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Resonant Multilevel Amplitude Damping Channels

We introduce a new set of quantum channels: resonant multilevel amplitude damping (ReMAD) channels. Among other instances, they can describe energy dissipation effects in multilevel atomic systems induced by the interaction with a zero-temperature bosonic environment. At variance with the already known class of multilevel amplitude damping (MAD) channels, this new class of maps allows the presence of an environment unable to discriminate transitions with identical energy gaps. After characterizing the algebra of their composition rules, by analyzing the qutrit case, we show that this new set of channels can exhibit degradability and antidegradability in vast regions of the allowed parameter space. There we compute their quantum capacity and private classical capacity. We show that these capacities can be computed exactly also in regions of the parameter space where the channels aren't degradable nor antidegradable.

While two-level quantum systems (qubit) represent the fundamental building block for any Quantum Information processing, there are indications that working with qudits (i.e. quantum systems with an Hilbert space of dimension d > 2) may bring in some advantages, both in terms of communication and cryptography (see e.g. [1] and references therein) and of computation (see e.g. [2] and references therein), with qutrits that recently made their first appearance on commercial quantum devices [3]. Despite this fact the landscape of mathematical models describing the physical noises affecting these systems is still relatively unexplored especially in terms of their associated information capacities, see [4][5][6][7][8][9][10][11] for results on basic noise models in higher dimensions. Capacities are figures of merit developed in the context of Quantum Shannon Theory [12][13][14][15][16][17][18][19], which allow one to quantitatively measure the level of deterioration that a given noise process induces on the quantum system it acts upon. A formal definition of such quantities is properly constructed in a specific communication scenario. There one describes the effect of the noise as an information loss during a signaling process that connects a sender (Alice) who is controlling the state of the system before the action of the noise, and a receiver (Bob), who instead has access to the deteriorated version of the qudit. In this context, depending on the type of messages one is considering (e.g. classical, private classical, or quantum) and on the type of side resources one allocates to the task (e.g. shared entanglement, two-way communication), the capacity of the noise channel is defined. Formally it corresponds to the optimal rate which gauges the maximum number of bits, secret bits, or qubtis that Alice can reliably transfer to Bob per use of the channel in the communication setting. Unfortunately for the vast majority of models such optimal rates are not computable neither analytically nor algorithmically [20][21][22] due to superadditivity and superactivation effects [23][24][25][26][27]. In this sense the literature has evolved to find capacities bounds and to find channel properties to be leveraged in order to overcome the hurdles of the direct computation or at least provide meaningful upper bounds. Among others, concerning the unassisted quantum and private classical capacities, we find: degradability [5], antidegradability [28], weak degradability [28], additive extensions [29], conjugate degradability [30], less noisy or more capable channels [31], partial degradability [32], approximate degradability [33], teleportation-covariant channels [34], unital channels [35], low noise approximations [36]. In this sense a corpus of literature is being built with the aim to produce efficiently computable bounds and approximations of these capacities, see e.g. [37][38][39][40][41][42][43][44]. From this perspective, this paper approaches the evaluation of the quantum capacity Q and the private classical capacity C p of a new class of qudit channels, providing their exact expression for a wide range of noise parameters.
Specifically, we introduce a family of noisy transformations that mimic energy loss in a multilevel quantum system S (e.g. an atom or a molecule coupled with a low temperature e.m. field). Our construction builds up from the Multilevel Amplitude Damping (MAD) proposal of [7] where a d-dimensional generalization of the qubit Amplitude Damping Channel (ADC) [16] is modeled as a collection of independent two-level processes that induce transitions from higher energy levels of the system to the lower ones. These transitions are mediated by an exchange interaction of the type σ − S ⊗ σ + E + σ + S ⊗ σ − E between S and its environment E (σ ± X representing raising and lowering operators for the system X). The major difference between the transformations we discuss here and the ones analyzed in [7], is that we now allow for the possibility that the transitions of S involving identical energy gaps will couple with the same type of excitations in the environment. We dub this class of the processes Resonant Multilevel Amplitude Damping (ReMAD) channels: they behave as usual MAD channels on the populations of each energy level of S, but exhibit different effects on the coherence terms of the system. Specifically due to the peculiar nature of the selected S − E coupling, under the action of a ReMAD channel the system S will be typically slightly less prone to dephasing than under the action of a MAD channel characterized by the same transition probabilities. At the mathematical level this corresponds to a net reduction of the minimal number of Kraus operators [12] required to describe the effect of the associated noise. This leads to some simplification in the characterization of their quantum capacities. What makes this noise model interesting is its simplicity and the fact that it can emerge in a variety of scenarios. To enumerate some of the ones relevant in quantum information processing and quantum communications we can mention: atomic systems in quantum memories and quantum repeaters [45][46][47], optical qudits transmitted through lines such as optical fibers with polarization dependent losses [48], optical qudits interacting with beamsplitters and qudits encoded in harmonic oscillators [49,50], see [51,52] and references therein for applications and implementations of bosonic codes, quantum computation and simulation with molecular spins [53] and references therein.
The article is structured as follows: in Sec. II we introduce ReMAD channels, their complementary channels and their composition rules; in Sec. III we briefly review the issues of degradability, antidegradability and the computation of quantum and private classical capacities Q and C p ; in Sec. IV we provide an analysis of Q and C p for qutrit ReMAD channels; conclusions are drawn in Sec. V.

II. DEFINITIONS
Let H S be the Hilbert space associated with a d-dimensional quantum system S characterized by an energetically ordered canonical basis. This basis is represented by a collection of orthonormal levels {|0 S , |1 S , · · · , |d − 1 S } with E j < E j+1 , being E j the energy associated with the j-th level. In this context we can formally describe energy damping processes by specifying a lower-triangular d × d transition whose elements γ j,k for j ≤ k ∈ {0, · · · , d − 1} are positive semidefinite quantities that define the transition probabilities, i.e. the probability for the energy level |j to be mapped into the lower energy level |k .
Consistency requirements are such that that for j ≥ 1 identify γ j,j = 1− j−1 k=0 γ j,k as the survival probability of the level j (notice that by construction γ 0,0 = 1, being |0 S a fixed point of the channels model). The MAD channels introduced in Ref. [7] assign to each matrix Γ a special Linear, Completely Positive, Trace Preserving (LCPTP) transformation Φ (MAD) Γ that, at the level of Stinespring representation [54], can be seen as a coupling with a zero-temperature external bath B absorbing each individual energy jump E j → E k into a distinct (orthogonal) degree of freedom (see left panel of Fig. 1).
Specifically, givenρ ∈ S(H S ) a generic density operator of S, we express its evolution under the action of Φ with |0 B being the ground state of the bath, Tr B [· · · ] the partial trace over the environment, andV Γ the unitary transformation that induces the mappingŝ In the above expression for j ∈ {1, · · · , d − 1} and k ∈ {1, · · · , j}, kets |j, k B describe a collection of vectors that are mutually orthogonal and orthogonal with |0 B as well. Since they represent the states where the environment stores the energy E j − E k lost by S when moving from level |j S to level |k S , the construction (4) implicitly assumes that the environment of the model is capable to discriminate among the different energy jumps. This condition is physically realized e.g. when the energy spectrum of S is composed by incommensurable levels. Notice also that this construction fixes the dimension of the Hilbert space H B of B, as well as the minimum number of Kraus operators needed to represent Φ (MAD) Γ in the operator-sum representation [55] By introducing ReMAD channels Φ Γ we now allow for the possibility that some of the transition events of S will excite the same internal degrees of freedom of the bath, a condition which can be achieved e.g. when the energy levels of the system are equally spaced, i.e. E j = j∆E for all j ∈ {0, · · · , d − 1}. In this case we can replace the unitary coupling (4) with the new interaction where {|0 E , |1 E , · · · , |d − 1 E } are a (possibly energetically ordered) orthonormal basis of the environment E. The resulting LCPTP transformation is hence obtained by replacing Eqs. (3) and (5) with whereK (i) is a Kraus set characterized by at most d non-zero elements. It is easy to check that wile for d = 2, the ReMAD channel Φ Γ coincides with its corresponding MAD counterpart Φ (MAD) Γ (indeed both reduce to the conventional qubit ADC). Also one can easily verify that for arbitrary d, both Φ Γ and Φ However for d ≥ 3 the two sets of transformations have rather different impact on the off-diagonal term of the input state. Consider for instance the qutrit (d = 3) scenario where, thanks to the constraint (2) the transition matrix Γ can be parametrized by 3 non-negative terms, e.g. γ 10 , γ 21 , and γ 20 , forming a 3-dimensional vector (γ 10 , γ 21 , γ 20 ) spanning the domain represented in Fig. 2 by the green rectangular right wedge delimited by the vertexes A, B, C, D, E, and F. In this case Eq. (7) reduces tô while, identifying the canonical basis states |0 S , |1 S , and |2 S with the column vectors (1, 0, 0) T , (0, 1, 0) T , and (0, 0, 1) T , we can express the associated Kraus operators (18) aŝ Accordingly the action of Φ Γ on a generic density matrixρ of S produces the output state of the form to be compared with the associated transformation induced by the MAD counterpart with ρ ij := S i|ρ|j S . As one can observe, while on the diagonal elements in both cases we have the usual decay of populations predicted by (10), the ReMAD output state (14) exhibits a transfer of coherence that mixes the terms ρ 12 and ρ 10 which is not contemplated in the MAD output (15). It is also worth noticing that in case either γ 21 = 0 or γ 10 = 0 the qutrit map Φ (MAD) Γ and the qutrit map Φ Γ describe the same physical process.
As a final remark we notice that a particular example of ReMAD channel is provided by the so called beamsplitter type ADC Ψ η introduced in Ref. [56]. This subclass of ReMAD channels describes the evolution of a qudit encoded in the first d states of the Fock basis of an harmonic oscillator passing through a beamsplitter of transmittance η. It's straightforward to verify that such mappings are a special instance of the ReMAD class characterized by a transition matrix Γ[η] whose elements can be parametrized by the formula  11) which identifies the set of ReMAD channels for d = 3 (green region, the grey volume corresponds to the inaccessible parameter region); the beamsplitter type ADC of Ref. [56] corresponds to the blue line that connects the vertexes D and B.

A. Complementary maps
At variance with what happens with MAD channels, the complementary map [12]Φ Γ of a generic ReMAD transformation Φ Γ is also a ReMAD channel (up to an isometry). This can be seen by recalling thatΦ Γ can be obtained from (8) by replacing the partial trace over E with a partial trace over S, i.e.
where nowQ Notice next that up to an isometryV SE mapping the energy levels of E into the corresponding levels of S (i.e.V SE |l E → |l S , ∀l),Q (i) Γ has exactly the same form of the operators (18) computed with a new lowertriangular transition matrixΓ. The non-zero elements ofΓ, row by row, are obtained by a simple reordering of those of Γ, i.e.γ Specifically we can writeV and henceV Again, as an example we provide the explicit form of the complementary channelΦ Γ of the qutrit ReMAD map Φ Γ of Eq. (14), i.e.
Notice finally that, as a consequence of Eq. (19), it follows that in the case of a beamsplitter type ADC Ψ η defined by Eq. (16), one getsγ j,k [η] = γ j,k [1 − η]. This, thanks to Eq. (21), allows us to recover the well known fact that the complementary mapΨ η of Ψ η is isometrically equivalent to Ψ 1−η .

B. Composition rules
In Ref. [7] it was shown that MAD channels are closed under composition rules, a property that proves useful in studying their information capacities. In this section we investigate whether ReMAD channels behave similarly. Maybe surprisingly, it turns out that it's not always the case. To begin with, let's observe that from the composition rules of beamsplitters one can easily verify that the following identity holds true (the symbol "•" represents channel composition). Apart from this special case, the analysis is slightly convoluted. We therefore focus on the simplest, nontrivial case of d = 3 qutrit systems defined by the inputoutput relations (14). By explicit computation it follows that being Γ and Γ two 3 × 3 transition matrices as in Eq.
(1) we have plus the Hermitian conjugate of off-diagonal elements. We are interested in understanding under which conditions a new transition matrix Γ of elements γ j,k exists such that Among the equations above, those referring to elements 00, 02, 11, 12, 22 are all consistent with setting (observe that such definitions are compatible with the request that γ j,k ∈ [0, 1], as well as with the normalization constraint j k=0 γ j,k = 1 for all j). Element 01 instead forces an additional constraint, i.e.
which is not necessarily granted. As a matter of fact by substituting in Eq. (27) the values for γ 21 and γ 10 obtained in Eq. (50) we get that in order to satisfy it we need This identifies a specific region for the parameters Γ and Γ where the composition Φ Γ • Φ Γ corresponds to a ReMAD channel. Notice in particular that, in agreement with (23), the identity (28)  . We can use this observation to claim that being Φ Γ and Φ Γ two qutrits ReMAD channels having the same transition probabilities connecting level 2 to level 1 and level 1 to level 0, but with γ 20 larger than or equal to γ 20 , then they can be connected via a third ReMAD channel as in Eq. (25).

C. Covariance
The ReMAD channels exhibit a covariance property under suitable unitary transformations. Specifically consider the set of unitary gatesÛ with θ real. It then follows that and similarly for the complementary channelΦ we get As we'll see in the following the identities above turn out to be extremely useful in the computation of the capacities of our channels.

III. REVIEW OF QUANTUM AND PRIVATE CLASSICAL CAPACITIES
This section is dedicated to reviewing some basic notions about channel capacities for those readers who may be not familiar with these quantities. While the plethora of capacities for quantum channels includes a large collection of different functionals, our analysis will be focused on the quantum capacity Q and the private classical capacity C p .

A. Quantum Capacity and Private Classical Capacity
The quantum capacity Q(Φ) of a quantum channel Φ defines the maximum rate of transmitted quantum information achievable per channel use, assuming Φ to act in the regime of i.i.d. noise [12][13][14][15][16][17][18][19]. Intuitively it tells you how faithfully a quantum state, possibly correlated with an external system, can be sent and received by two communication parties if the communication line is noisy. Recently Q has also been showed to provide a lower bound to the space overhead necessary for fault tolerant quantum computation in presence of noise [57]. The formal definition of Q relies on the coherent information I coh [58]. Assuming n uses of the channel Φ and a generic stateρ (n) ∈ S(H ⊗n S ) we have and by regularization of this expression we get the maximized coherent information per channel use, that is the quantum capacity Q(Φ) [59-61] The private classical capacity C p (Φ) quantifies the maximum rate of classical information achievable per channel use assuming also the privacy of communication (where privacy is intended as limiting to arbitrarily small the amount of information that an eavesdropper can extract from the environment during the communication). The fundamental tool needed to compute C p (Φ) is the Holevo information functional χ [62].
Through the Holevo information we define next the private information for n uses C (n) p (Φ), that involves a maximization over all ensembles E n from which finally C p (Φ) can be computed via regularization over n, i.e. [61,63]: We conclude our brief review by recalling that Q and C p obey data processing inequalities [13]. This means that, for (any) channel Φ that can be expressed as a composition of other two LCPTP maps Φ 1 and Φ 2 (i.e.

B. Degradability and antidegradability
The need of regularization in the evaluation of Q(Φ) and C p (Φ) poses a well known problem which ultimately is the underlying reason of our efforts here. An exception to this predicament is given by degradable [5] and antidegradable [28] channels, whose definitions and properties are reviewed here. A quantum channel Φ : while it's said antidegradable if it exists a LCPTP map A : In case Φ is mathematically invertible, a simple direct way to determine whether it is degradable or not is to formally invert the composition in Eq. (41). This is done (if possible) by constructing the super-operator D =Φ • Φ −1 and by checking whether such object is LCPTP [64,65]. The check can be performed by studying the positivity of its associated Choi matrix C D , i.e.
where given |Γ RS = d−1 i=0 |i R |i S we set C D := (I R ⊗ D S ) |Γ Γ| RS (see App. A for details). A completely similar argument can be made for antidegradability: in this case we can claim where now C A is the Choi operator of the map A = Φ •Φ −1 .
As already mentioned the evaluation of the quantum a private classical capacity simplifies for degradable channels. To begin with, in this case Q and C p result to be additive, so the regularization over n in Eq. (35) isn't needed, leading to the single-letter formula [66] Another important simplification arises from the fact that under degradability conditions the coherent information functional is concave w.r.t. the input state [67]. Accordingly if Φ andΦ turn out to be covariant under the action of some unitary group, the maximization in Eq. (45) can be further simplified. In the case of ReMAD channels where the properties (31) and (32) hold true, under degradability conditions we can write (46) where in the last equality we made use of the invariance of von Neumann entropy under unitary transformations. Observing then that we can conclude that for degradable ReMAD channels the maximization over of (45) can be restricted on the set of statesρ (diag) which are diagonal in the canonical basis, i.e.
For antidegradable channels instead, due to a no-cloning argument [68], Q(Φ) = 0. Similarly, C p (Φ) = 0: the environment can reconstruct the channel output simply by applying the antidegrading channel, so no private information can be transmitted. Therefore for channels exhibiting antidegradability no maximizations are needed.

IV. DEGRADABILITY AND ANTIDEGRADABILITY REGIONS FOR d = 3 REMAD CHANNELS
In this section we study the degradability/antidegradability properties for qutrits ReMAD channels. Mimicking the approach used for other amplitude damping channels, we tackle the problem working under the heuristic assumption that if a degrading (or antidegrading) channel of a ReMAD channel exists, it is itself a ReMAD channel. While this choice is potentially suboptimal, numerical tests based on the more rigorous (but analytically impractical) matrix inversion method discussed in Sec. III B, reveals that this is not the case.
Reminding the isometric connection given in Eq. (21), we hence translate the degradability condition on Φ Γ into the problem of identifying a transition matrix Γ s.t.
Imposing now the vector (γ 10 , γ 21 , γ 20 ) to belong to the domain D 3 of Eq. (11) we get the degradability region which we depict as the yellow region of Fig. 3. In such region we can compute exactly the value of Q(Φ Γ ) and C p (Φ Γ ) using the single-letter formula (48) only involving diagonal input matrices. Results of such optimization are reported in Fig. 4 for different values of the parameters γ 10 , γ 21 and γ 20 . In the cases of γ 21 = 0 and γ 10 = 0 the obtained values were already known in the whole resulting parameter space (including some non-degradable regions), since there ReMAD channels reduce to double decay MAD channels, already studied in [7].
To check antidegradability of a ReMAD channel we follow the same path, looking for a Γ that fulfills the identity leading to a constraint that can be obtained from the one in Eq. (50) by exchanging γ j,k withγ j,k . This leads finally to γ 10 = 2γ 10 − 1 γ 10 , Imposing to have (γ 10 , γ 21 , γ 20 ) in D 3 brings us to the antidegradability region depicted in blue in Fig. 3. The white regions correspond to points associated with not degradable (nor antidegradable) channels, grey regions are those outside the allowed parameter space D 3 . As we'll see in Sec. IV A, at least for γ 10 = 0.000 (left-top panel) Q and C p can also be extended in the white region.

A. Capacities in non-degradable and non-antidegradable regions
In those regions of parameters for which degradability or antidegradability are not achieved a closed accessible expression for Q and C p is lacking. A notable exception is provided by the two planar regions identified respectively by γ 10 = 0 (triangular surface of Fig. 2, identified by the vertexes ADE) and γ 21 = 0 (rectangular region ABCD). As one can observe from Fig. 3 they overlap only partially with the degradable and antidegradable regions: yet, thanks to the fact that there Φ Γ and Φ (MAD) Γ coincide (see comments in Sec. II), we can use the results of [7] to compute the quantum capacity of the associated ReMAD channels. Even for those points that aren't explicitly degradable (or antidegradable), specifically: Planar region γ 10 = 0: here the channel Φ Γ is neither degradable nor antidegradable for all points with γ 12 + γ 20 ≥ 1/2. Exploiting the results of [7] we can conclude that in this region the capacities are constant and equal to 1, i.e.  [7] we can conclude that in these regions the capacities assume symmetric values, i.e. give 0 ≤ x ≤ 1/2 ≤ y ≤ 1, we have with see right panel of Fig. 5. In the first expression the maximization is performed over the populations p 1 , p 2 ∈ [0, 1] of the input state with respect to levels |1 S and |2 S under the consistency constraint p 1 + p 2 ≤ 1; the second identity instead follows from the observation that for p 0 fixed, the maximum of the r.h.s. term is always achieved by p 2 = 0 (no population assigned to the second excited level). Observe that the final expression (61) exactly matches the quantum capacity of a qubit ADC channel with damping probability equal to 1 − x [69]. With a little additional effort, by comparison with the results of [7] we can also determine the exact value of Q(Φ Γ ) for the elements on the boundary of the D 3 region identified by the constraints (notice that while for γ 21 ≤ 1/2 the points belong to the antidegradable region, for γ 21 > 1/2 they are neither degradable nor antidegradable). For these values Eq. (14) becomes for which capacities Q and C p have been computed for all values of γ 10 [7]. The explicit connection between (58) and (59) follows by observing that, setting γ 10 = 1 − γ 21 we can map the former into the latter via a unitary transformation on the input of the channel via the identity beingV S the unitary operator reordering the canonical basis |0 S , |1 S , |2 S , into |1 S , |2 S , |0 S . Accordingly using Eq. (33) of the Supplementary Material of [7], in the region 1 2 ≤ γ 21 ≤ 1 we can write in the region 0 ≤ γ 21 ≤ 1 2 instead, Q = C p = 0 as the channel is antidegradable. The overall profile of the capacities in this case reduces then to that of a qubit ADC with damping parameter 1 − γ 21 [69]. We report its evaluation in Fig. 6, left panel. Notice that in the first expression the maximization is performed over the populations p 0 , p 1 ∈ [0, 1] of the input state with respect to levels |0 S and |1 S under the consistency constraint p 0 + p 1 ≤ 1. The second identity then follows by noticing that, similarly to what happens in Eq. (56), for p 0 fixed the maximum of the r.h.s. term is always achieved by p 1 = 0 (no population assigned to the first excited level). Comparing Eq. (61) with Eq. (55) we observe that on the edges BC and BF, that  (62), in the region γ 20 ≥ 1 2 the capacity is found for provably not degradable nor antidegradable channels.
bound D 3 on the plane γ 10 = 1, the capacity of the channel takes the same values as those for fixed γ 20 . Indeed we get which again is the capacity profile of a qubit ADC with damping parameter γ 20 . See 6, right panel.
In the other sectors of the parameter space it is still possible to exploit some information theoretic properties to obtain computationally efficient upper bounds. For instance exploiting the considerations underlined in the final paragraphs of Sec. II B and the data processing inequality (39) we can claim that, for fixed values of γ 10 and γ 21 , both Q(Φ Γ ) and C p (Φ Γ ) are non-increasing functions of γ 20 , e.g.

V. CONCLUSIONS
We introduced and characterized a new class of physically relevant noise models for high-dimensional quantum systems. ReMAD channels enlarge the limited class of quantum channels for which an exact analytical or numerical approach in terms of quantum and private classical capacities is feasible. This has been shown by focusing on the simplest, yet nontrivial, case in which the system of interest is a qutrit. In such case we have shown the existence of a substantial region of the noise parameter space where the channels Φ Γ are degradable/antidegradable, allowing hence the direct expression of the associated quantum and private classical capacities. Also, by resorting to formal mapping to the case of MAD channels discussed in [7], we computed the value of these capacities in regions where Φ Γ is provably neither degradable nor antidegradable. The full characterization in terms of information capacities, such as for instance classical capacity and two-way capacities, is still missing for ReMAD channels and will require further investigations. A semi-analytical treatment is instead achievable for the characterization of entanglement assisted quantum and classical capacities Q E and C E of ReMAD channels, see Appendix B.
we have: where the definition of Q E (Φ) is justified by the fact that in presence of entanglement a qudit quantum state can be teleported 'spending' two classical dits (quantum teleportation) and viceversa two classical dits can be communicated by sending a single qudit (superdense coding).
To effectively compute these quantities we notice that the Shannon entropy is always concave w.r.t. the input state but, as we saw before, the coherent information can be proved concave only when the channel is degradable. Nevertheless the quantum mutual information is always concave in the input state [13, chapter 13.4.2]. Therefore also I(Φ) can be maximized just over diagonal states if Φ is a covariant channel by following steps as in Eq. (46). We report then in Fig. 7 the evaluation of C E (Φ Γ ) for a qutrit ReMAD channel at varying γ 20 and γ 21 for some instances of γ 10 values.