hand-side of classical capacity or the two-way quantum capacity52,56.We Eq. (16) and run them also outside the degradability region, in finally conclude by noticing that the MAD channel scheme dis- synthesis evaluating the maximum of the coherent information of cussed in the present paper can be also easily adapted to include D γ ;γ ; on the diagonal sources. Notice that since the map is not 52 ð 1 2 0Þ generalizations of the (qubit) generalized ADC scheme ,by degradable, the coherent information is not necessarily concave allowing reverse damping processes which promote excitations and the restriction to diagonal sources does not even guarantee from lower to higher levels that could mimic, e.g., thermalization that the computed expression corresponds to the true events.

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! γ0 γ0 ; γ0 ; γ0 fi and ¼ð 1 2 3Þ two rate vectors ful lling the conditions in Eq. (6), we have D! D! ¼D!; γ0 γ00 γ ð44Þ

with !γ ¼ðγ ; γ ; γ Þ a new rate vector of components 1 2 3 8 > γ γ00 γ0 γ0 γ00; < 1 ¼ 1 þ 1 À 1 1 γ ¼ γ00ð1 À γ0 À γ0 Þþγ0 ð1 À γ00Þ; ð45Þ :> 2 2 1 2 2 3 γ γ00 γ00 γ0 γ0 γ0 γ00 : 3 ¼ 3 þ 2 ð 1 À 3Þþ 3ð1 À 3 Þ which also satisfies Eq. (6). The importance of Eq. (44) for the problem we are facing stems from channel data-processing inequalities (or bottleneck) inequalities7,51,52, according to which, any information capacity functional Γ (ref. 2) such as the quantum capacity Q, the classical capacity C, the private classical capacity Cp, the entanglement-assisted classical capacity CE etc., computed for a CPTP map Φ ¼ Φ0  Φ00 obtained by concatenating channel Φ0 with channel Φ″, must fulfill the following relation ΓðΦÞ ≤ minfgΓΦðÞ0 ; ΓΦðÞ00 : ð46Þ Applied to Eq. (44), the above inequality can be used to predict monotonic Γ !γ behaviors for the capacity ðD!γ Þ as a function of the rate vector , 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 γ component of the rate vector is different from zero (say 1), we get

D γ0 ; ; Dγ00 ; ; ¼Dγ00 ; ; Dγ0 ; ; ¼Dγ ; ; ; ð47Þ ð 1 0 0Þ ð 1 0 0Þ ð 1 0 0Þ ð 1 0 0Þ ð 1 0 0Þ γ fi with 1 as in the rst identity of Eq. (45). Accordingly, we can conclude that all the capacities ΓðD γ ; ; Þ should be non-increasing functionals of the parameter γ , i.e. ð 1 0 0Þ 1 0 ΓðD γ ; ; Þ ≥ ΓðD γ0 ; ; Þ; 8γ ≤ γ ; ð48Þ ð 1 0 0Þ ð 0 0Þ 1

(the same expressions and conclusions apply also for D ;γ ; and D ; ;γ ). ð0 2 0Þ ð0 0 3 Þ 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, !γ γ ; γ ; γ for an arbitrary choice of the rate vector ¼ð 1 2 3Þ in the allowed CPTP domain, the MAD channel D γ ;γ ;γ can be expressed as ð 1 2 3Þ

D γ ;γ ;γ ¼D ; ;γ D ;γ ; Dγ ; ; ð49Þ ð 1 2 3Þ ð0 0 3Þ ð0 2 0Þ ð 1 0 0Þ

¼D ;γ ; D ; ;γ Dγ ; ; ; ð50Þ ð0 2 0Þ ð0 0 3Þ ð 1 0 0Þ with γ γ γ 3 ; γ 2 ; 3  γ 2  γ ð51Þ 1 À 2 1 À 3 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 γ ;γ ;γ Þ ð 1 2 3 Þ must be non-increasing functionals of all the cartesian components of rate vector !γ , i.e. 0 ΓðD γ ;γ ;γ Þ ≥ ΓðD γ0 ;γ0 ;γ0 Þ; 8γ ≥ γ ; ð52Þ ð 1 2 3 Þ ð 1 2 3 Þ i i and must be restricted by the upper bound no  ΓðD γ ;γ ;γ Þ ≤ min ΓðD γ ; ; Þ; D ;γ ; ; D ; ;γ : ð53Þ ð 1 2 3 Þ ð 1 0 0Þ ð0 2 0Þ ð0 0 3 Þ fi γ = As a further re nement notice that, setting 2 0 in Eqs. (49) and (50), we get

D γ ; ;γ ¼Dγ ; ; D ; ;γ ¼D ; ;γ Dγ ; ; ; ð54Þ ð 1 0 3Þ ð 1 0 0Þ ð0 0 3 Þ ð0 0 3Þ ð 1 0 0Þ which replaced back into Eq. (50) gives us

Fig. 10 Entanglement-assisted quantum capacity for double-decay D γ ;γ ;γ ¼D ;γ ; Dγ ; ;γ ; ð55Þ ð 1 2 3Þ ð0 2 0Þ ð 1 0 3 Þ channels. a Entanglement-assisted quantum capacity QE of the channel which allows us to replace Eq. (53) with the stronger requirement Dðγ ;γ ; Þ w.r.t. the damping parameter γ1, γ2, the parameters region (γ1, γ2,0) no 1 2 0 ΓðD γ ;γ ;γ Þ ≤ min ΓðD ;γ ; Þ; ΓðD γ ; ;γ Þ : ð56Þ corresponds to the CADF square region of Fig. 1. b Entanglement-assisted ð 1 2 3Þ ð0 2 0Þ ð 1 0 3 Þ Q quantum capacity E of the channel Dðγ ;0;γ Þ w.r.t. the damping parameter γ = 1 3 Similarly by setting 1 0, we get γ1, γ3, the parameters region (γ1,0,γ3) corresponds to the ABED region of D ;γ ;γ ¼D ; ;γ D ;γ ; ¼D ;γ ; D ; ;γ ; ð57Þ Fig. 1. c Entanglement-assisted quantum capacity QE of the channel Dð ;γ ;γ Þ ð0 2 3Þ ð0 0 3Þ ð0 2 0Þ ð0 2 0Þ ð0 0 3 Þ 0 2 3 w.r.t. the damping parameter γ2, γ3, the parameters region (0, γ2, γ3) that yields corresponds to the DEF region of Fig. 1. D γ ;γ ;γ ¼D ;γ ;γ Dγ ; ; ; ð58Þ ð 1 2 3Þ ð0 2 3Þ ð 1 0 0Þ and no ΓðD γ ;γ ;γ Þ ≤ min ΓðD ;γ ;γ Þ; ΓðD γ ; ; Þ : ð59Þ Methods ð 1 2 3Þ ð0 2 3Þ ð 1 0 0Þ Composition rules, data-processing, bottleneck inequalities. It is relatively easy γ = to verify that the set of qutrit MAD channels in Eq. (7) is closed under con- Finally setting 3 0 in Eq. (49), we get ! 00 00 00 00 ; catenation. Specifically we notice that given D! and D! with γ ¼ðγ ; γ ; γ Þ Dðγ ;γ ;0Þ ¼Dð0;γ ;0Þ Dðγ ;0;0Þ ð60Þ γ0 γ00 1 2 3 1 2 2 1

COMMUNICATIONS PHYSICS | (2021) 4:22 | https://doi.org/10.1038/s42005-021-00524-4 | www.nature.com/commsphys 9 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00524-4 that leads to identity must apply ^ ^ ^ D γ ;γ ;γ ¼D ; ;γ Dγ ;γ ; ; ð61Þ MΦ~ ¼ M MΦ; ð75Þ ð 1 2 3Þ ð0 0 3Þ ð 1 2 0Þ N ^ and with MN the matrix representation of the CPTP connecting channel N , implying no ~ À1 ^ ^ À1 that the super-operator Φ  Φ is now represented by matrix MΦ~ MΦ . ΓðD γ ;γ ;γ Þ ≤ min ΓðD γ ;γ ; Þ; D ; ;γ : ð62Þ ð 1 2 3 Þ ð 1 2 0Þ ð0 0 3Þ Covariance of the channel. Besides allowing for the single-letter simplification in Complementary channels and degradability. A CPTP map Φ : LðH Þ! Eq. (15), another important consequence of the degradability property of Eq. (70) A is the fact that, for channels fulfilling such condition, the coherent information in LðH Þ can be seen as the evolution induced by an isometry V^ : H !H H B A B E Eq. (11) is known to be concave62 with respect to the input state ^ρ, i.e. involving an environment E, called Stinespring dilation57,58. Specifically for all ! ρ S X X input states A 2 A we can write J Φ; p ^ρ ≥ p JðΦ; ^ρ Þ; ð76Þ y k k k k Φ ^ρ ^^ρ ^ : ð63Þ k k ð AÞ¼TrE½V AV Š ÈÉ ; ^ρ If instead we trace out the degrees of freedom in B, we obtain the for all statistical ensemble of input states pk k . This last inequality allows for Φ~ : fi Φ complementary (or conjugate) channel LðHAÞ ! LðHEÞ, i.e. some further drastic simpli cation in particular when the channel is covariant under a group of unitary transformations as in Eq. (68). Indeed, thanks to ref. 59 Φ~ ^ρ ^^ρ ^ y : ð AÞ¼TrB½V AV Š ð64Þ and the invariance of the von Neumann entropy under unitary operations, we can ^ Φ now observe that Being Mk the Kraus operators generating and jik E a basis for the  ^ A A y A A y A A y environment, the operator V can be written as: JðΦ; U^ ^ρU^ Þ¼ S ΦðU^ ^ρU^ Þ À SðΦ~ðU^ ^ρU^ ÞÞ X g g g g g g V^ ¼ M^ jik ; ^ B Φ ^ρ ^ B y ^ E Φ~ ^ρ ^ E y ð77Þ k E ð65Þ ¼ SðUg ð ÞUg ÞÀSðUg ð ÞUg Þ k ¼ JðΦ; ^ρÞ; and being X for all input states and for all elements g of the group. Given then a generic input ^^ρ ^ y ^ ^ρ ^ y ; ^ρ V AV ¼ Mi AMj jii hjj E ð66Þ state of the system, construct the following ensemble of density matrices ; μ ; ^ρ μ fi i j fd ðgÞ g g with d (g) some properly de ned probability distribution on G and ^ρ ^ A ^ρ ^ A y fi it is straightforward to verify that Eq. (64) can be equivalently expressed as with g  Ug Ug .De ning then X Z Z Φ~ ^ρ ^ ^ρ ^ y : Λ ^ρ μ ^ρ μ ^ A ^ρ ^ A y; ð AÞ¼ TrB½Mi AMj Šjii hjj E ð67Þ G½ Š d g g ¼ d g Ug Ug ð78Þ i;j μ ; ^ρ Φ A fact that it is worth mentioning, as it will play a fundamental role in our the average state of fd ðgÞ g g we notice that if is degradable the following analysis, is that59 for a channel Φ that is covariant under a unitary representation inequality holds true: of some group G, i.e. Z A A y JðΦ; Λ ½^ρŠÞ ≥ dμ JðΦ; U^ ^ρU^ Þ¼JðΦ; ^ρÞ; ð79Þ Φ ^ A ^ρ ^ A y ^ B Φ ^ρ ^ B y; ^ρ S ; ; G g g g ðUg Ug Þ¼Ug ð ÞUg 8 2 ðHÞ 8g 2 G ð68Þ where in the last passage we used the invariance in Eq. (77). Accordingly, we can then also the complementary channel Φ~ is covariant under the same ^ρ now restrict the maximization in Eq. (9) to only those input states G which result transformations, i.e. from the averaging operation of Eq. (78), i.e. A A y E E y Φ~ðU^ ^ρU^ Þ¼U^ Φ~ð^ρÞU^ ; 8^ρ 2 SðHÞ; 8g 2 G; ð69Þ QðΦÞ¼Qð1ÞðΦÞ¼max JðΦ; ^ρ Þ: g g g g ^ρ G ð80Þ G = ^ X “ ”— where for X A, B, E, Ug is the unitary operator that represents the element g of For the special case of the MAD channels D introduced in Methods (MAD the group G in the output space X. channels and composition rules), thanks to Eq. (4) we can identify the group G We finally recall the definition of degradable and anti-degradable channels48.A with the set of unitary operations which are diagonal in the computational basis Φ : fgjii . Taking dμ a flat measure, Eq. (78) allows us to identify Λ ½^ρŠ with quantum channel is said degradable if a CPTP map N LðHBÞ ! LðHEÞ exists i¼0;ÁÁÁ;dÀ1 g G s.t. the density matrices of A which are diagonal as well, i.e. ~ Λ ^ρ ^ρ ; Φ ¼NΦ; ð70Þ G½ Š¼diag½ Š ð81Þ : and therefore to derive from Eq. (80) the following compact expression: while it is said antidegradable if it exists a CPTP map M LðHEÞ ! LðHBÞ s.t. ~ QðDÞ ¼ Qð1ÞðDÞ ¼ max JðD; ^ρ Þ; Φ ¼MΦ; ð71Þ ^ρ diag ð82Þ diag “∘” Φ = (the symbol representing channel concatenation). Notice that in case is which for dC 3 reduces to Eq. (16) of the main text. For completeness, we mathematically invertible, a simple direct way to determine whether it is report also an alternative, possibly more explicit way to derive Eq. (82). This is degradable or not is to formally invert Eq. (70) constructing the super-operator obtained by observing that a special instance of the unitaries which are diagonal ~ Φ  ΦÀ1 and check whether such object is CPTP (e.g. by studying the positivity of in the computational basis of a MAD channel and hence fulfill the identity in Eq. 60,61 its Choi matrix) , i.e. explicitly (4), is provided by the subgroup ODðdÞ formed by the operators represented by Φ Φ Φ~ ΦÀ1 : the diagonal d × d matrices for which all the non-zero (and diagonal) elements invertible ) degradable iff  is CPTP ð72Þ 1^ are ±1. Clearly the identity operator is an element of ODðdÞ andthegroupis Concretely this can be done by using the fact that since quantum channels are fi d ^ρ nite with 2 elements.ÈÉ Given then an arbitrary input state of A, construct then linear maps connecting vector spaces of linear operators, they can in turn being ; ^ρ ^ρ ^ ^ρ^ y ^ the ensemble pk k formed by the density matrices k  Ok Ok,withOk being represented as matrices acting on vector spaces. This through the following fl = d vectorization isomorphism: the k-th element of ODðdÞ,andbyaÈÉat probability set pk 1/2 .Itcanbe 63 ; ^ρ X X shown that the average state of pk k is diagonal in the computational basis, ^ρ ρ ρ ρ 2 A ¼ ijjii Ahjj À! jii¼ ijjii A jij A 2HA i.e. ij ij ð73Þ 2Xd À1 Φ ^ρ ^ ρ ; 1 ^ ^ y ð AÞÀ! MΦjii O ^ρO ¼ diag ð^ρÞ ; ð83Þ 2d k k ^ 2 ´ 2 2 2 k¼0 where now MΦ is a d B dA matrix connecting H A and H B (dA andÈÉdB being, from which Eq. (82) can once more be derived as a consequence of Eq. (80)for respectively, the dimensions of H and H ), which given a Kraus set M^ for Φ A B k k all degradable D. it can be explicitly expressed as X ^ ^ ^Ã: MΦ ¼ Mk Mk ð74Þ Data availability k Data used to produce the figures shown in the manuscript are available from the Following Eq. (70), we have, hence, that for a degradable channel the following corresponding author upon request.

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Received: 29 August 2020; Accepted: 5 January 2021; 34. Brußand, D. & Macchiavello, C. Optimal eavesdropping in cryptography with three-dimensional quantum states. Phys. Rev. Lett. 88, 127901 (2002). 35. Klimov, A. B., Guzmán, R., Retamal, J. C. & Saavedra, C. Qutrit quantum computer with trapped ions. Phys. Rev. A 67, 062313 (2003). 36. Kaszlikowsky, D. et al. Quantum cryptography based on qutrit Bell inequalities. Phys. Rev. A 67, 012310 (2003). References 37. Bartůšková, L. et al. Optical implementation of the encoding of two qubits to a 1. Shannon, C. E. A mathematical theory of communication. Bell Syst. Tech. J. single qutrit. Phys. Rev. A 74, 022325 (2006). 27, 379–423 (1948). 38. Lanyon, B. P. et al. Manipulating biphotonic qutrits. Phys. Rev. Lett. 100, 2. Holevo, A. S. & Giovannetti, V. Quantum channels and their entropic 060504 (2008). ’ characteristics. Rep. Prog. Phys. 75, 4 (2012). 39. Bocharov, A., Roetteler, M. & Svore, K. M. Factoring with qutrits: Shor s 3. Bennett, C. H. & Shor, P. W. Quantum information theory. IEEE Trans. algorithm on ternary and metaplectic quantum architectures. Phys. Rev. A 96, Inform. Theory 44, 2724 (1998). 012306 (2017). 4. Holevo, A. S. Quantum Systems, Channels, Information (De Gruyter, 2019). 40. Shlyakhov, A. R. et al. Quantum metrology with a transmon qutrit. Phys. Rev. 5. Wilde, M. Quantum Information Theory (Cambridge University Press, A 97, 022115 (2018). 2017). 41. Luo, Y.-H. et al. Quantum teleportation in high dimensions. Phys. Rev. Lett. 6. Watrous, J. The Theory of Quantum Information (Cambridge University, 123, 070505 (2019). fi 2018). 42. Bækkegaard, T. B. et al. Realization of ef cient quantum gates with a 7. Nielsen, M. & Chuang, I. Quantum Computation and Quantum Information: superconducting qubit-qutrit circuit. Sci. Rep. 9, 13389 (2019). 10th Anniversary Edition 10th edn (Cambridge University Press, 2011). 43. Li, B., Yu, Z. & Fei Geometry of quantum computation with qutrits. Sci. Rep. 8. Gyongyosi, L., Imre, S. & Nguyen, H. V. A survey on quantum channel 3, 2594 (2013). capacities. IEEE Commun. Surv. Tutor. 20, 1149–1205 (2018). 44. Lloyd, S. Capacity of the noisy quantum channel. Phys. Rev. A 55, 1613 (1997). 9. Giovannetti, V. & Fazio, R. Information-capacity description of spin-chain 45. Shor, P. W. The quantum channel capacity and coherent information. Lecture correlations. Phys. Rev. A 71, 032314 (2005). Notes, MSRI Workshop on Quantum Computation (Quantum Information 10. D’Arrigo, A., Benenti, G., Falci, G. & Macchiavello, C. Classical and quantum and Cryptography) https://www.msri.org/workshops/203/schedules/1181 capacities of a fully correlated amplitude damping channel. Phys. Rev. A 88, (2002). 042337 (2013). 46. Devetak, I. The private classical capacity and quantum capacity of a quantum – 11. Jahangir, R., Arshed, N. & Toor, A. H. Quantum capacity of an amplitude- channel. IEEE Trans. Inform. Theory 51,44 55 (2005). damping channel with memory. Quantum Inf. Process. 14, 765–782 (2015). 47. Cai, N., Winter, A. & Yeung, R. W. Quantum privacy and quantum wiretap – 12. D’Arrigo, A., Benenti, G., Falci, G. & Macchiavello, C. Information channels. Probl. Inf. Transm. 40, 318 336 (2004). transmission over an amplitude damping channel with an arbitrary degree of 48. Devetak, I. & Shor, P. W. The capacity of a quantum channel for simultaneous – memory. Phys. Rev. A 92, 062342 (2015). transmission of classical and quantum information. Math. Phys. 256, 287 303 13. Ouyang, Y. Channel covariance, twirling, contraction, and some upper bounds (2005). on the quantum capacity. Quantum Inf. Comput. 14, 0917–0936 (2014). 49. Caruso, F. & Giovannetti, V. Degradability of Bosonic Gaussian channels. 14. Macchiavello, C. & Sacchi, M. F. Efficient accessible bounds to the classical Phys. Rev. A 74, 062307 (2006). capacity of quantum channels. Phys. Rev. Lett. 123, 090503 (2019). 50. Smith, G. The private classical capacity with a symmetric side channel and its 15. Muthukrishnan, A. & Stroud, C. R. Multivalued logic gates for quantum application to quantum cryptography. Phys. Rev. A 78, 022306 (2008). computation. Phys. Rev. A 62, 052309 (2000). 51. Keyl, M. Fundamentals of quantum information theory. Phys. Rep. 369, 431 16. Lanyon, B. et al. Simplifying quantum logic using higher-dimensional Hilbert (2002). spaces. Nat. Phys. 5, 134–140 (2009). 52.Khatri,S.,Sharma,K.&Wilde,M.M.Information-theoreticaspectsof 17. Ralph, T. C., Resch, K. J. & Gilchrist, A. Efficient Toffoli gates using qudits. the generalized amplitude-damping channel. Phys.Rev.A102,012401 Phys. Rev. A 75, 022313 (2007). (2020). 18. Ivanov, S. S., Tonchev, H. S. & Vitanov, N. V. Time-efficient implementation 53. Cubitt, T., Ruskai, M. & Smith, G. The structure of degradable quantum of quantum search with qudits. Phys. Rev. A 85, 062321 (2012). channels. J. Math. Phys. 49, 102104 (2008). 19. Gedik, Z. et al. Computational speed-up with a single qudit. Sci. Rep. 5, 14671 54. Bennett, C. H., Shor, P. W., Smolin, J. A. & Thapliyal, A. V. Entanglement- (2015). assisted classical capacity of noisy quantum channels. Phys. Rev. Lett. 83, 3081 20. Kiktenko, E. O., Nikolaeva, A. S., Xu, P., Shlyapnikov, G. V. & Fedorov, A. K. (1999). Scalable quantum computing with qudits on a graph. Phys. Rev. A 101, 022304 55. Bennett, C. H., Shor, P. W., Smolin, J. A. & Thapliyal, A. V. Entanglement- (2020). assisted capacity of a quantum channel and the reverse Shannon theorem. – 21. Cozzolino, D., DaLio, B., Bacco, D. & Oxenlowe, L. K. High-dimensional IEEE Trans. Inform. Theory 48, 2637 2655 (2002). quantum communication: benefits, progress, and future challenges. Adv. 56. Pirandola, S., Laurenza, R., Ottaviani, C. & Banchi, L. Fundamental limits of Quantum Technol. 2, 1900038 (2019). repeaterless quantum communications. Nat. Commun. 8, 15043 (2017). 22. Looi, S. Y., Yu, L., Gheorghiu, V. & Griffiths, R. B. Quantum-error-correcting 57. Choi, M. D. Completely positive linear maps on complex matrices. Linear Algebra Appl. 10, 285–290 (1975). codes using qudit graph states. Phys. Rev. A 78, 042303 (2008). * 23. Goyal,S.K.,Boukama-Dzoussi,P.E.,Ghosh,S.,Roux,F.S.&Konrad,T. 58. Stinespring, W. Positive functions on C -algebras. Proc. Am. Math. Soc. 6, – Qudit-teleportation for photons with linear optics. Sci. Rep. 4,4543 211 216 (1955). (2014). 59. Holevo, A. Complementary channels and the additivity problem. Theory – 24. Muralidharan, S., Zou, C.-L., Li, L., Wen, J. & Jiang, L. Overcoming erasure Probab. Appl. 51,92 100 (2007). errors with multilevel systems. New J. Phys. 19, 013026 (2017). 60. Smith, G. & Smolin, J. Degenerate quantum codes for Pauli channels. Phys. 25. Lima, G. et al. Experimental quantum tomography of photonic qudits via Rev. Lett. 98, 030501 (2007). mutually unbiased basis. Opt. Express 19, 3542–3552 (2011). 61. Wolf, M. & Perez-Garcia, D. Quantum capacities of channels with small 26. Nisbet-Jones, P. et al. Photonic qubits, qutrits and ququads accurately environment. Phys. Rev. A 75, 012303 (2007). prepared and delivered on demand. New J. Phys. 15, 053007 (2013). 62. Yard, J., Hayden, P. & Devetak, I. Capacity theorems for quantum multiple- 27. Marques, B. et al. Experimental simulation of decoherence in photonics access chan-nels: classical-quantum and quantum-quantum capacity regions. – qudits. Sci. Rep. 5, 16049 (2015). IEEE Trans. Inform. Theory 54, 3091 3113 (2008). 28. Babazadeh, A. et al. High-dimensional single-photon quantum gates: concepts 63. Servaes. Effect of diagonal orthogonal matrices (version: 2018-12-12). https:// and experiments. Phys. Rev. Lett. 119, 180510 (2017). math.stackexchange.com/q/3036627 (2018). 29. Kues, M. et al. On-chip generation of high-dimensional entangled quantum states and their coherent control. Nature 546, 622–626 (2017). 30. Moreno-Pineda, E. et al. Molecular spin qudits for quantum algorithms. Chem. Soc. Rev. 47, 501–513 (2018). Acknowledgements “ 31. Giordani, T. et al. Experimental engineering of arbitrary qudit states with We acknowledge support from PRIN 2017 Taming complexity with quantum strate- ” discrete-time quantum walks. Phys. Rev. Lett. 122, 020503 (2019). gies . S.C. thanks P. Novelli for fruitful discussions. 32. Sawant, R. et al. Ultracold polar molecules as qudits. New J. Phys. 22, 013027 (2020). 33. Chessa, S. & Giovannetti, V. Partially coherent direct sum channels. Preprint Author contributions at https://arxiv.org/abs/2008.00494 (2020). S.C. and V.G. contributed equally to the conception and writing of the manuscript.

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