Detecting positive quantum capacities of quantum channels

Determining whether a noisy quantum channel can be used to reliably transmit quantum information is a challenging problem in quantum information theory. This is because it requires computation of the channel’s coherent information for an unbounded number of copies of the channel. In this paper, we devise an elementary perturbative technique to solve this problem in a wide variety of circumstances. Our analysis reveals that a channel’s ability to transmit information is intimately connected to the relative sizes of its input, output, and environment spaces. We exploit this link to develop easy tests which can be used to detect positivity of quantum channel capacities simply by comparing the channels’ input, output, and environment dimensions. Several noteworthy examples, such as the depolarizing and transpose-depolarizing channels (including the Werner-Holevo channel), dephasing channels, generalized Pauli channels, multi-level amplitude damping channels, and (conjugate) diagonal unitary covariant channels, serve to aptly exhibit the utility of our method. Notably, in all these examples, the coherent information of a single copy of the channel turns out to be positive.


Introduction
The capacity of a noisy communication channel quantifies the fundamental physical limit on noiseless communication through it.Shannon proved that every noisy classical channel has a unique capacity which is given in terms of the mutual information between the random variables characterizing the channel's input and output [Sha48].In contrast, a quantum channel has many different kinds of capacities.These depend on a variety of factors, for example, on the type of information (classical or quantum) which is being transmitted, whether or not this information is private, the nature of the input states (entangled or not), and whether any auxiliary resource is available to assist the transmission 1 .
The quantum capacity of a quantum channel quantifies the maximum rate of noiseless and coherent quantum communication through it, in the absence of any auxiliary resource.In general, it is not possible to explicitly compute the quantum capacity of a given channel.This is because Lloyd [Llo97], Shor [Sho02], and Devetak [Dev05] have shown, with increasing levels of rigour, that the quantum capacity of a channel Φ is given by the following regularized formula (as opposed to a single letter expression): where In the above definition, Φ c denotes a channel which is complementary to Φ (see Section 2.1), and for any quantum state ρ, S(ρ) := − Tr(ρ log ρ) denotes its von Neumann entropy.The quantity Q (1) (Φ) is called the coherent information of Φ. Usually, it is superadditive: Q (1) (Φ ⊗n ) > nQ (1) (Φ), and hence the quantum capacity of Φ is larger than its coherent information and notoriously difficult to compute.Some exceptional classes of channels which are currently known to have additive coherent information (i.e. for which Q (1) (Φ ⊗n ) = nQ (1) (Φ)) are strictly contained in the set of more capable quantum channels [Wat12], whose defining property is that their complementary channels have zero quantum capacity.Furthermore, it has been shown that there exist pairs of quantum channels (say Φ 1 and Φ 2 ), each of which has zero quantum capacity, but which can be used in tandem to transmit quantum information, i.e.Q(Φ 1 ⊗ Φ 2 ) > 0. This startling effect (known as superactivation [SY08]), is a purely quantum phenomenon because classically, if two channels have zero capacity, the capacity of the joint channel must also be zero.
1.1.Summary of main results.In light of the above discussion, it is of utmost importance to determine when a given quantum channel has positive quantum capacity.However, to date, there is no known procedure or algorithm which solves this problem, except in a handful of special cases [SS12].The main result of this paper makes a significant step forward in this direction by providing a powerful sufficient condition to guarantee positivity of quantum capacities for a wide variety of quantum channels and their complements; see Theorem 3.1 in the main text, which is also stated below for convenience.
An equivalent formulation of the above theorem yields a first-of-its-kind necessary condition for membership in the set of zero capacity quantum channels; see Theorem 3.8.As a sanity check, we prove that some of the few known classes of channels with zero capacity satisfy the necessary condition of Theorem 3.8; see Lemma 3.9.Notably, the main result in [Sid20] arises as an immediate consequence of Theorem 3.1, which can be applied to detect positive quantum capacity of a given channel (or its complement) if the channel's output and environment dimension are unequal and if there exists a pure state whose image under the channel has the maximum possible rank; see Corollary 3.2.Notwithstanding the difficulty in ascertaining the existence of the aforementioned pure state for any given channel, we prove that the desired pure state exists for all channels with sufficiently large input dimensions, so that any such channel (or its complement) with unequal output and environment dimension must have positive quantum capacity, irrespective of the specific form of the channel; see Corollary 3.6.Similar results were obtained in [Sid20] for channels with large output and environment dimensions, which are derived again in Corollary 3.4 with simpler proofs.
Meanwhile, Theorem 3.1 is completely unaffected by the above considerations, and hence can be applied to a much broader spectrum of quantum channels, including the ones which have equal output and environment dimensions and the ones for which no pure state gets mapped to an output state with maximal rank; see Remark 3.3.
Using our main results, we are able to establish positivity of the quantum capacities of numerous important quantum channels and their complements in Section 4. The following is a list of main results in this direction.
• In [LW17], the qubit depolarizing channel D p : M 2 → M 2 was shown to have positive complementary quantum capacity 2 for all non-zero values of the noise parameter p > 0. We extend this result to show that the qudit depolarizing and transpose-depolarizing channels • In [LW17], the qubit Pauli channel Φ P : M 2 → M 2 was also shown to have positive complementary quantum capacity whenever the defining probability matrix P ∈ M 2 has at least three non-zero entries.An extension of this result for generalized qudit Pauli channels Φ P : M d → M d with a much simpler proof is presented in Theorem 4.4.• For a multi-level amplitude damping channel Φ γ : M d → M d , we establish simple constraints on the decay rate vector γ ∈ R d(d−1)/2 which ensure the positivity of its quantum capacity and its complementary quantum capacity; see Theorem 4.6.• We show that a generalized dephasing or Hadamard channel Φ B : M d → M d (parametrized by a correlation matrix B ∈ M d ) has zero quantum capacity if and only if it is entanglementbreaking; see Theorem 4.12.• Recently, the family of (conjugate) diagonal unitary covariant (dubbed (C)DUC) quantum channels was introduced in [SN20a].A rich variety of channels, such as the depolarizing and transpose depolarizing channels, amplitude damping channels, dephasing channels etc. are known to belong in this family.In Proposition 4.10, we completely characterize the class of (C)DUC channels for which there exists a pure input state which gets mapped to a maximal rank output state, so that Corollary 3.2 can be applied to infer positivity of the quantum capacities of these channels and their complements; see Theorem 4.11.Moreover, Theorem 3.1 has various interesting ramifications, which are studied in Section 5.For instance, it leads to simplified proofs of certain existing structure theorems for the class of degradable quantum channels, and an extension of their applicability to the larger class of more capable quantum channels.1.2.Mathematical methods.The primary tool employed in the proof of Theorem 3.1 is a seminal result (by Rellich) from the theory of analytic perturbations of Hermitian matrices.Put simply, if a Hermitian matrix A 0 with an eigenvalue λ of multiplicity n is subjected to a linear (Hermitian) perturbation A( ) = A 0 + A 1 in a real parameter , then the perturbed matrix has exactly n eigenvalues converging to λ as → 0, all of which, rather remarkably, admit convergent power series expansions in a neighborhood of = 0; see Theorem A.1.Now, for a given channel Φ, it is trivial to see that I c (ρ; Φ) = 0 for all pure input states ρ = |ψ ψ|.It turns out that if we choose ρ( ) = (1 − ) |ψ ψ| + σ as a slight perturbation of an arbitrarily chosen pure state |ψ ψ| with an arbitrary mixed state σ, then the first order correction terms in the eigenvalue expansions of the perturbed outputs Φ[ρ( )] and Φ c [ρ( )] can be analyzed to yield a simple condition which guarantees that I c (ρ( ); Φ) > 0 for sufficiently small values of , so that 2 The complementary quantum capacity of a channel refers to the quantum capacity of anyone of its complements.

Prerequisites
Let us start our journey by briefly reviewing the basics of quantum channels and their capacities in this section.A more thorough discussion on these topics can be found in [Wat18].
2.1.Quantum channels.We denote the set of all d 1 × d 2 complex matrices by M d 1 ×d 2 .When A quantum channel is a completely positive and trace preserving linear map Φ : is a unital completely positive linear map defined uniquely by the following relation: Two quantum channels Φ : M d → M dout and Φ c : M d → M denv are complementary to each other if there exists an isometry (often called a Stinespring isometry) V : In the above scenario, all input states are first isometrically embedded into the combined outputenvironment space, with the action of Φ and Φ c then retrieved by partially tracing out the environment and the output space, respectively [KMNR07,Hol07].For a given channel Φ : M d → M dout , the collection of all channels that are complementary to Φ is denoted by C Φ .
Remark 2.1.To every channel Φ : M d → M dout , we can uniquely associate two sets of channels C Φ and C Φc , where Φ c ∈ C Φ is complementary to Φ.These sets are non-empty, since the Stinespring dilation theorem guarantees that the given channel Φ can be expressed as in Eq. (5) for some isometry We define the minimal environment and output dimension of Φ as respectively, where Φ c ∈ C Φ .Lemma 2.2 tells us that the above definition does not depend on the choice of Φ c .We say that two complementary channels Φ : It is clear from the definition that if two complementary channels Φ and Φ c are minimally defined, then they are isometrically related to every channel in the sets C Φc and C Φ , respectively, in the sense of Remark 2.1.
The Choi matrix of a quantum channel Φ : denotes a maximally entangled state and id : M d → M d is the identity map.The rank of the Choi matrix of a channel is called the Choi rank of the channel3 .The minimal environment dimension of a channel is related to its Choi rank via the following crucial lemma.
Lemma 2.2.For a channel Φ : M d → M dout and some complementary channel Proof.Let us begin with the expression for the minimal environment dimension.For any channel Φ : M d → M dout , the Stinespring dilation theorem tells us that there exists an isometry V : [Wat18,Corollary 2.27].Moreover, any other isometry V : C d → C dout ⊗ C denv that defines the channel as above must be such that rank J(Φ) ≤ d env , so that d * env (Φ) = rank J(Φ).Now, let r = rank Φ c (1 d ) for some Φ c ∈ C Φ .Then, since all complementary channels Φ c : M d → M denv are isometrically related (see Remark 2.1), we have r = rank Φ c (1 d ) ≤ d env for all Φ c ∈ C Φ , so that r ≤ d * env (Φ).To establish the reverse inequality, observe that for any positive semi-definite X ∈ M d and Φ c : M d → M denv complementary to Φ, there exists a small enough δ > 0 such that Finally, the expression for the minimal output dimension can be derived by applying the formula for the minimal environment dimension to any complementary channel Φ c ∈ C Φ .
Remark 2.3.We note some straightforward consequences of Lemma 2.2 below: where Φ c ∈ C Φ is complementary to Φ and S(ρ) = − Tr(ρ log ρ) denotes the von Neumann entropy of ρ ∈ S d .The coherent information Φ is then defined to be Notice that since the entropy S is continuous over S d which is a compact set (with respect to the standard topology induced by the operator norm, say), the maximum in the above definition is indeed achieved at some ρ ∈ S d .The seminal works of Lloyd [Llo97], Shor [Sho02], and Devetak [Dev05] show that the quantum capacity of a channel Φ is given by the following regularized expression: where, for n ∈ N, the coherent information of the product channel Q (1) (Φ ⊗n ) is called the n−shot coherent information of Φ.If the maximum in Eq. ( 8) is attained at ρ ∈ S d , then it is easy to see that I c (ρ ⊗n ; Φ ⊗n ) = nQ (1) (Φ), so that Q (1) (Φ ⊗n ) ≥ nQ (1) (Φ).However, it may happen that the coherent information is superadditive: Q (1) (Φ ⊗n ) > nQ (1) (Φ), in which case the task of evaluating Q(Φ) becomes intractable [DSS98, CEM + 15].The regularization in Eq. ( 9) is not required only for certain special channels for which the coherent information is additive: Q (1) (Φ ⊗n ) = nQ (1) (Φ), and consequently, the quantum capacities of these channels admit nice single-letter expressions in terms of the channels' coherent information Remark Quantum channels that are complementary to degradable channels are known as anti-degradable.
In other words, a channel Φ : M d → M dout is said to be anti-degradable if there exists a channel If such a channel could be used for reliable quantum communication, then its environment would be able to replicate all the transmitted information by applying the anti-degrading map N , thus violating the no-cloning theorem [BDS97, SS12].Hence, anti-degrading channels have zero quantum capacity.More generally, all the currently known classes of channels with additive coherent information lie within the strict superset of more capable channels [Wat12], which are defined by the property that their complementary channels have zero quantum capacity.
The set of quantum channels with zero quantum capacity: PPT and anti-degradable (ADG) channels are the only kinds of channels which are currently known to belong in this set.Entanglement-breaking (EB) channels form a strict subset of the intersection of the PPT and anti-degradable sets of channels.The question mark indicates that it is not currently known if there exist quantum channels which are neither anti-degradable nor PPT but still have zero quantum capacity.
Apart from anti-degradable channels, only PPT channels [HHH00] are known to have zero capacity.A channel Φ : M d → M dout is said to be PPT if • Φ is again a quantum channel, where : M dout → M dout is the transpose map.Equivalently, Φ is PPT if and only if its Choi matrix J(Φ) ∈ M dout ⊗ M d is positive under partial transposition (PPT).Such channels cannot have positive capacity because if they did, one would be able to distill maximally entangled pure states from the PPT Choi matrices of these channels, which is impossible [HHH98].The well-known family of entanglement-breaking channels [HSR03] -which consists of channels whose Choi matrices are separable -is strictly contained within the the intersection of the anti-degradable and PPT families.
Transpose degradable and Transpose anti-degradable channels.
Remark 2.5.In [BDHM10], the authors introduced another class of quantum channels with additive coherent information consisting of conjugate degradable channels.According to their definition, a quantum channel is said to be conjugate degradable if its environment can be simulated up to complex conjugation via its output.More precisely, a channel Φ : where C denotes (entrywise) complex conjugation on M denv and Φ c : M d → M denv is complementary to Φ.In an analogous fashion, the class of conjugate anti-degradable channels was defined, and these channels were claimed to have zero quantum capacity.However, there is a fundamental problem with the stated definitions, which stems from the fact that the map C is anti-linear.Therefore, Eq. ( 10) is invalid, since it is equating an anti-linear map on its left hand side with a linear map on its right hand side.This issue can be easily resolved if we just replace the anti-linear operation of complex conjugation by the linear operation of transposition.Hence, we are led to the following definitions of transpose degradable and transpose anti-degradable channels.
We say that a channel Φ : where denotes the transpose map on the relevant matrix spaces.It is easy to show that transpose degradable channels have additive coherent information, while transpose anti-degradable channels have zero quantum capacity.In this regard, note that all transpose anti-degradable channels are PPT as well.Whether these classes are actually different from their non-transposed counterparts is currently unknown.

Figure 2. [Wat12]
The set of more capable (MC) quantum channels, i.e. channels with zero complementary quantum capacity.Degradable (DG) and transpose degradable (TDG) channels lie strictly within this set.The relationship between the sets of DG and TDG channels is currently unestablished.

Main results
We have seen that an explicit computation of the quantum capacity of a quantum channel is usually intractable.So, is it at least possible to determine if a given channel Φ has positive quantum capacity or not?The question is clearly of immense significance, as is evident from our discussions in Sections 1 and 2.2.The answer, however, has proved to be quite elusive.The natural first step to tackle this problem would be to show that the coherent information of Φ is positive, which, according to our discussion in Section 2.2, trivially provides a lower bound to the quantum capacity.Moreover, for a pure input state |ψ ψ| ∈ S d and Φ c ∈ C Φ , Lemma A.3 informs us that the nonzero eigenvalues of Φ(|ψ ψ|) and Φ c (|ψ ψ|) (counted with multiplicities) are identical, and hence I c (|ψ ψ| ; Φ) = 0. Now, let's see if we can slightly perturb our pure input state so as to obtain a positive value of the coherent information.To this end, we define a one-parameter family of input states ρ( ) = (1 − ) |ψ ψ| + σ, where σ ∈ S d is an arbitrary state and ∈ [0, 1] is the perturbation parameter.Let us focus on the zero eigenvalue of Φ(|ψ ψ|) and Φ c (|ψ ψ|) with multiplicities κ = dim ker Φ(|ψ ψ|) and κ c = dim ker Φ c (|ψ ψ|), respectively.Once we turn on the perturbation, will have exactly κ and κ c eigenvalues, respectively, which converge to zero as → 0, see Theorem A.1.Remarkably, these eigenvalues admit convergent power series expansions in the perturbation parameter (in a neighborhood of = 0).Furthermore, if K ψ and K c ψ denote the orthogonal projections onto the unperturbed eigenspaces ker Φ(|ψ ψ|) and ker Φ c (|ψ ψ|), respectively, then the non-zero eigenvalues of K ψ Φ(σ)K ψ and K c ψ Φ(σ)K c ψ determine the first order correction constants in the aforementioned eigenvalue expansions, thus giving rise to the following crucial term in the derivative of the coherent information4 (see Appendix A): All the higher order corrections above can be shown to be bounded in , so that the derivative becomes positive for sufficiently small values of if Tr(K ψ Φ(σ)) > Tr(K c ψ Φ c (σ)).Hence, I c (ρ( ); Φ) is strictly increasing when 0 < < δ (for some small δ > 0), which implies that it has a positive value in the stated range (recall that at = 0, I c (ρ( ); Φ) = 0).
In order to formulate the above discussion in a mathematically rigorous fashion, a significant number of non-trivial intermediate steps are required, which are presented in Appendix A. The final theorem stated below is the primary result of this paper.
Theorem 3.1.Let Φ : M d → M dout and Φ c : M d → M dout be complementary channels.For a pure input state |ψ ψ| ∈ S d , denote the orthogonal projections onto ker Φ(|ψ ψ|) and ker Φ c (|ψ ψ|) by K ψ and K c ψ , respectively.Then, ).Before proceeding further, let us note that for a given channel Φ : M d → M dout and an arbitrary pure state |ψ ψ| ∈ S d , Lemma 2.2 and Lemma A.3 can be exploited to deduce that the maximal possible rank of However, it is not easy to determine if there exists a pure state for which the above bound gets saturated.We will get back to this issue after proving our next result, Corollary 3.2, which is a to the trace factor in Eq. ( 12) are referred to as the rates of the -log singluarities of the output entropies S[Φ(ρ( ))] and S[Φc(ρ( ))], albeit without any mention of the explicit power series expansions of the relevant eigenvalues.
simple but very useful consequence of Theorem 3.1.We should mention that this result has been derived independently in [Sid20].
Corollary 3.2.Let Φ : M d → M dout be a channel such that there exists a pure state Proof.We only prove the second part here and leave a similar proof of the first part to the reader.
Remark 3.3.Theorem 3.1 is more general than Corollary 3.2, since it can be used to detect positivity of the quantum capacity of channels which lie outside the realm of applicability of Corollary 3.2, especially for channels Φ with d * out (Φ) = d * env (Φ).In Theorem 4.12, we show that the class of dephasing or Hadamard channels contains such examples; see also Example 4.6.Moreover, there exist channels Φ for which even though d * out (Φ) = d * env (Φ), there is no pure state that gets mapped to an output state with maximal rank, so that Corollary 3.2 cannot be applied.The well-known Werner-Holevo channel provides such an example.Using Theorem 3.1, we obtain positivity of the quantum capacity of its complement for all input dimensions d ≥ 4 in Theorem 4.3.
for some channel Φ, Corollary 3.2 can be applied if the existence of a pure input state that gets mapped to a state with maximal rank in the output space can be guaranteed, see Eq. (13).However, this condition can be difficult to check in practice.In order to see why, let us consider a pair of complementary channels Φ : Then, the associated isometry V : C d → C dout ⊗C denv which defines these channels (see Eq. ( 5)) identifies the input space C d with a d−dimensional subspace range V ⊆ C dout ⊗ C denv , which can further be identified with a d−dimensional matrix subspace vec −1 (range V ) ⊆ M dout×denv via the inverse of the vectorization map, see Eq. (3).Since establishing the existence of a pure state with maximal output rank is equivalent to establishing the existence of a full rank matrix in the subspace vec −1 (range V ) ⊆ M dout×denv , which is known to be hard [Lov89,Gur03].This is why the simple dimensional inequalities given below are so convenient.Put simply, they ensure the existence of the desired pure input state whenever d * out (Φ) or d * env (Φ) is sufficiently large for an arbitrary channel Φ.We should point out that the following inequalities were also derived in [Sid20], albeit in a different way.
An identical argument can be used to prove the first part as well.
By exploiting an intriguing matrix-theoretic result (Lemma 3.5), we next derive a new dimensional inequality in Corollary 3.6, which ensure the existence of the sought-after pure state (i.e. the state which gets mapped onto an output state with maximal rank) for any given channel.Unlike Corollary 3.4, the new inequality in Corollary 3.6 implies that all channels with sufficiently large input dimension must either have positive quantum capacity or positive complementary quantum capacity.To see how these two corollaries nicely complement each other, see Remark 3.7.
Proof.This result was first proven in [Fla62] for matrix spaces over fields F with cardinality |F | ≥ r + 1, and was later generalized to work for arbitrary fields in [Mes85].We nevertheless provide a simple proof for the case when F = C is the field of complex numbers.By padding all the matrices in S with extra zero rows, we can assume that d 1 = d 2 = d.Now, without loss of generality, we can further assume that the following block matrix is contained in S where 1 r ∈ M r is the identity matrix.If S doesn't contain such a matrix, then it is easy to find non-singular matrices P, Q ∈ M d such that the subspace P SQ := {P XQ : X ∈ S}, which has the same dimension as S, contains the aforementioned matrix.Now, let Then, for 0 = X ∈ S ∩ S, where the implication holds for all c ∈ C.However, a simple application of the Schur complements of block matrices reveals that rank(X + cI) c .This leads to a contradiction, and hence S ∩ S = {0}.Hence, we arrive at the desired conclusion: We only prove the first part here, and leave an analogous proof of the second part to the reader.As usual, we assume that the channels Φ : , where the equivalence of the latter two operator inequalities in M d is a consequence of the fact that both Φ * and Φ * c are unital, R ψ + K ψ = 1 dout and R c ψ + K c ψ = 1 denv .As a sanity check, the next lemma shows that anti-degradable and transpose anti-degradable channels (which are known to have zero capacity, see Section 2.2) satisfy the necessary condition stated in Theorem 3.8.Lemma 3.9.Let Φ : M d → M dout be an anti-degradable or transpose anti-degradable channel and Φ c : M d → M denv be complementary to Φ.Then, , where R ψ and R c ψ are as defined in Corollary 3.8.Proof.We prove the result for anti-degradable channels and leave an almost identical proof of the transposed version to the reader.It suffices to obtain the result for n = 1, since if Φ is antidegradable, then Φ ⊗n is also anti-degradable for all n ∈ N. To begin with, note that the antidegradability of Φ : M d → M dout implies that there exists a channel N : Hence, for every pure state |ψ ψ| ∈ S d , we can exploit Lemma B.1 to obtain the following sequence of implications: (16) Observe that while obtaining the implications above, we used the fact that for A, B ≥ 0, AB = 0 ⇐⇒ Tr(AB) = 0. Now, since N * is unital and completely positive (being the adjoint of a quantum channel), it is contractive in the operator norm (see [Bha15, Theorem 2.3.7]),which implies that N * (K ψ ) ≤ K ψ ≤ 1. Combining Eq. ( 16) with the previous result, we obtain ). Remark 3.10.It would be desirable to obtain a result analogous to Lemma 3.9 for PPT channels, which also have zero quantum capacity.Observe that Lemma 3.9 already contains the desired result for transpose anti-degradable channels, which form a subclass of PPT channels.We should emphasize, however, that the relationship between a (transpose) anti-degradable channel and its complement -which is crucially exploited in the proof of Lemma 3.9 -does not hold for PPT channels in general.Hence, the aforementioned proof would not work in this case.

Examples
In this section, we apply the results derived in Section 3 to several different families of quantum channels.Before starting with the examples, let us recall from Section 2.2 that a channel Φ is said to be more capable if any complementary channel Φ c ∈ C Φ has zero capacity Q(Φ c ) = 0.Moreover, all channels which are currently known to have additive coherent information are of the above type.Thus, whenever we show that a channel's complement has positive quantum capacity, we show that the channel itself is not more capable.In particular, such a channel cannot be degradable or transpose degradable.
and are defined as follows: These families of channels are arguably two of the most prominent noise models in quantum information theory.While considerable effort has gone into computing the quantum capacity of depolarizing channels [SS08, SSWR17, LDS18], an analysis of their complementary quantum capacities has largely evaded the spotlight.Recently, by carefully scrutinizing the coherent information of the qubit depolarizing channels D p : M 2 → M 2 , Watrous and Leung have shown that the these channels have positive complementary quantum capacities for all p > 0 [LW17, Theorem 1].A similar analysis has also been performed for the qubit D q : M 2 → M 2 and qutrit D q : M 3 → M 3 transpose-depolarizing channels to establish positivity of their complementary quantum capacities for certain values of the parameter q, see [Brá15, Section 3.1].We now substantially generalize the above results to obtain positivity of the complementary quantum capacities of depolarizing and transpose-depolarizing channels in arbitrary dimensions while also providing much simpler proofs.Proof.Firstly, observe that for all allowed p, q, both D p and D q are unital so that d * out (D p ) = d * out (D q ) = d, see Remark 2.3.Moreover, the Choi matrices of these channels can be written as |ii jj| and These are, respectively, the well-known Isotropic [HH99] and Werner [Wer89] matrices, which admit nice block diagonal decompositions so that their ranks can be easily computed (see [SN20a, Example 3.3 and Corollary 4.2] or Lemma 4.9): Now, since for all p > 0 and the desired results hold because of Corollary 3.4.For q = d d+1 , it is easy to see that D q maps any pure state to a full rank state on the output, so that Corollary 3.2 can be applied to obtain positivity of the complementary quantum capacity.
Remark 4.2.The depolarizing-and transpose-depolarizing channels have the following covariance property for all U ∈ U d and X ∈ M d : where U d is the unitary group in M d .Upon relaxing the above covariance condition to hold only for the smaller group of diagonal unitary matrices DU d ⊂ U d , we obtain substantially bigger classes of channels which are defined by ∼ d 2 real parameters (as opposed to the single real parameters p and q in the depolarizing and transpose-depolarizing families).Theorem 4.11 serves as an analogue of Theorem 4.1 for these larger classes of channels.
Let us now analyse the depolarizing-and transpose-depolarizing channels present at the extreme ends of their respective parameter ranges.For p = 0, D p = id : M d → M d is the identity channel and d * env (D p ) = 1 for all d ∈ N, so that any complementary channel has zero quantum capacity.The q = d d−1 case of the transpose-depolarizing channel D q is much more subtle.For d = 2, the channel D 2 : M 2 → M 2 has d * env (D 2 ) = 1 so that any complementary channel has zero capacity.When d = 3, it is known that D 3/2 : M 3 → M 3 is both degradable and anti-degradable [CRS08] (in fact, there exists a complementary channel [D 3/2 ] c such that [D 3/2 ] c = D 3/2 ) and hence any complementary channel again has zero capacity.For all d ≥ 4, Theorem 4.3 below settles the question.It is perhaps worthwhile to point out that the transpose-depolarizing channel at the extreme parameter value q = d d−1 is more widely known as the Werner-Holevo channel.
where F : Clearly, range Φ c (|0 0|) lies within the anti-symmetric subspace of then the orthogonal projections onto the symmetric and anti-symmetric subspaces of C d ⊗ C d can be written as respectively.Using Eq. ( 23), we can express the orthogonal projection onto ker Φ c (|0 0|) as follows , the required trace expressions can be easily calculated.We have Tr(K 0 Φ(1 d )) = 1 and for all d ≥ 4, A direct application of Theorem 3.1 suffices to conclude that Q (1) (Φ c ) > 0.
4.2.Generalized Pauli (or Weyl Covariant) Channels.For an entrywise non-negative matrix P ∈ M d satisfying ij p ij = 1, the generalized Pauli channel Φ P : M d → M d associated with P is a mixed unitary channel defined as follows: where is the unitary orthogonal basis of M d formed by the discrete Heisenberg-Weyl operators: are the so-called shift and clock matrices, respectively.Note that in the above definition, addition inside kets happens modulo d and ω = e 2πi/d is the d th root of unity.The orthogonality of the unitary basis {U ij } d−1 i,j=0 can be easily verified by showing that Tr(U † ij U kl ) = dδ ik δ jl .When d = 2, these unitary matrices are identical to the familiar 2 × 2 pauli matrices.In this case, a sophisticated analysis of the coherent information of the qubit Pauli channels reveals that these channels have positive complementary quantum capacity whenever the associated matrix P ∈ M 2 has at least 3 non-zero entries [LW17, Theorem 2].Our next theorem extends this result for generalized Pauli channels acting in arbitrary dimensions.
Theorem 4.4.Let Φ P : M d → M d be a generalized Pauli channel associated with P ∈ M d .If P has at least d + 1 non-zeros entries distributed in such a way that either every row or every column of P is non-zero, then any complementary channel Φ c P ∈ C Φ P has positive quantum capacity.Proof.Firstly, since Φ P is unital, we have d * out (Φ P ) = d.In addition, if P has at least d + 1 non-zero entries, then d * env (Φ P ) ≥ d + 1 (see Remark 2.3), so that Corollary 3.2 can be applied to obtain the desired result if there exists a pure state |ψ ψ| ∈ S d such that Φ P (|ψ ψ|) ∈ S d has full rank.Now, if P has a non-zero entry p ik i = 0 for each i ∈ {0, 1, . . ., d − 1}, then every power of the shift matrix X is present in the Kraus decomposition of Φ P (see Eq. ( 25)).Hence, by choosing |ψ to be an eigenvector of the clock matrix Z (say |ψ = |0 ), we get a full rank state on the output: Similarly, if every column of P has a non-zero entry, then an eigenvector of the shift matrix X, say |ψ = 1 i=0 |i , would result in a full rank output.

4.3.
Multi-level amplitude damping channels.Multi-level amplitude damping (MAD) channels are employed to model the decay dynamics of a particle in a d-level quantum system with the associated Hilbert space C d = span{|i } d−1 i=0 .A total of d(d − 1)/2 real numbers {γ j→i } 0≤i<j≤d−1 are used to parameterize the decay rates of the particle from a higher j th level to a lower i th level, so that the MAD channel Φ γ : M d → M d admits the following Kraus representation where Complete positivity and trace preserving property of Φ γ implies that ∀i < j : γ j→i ≥ 0 and ∀j ∈ [d] : We say that a level j ∈ {1, . . ., d − 1} gets totally depleted under the channel Φ γ if for all k > j, γ k→j = 0 (i.e., no higher level has a positive rate of decaying to the j th level) and i<j γ j→i = 1 (i.e., the sum of decay rates from the j th level to all the lower levels equals one).Since there are no levels below the ground level j = 0, it can never get totally depleted.Given Φ γ , we denote the number of levels which do not get totally depleted (including the ground level) by n 1 ( γ).We denote the total number of non-zero decay rates in γ by n 2 ( γ).The following lemma relates these newly defined parameters with the minimal output and environment dimensions of Φ γ .
To obtain the expression for d * env (Φ γ ), observe that the number of linearly independent Kraus operators in the Kraus representation of Φ γ (see Eq. ( 27) 3. The quantum capacity of the 2-level (or qubit) amplitude damping channels is quite well understood [GF05].This is because any such channel is completely determined by a single decay rate γ ∈ R, so that d * env (Φ γ ) ≤ 2 and these types of channels are known to be either degradable or anti-degradable [WPG07].Unfortunately, the same cannot be said for MAD channels in higher dimensions.Hence, their capacity analysis has recently started attracting significant interest from the community, see [CG21] and references therein.However, to the best of our knowledge, virtually nothing is known about the complementary quantum capacities of these channels.In the following theorem, we obtain the first results in this direction by deriving simple sufficient conditions on the decay rate vector γ ∈ R d(d−1)/2 which ensure that the associated MAD channel Φ γ has positive (complementary) capacity.
and the conditions stated in the theorem ensure that .
Then Lemma 4.5 and Corollary 3.2 can be readily applied to obtain the required results.
Remark 4.7.Theorem 4.6 can be considered as a special case of a more general result which holds for the class of (conjugate) diagonal unitary covariant channels, see Proposition 4.10 and Theorem 4.11.
It is easy to see that if Φ is DUC (resp.CDUC), then for all U ∈ DU d , These channels and their corresponding Choi matrices have recently been examined quite thoroughly in [JM19,SN20a,SN20b,Sin21].In particular, it has been shown that the action of a DUC (resp.CDUC) channel can be parameterized by two matrices A, B ∈ M d with equal diagonals: where B = B − diag B and denotes the entrywise or Hadamard matrix product.
• resp.A 0, B = B † , and A plethora of different classes of quantum channels can be shown to belong to the families of DUC and CDUC channels; see [SN20a, Section 7].In particular, the previously considered classes of depolarizing-and MAD channels are CDUC, while the transpose-depolarizing channels are DUC.
In this section, we derive very general sufficient conditions on the matrices A, B ∈ M d which ensure that the associated DUC and CDUC channels have positive complementary quantum capacity.Theorems 4.1 and 4.6 would then pop out as special cases of these more general results.To this end, let us first see how the minimal output and environment dimensions of these channels can be obtained from the associated matrices A, B ∈ M d .Lemma 4.9.For a DUC channel Ψ A,B : Proof.Apply the channels Φ A,B and Ψ A,B on the identity matrix 1 d ∈ M d to conclude that their minimal output dimensions are both equal to the number of non-zero rows present in A, see Lemma 2.2.The expressions of the minimal environment dimensions can be derived by computing the relevant Choi ranks, see [SN20a, Theorem 6.4 and Corollary 4.2] and Lemma 2.2.
Equipped with the necessary background, we are now ready to characterize the class of (C)DUC channels Φ for which there exist a pure state |ψ ψ| such that rank Φ(|ψ ψ|) = min{d * out (Φ), d * env (Φ)}, so that Corollary 3.2 can be invoked to obtain positivity of the (complementary) capacities of these channels whenever d * out (Φ) = d * env (Φ).Recall the discussion following Remark 3.3, where the difficulty in obtaining such a characterization for arbitrary channels is clearly highlighted.More often than not, one can only hope to obtain some sufficient conditions on the channel under consideration (like the one presented in Theorem 4.4) to guarantee that the required pure state exists.This fact makes the following Proposition all the more important, since it contains a simple necessary and sufficient condition to ensure the existence of the desired pure state for arbitrary (C)DUC channels.Proof.In order to prove the proposition, it suffices to show that for a DUC or a CDUC channel Φ, if rank Φ(|e e|) < r, then for all |ψ ψ| ∈ S d , rank Φ(|ψ ψ|) < r as well.Moreover, since a positive semi-definite matrix X ∈ M d has rank X < r if and only if all its r × r principal submatrices are singular, we can choose to work with r × r principal submatrices corresponding to an arbitrary selection of r rows and columns indexed by some I ⊆ [d] with |I| = r.Hence, we only need to prove that (see Appendix C for the relevant notation): The following proof is split into two cases, and can be found in Appendix C.Then, Proof.The proof is a straighforward consequence of Proposition 4.10 and Corollary 3.2.4.5.Generalized dephasing or Hadamard channels.Generalized dephasing or Hadamard channels (also known as Schur multipliers) Φ B : M d → M d have the effect of diminishing the magnitude of the off-diagonal entries of the input states while perfectly preserving their diagonal elements.These channels are parameterized by a correlation matrix B ∈ M d (i.e., B ≥ 0 and all its diagonal entries are equal to one): Note that above denotes the Hadamard or entrywise matrix product.It is clear that a dephasing channel Φ B = Φ A,B is CDUC (see Eq. ( 32)) with A = 1 d .Complementary channels to the dephasing channels are known to be entanglement-breaking (and hence also antidegradable); so they have zero quantum capacity.This, in particular, implies that the dephasing channels themselves are all degradable and hence their quantum capacities admit a single-letter expression in terms of the channels' coherent information In the following theorem, we obtain a complete characterization of the set of zero capacity dephasing channels.
Theorem 4.12.Let Φ B : M d → M d be a dephasing channel associated with B ∈ M d .Then, It is well-known that the final three equivalences hold, see [SN20a, Example 7.3] for instance.Moreover, if Φ B is entanglement-breaking, then it clearly has zero capacity.Hence, to prove the theorem, it suffices to show that if In the above calculation, we have used the fact that V V † = d−1 l=0 |lφ l lφ l | acts as the orthogonal projection onto range V ⊆ C d ⊗ C d .As φ j |φ i = 0, the number of terms in the second sum above is at least two, so that V (Φ c B ) * (R c ψ )V † has rank at least two.Therefore, of unit rank.Theorem 3.8 then tells us that Q (1) (Φ B ) > 0, and the proof is complete.4.6.A family of channels with equal output and environment dimensions.Consider a unitary operator V : C 4 → C 2 ⊗ C 2 , and a pure state |ψ ∈ C 4 such that V |ψ = |0 ⊗ |1 , where {|0 , |1 } denotes the computational basis of C 2 .Then, for the quantum channel Φ : M 4 → M 2 and its complement Φ c : M 4 → M 2 defined as follows, the orthogonal projections onto range Φ(|ψ ψ|) and range Φ c (|ψ ψ|) are given by R ψ = |0 0| and R c ψ = |1 1| , respectively.It is clear that the adjoints of Φ and Φ c act on these projectors as follows V. Now, by using the fact that V V † = 1 4 (since V is unitary), we obtain . By Theorem 3.8, we then conclude that such a channel Φ has Q (1) (Φ) > 0. The same argument applied to the complementary channel shows that Q (1) (Φ c ) > 0 as well.
By considering unitary mappings V : C d 2 → C d ⊗ C d and following the same steps as above, it is easy to infer that for any channel Φ : In particular, such a channel can neither be degradable nor anti-degradable.

Structure theorems for more capable quantum channels
Cubitt et al proved two structure theorems for degradable channels (see [CRS08, Theorem 3 and 4]), which we extend in this section to the strictly larger class of more capable quantum channels.Their proofs follow simply from our results in Section 3.
becomes a unitary map, which implies that Q (1) (Φ c ) > 0 and hence leads to a contradiction (see Section 4.6).Thus, d ≤ 3 and the proof is complete.

Conclusion and open problems
In this paper, we have employed some basic techniques from analytic perturbation theory of Hermitian matrices (Theorem A.1) to derive a powerful sufficient condition for a quantum channel (or its complement) to have positive quantum capacity (Theorem 3.1).An equivalent formulation of this condition equips us with a first-of-its-kind necessary condition for membership in the set of zero capacity quantum channels (Theorem 3.8).These are significant results because to date, no systematic procedure is known to check if a given quantum channel can be used to reliably transmit quantum information, and this is precisely because of our limited understanding of the set of zero capacity quantum channels.Notably, the main result in [Sid20] pops out as an immediate consequence of Theorem 3.1, which can be used to detect positive quantum capacities of channels (or their complements) with unequal output and environment dimensions for which there exists a pure input state which gets mapped to a maximal rank output state (Corollary 3.2).Obtaining a complete characterization of channels for which such a pure state exists is a rather formidable task.Nevertheless, we have derived simple inequalities between the input, output and environment dimensions of a given channel that suffice to ensure the existence of the sought-after pure state, irrespective of the specific structure of the channel (Corollaries 3.4 and 3.6).By exploiting our main results, we have shown that a variety of interesting examples of quantum channels have positive quantum capacity (Section 4).Moreover, our results lead to simplified proofs of certain existing structure theorems for the class of degradable quantum channels, and an extension of their applicability to the larger class of more capable quantum channels.
Listed below are some of the open problems that stem from our research.
• Show that PPT channels satisfy the necessary condition of Theorem 3.8 for membership in the set of zero capacity quantum channels.• Look for other quantum channels which satisfy the condition stated in Theorem 3.8.This could potentially lead to new examples of channels which are neither anti-degradable nor PPT, but still have zero quantum capacity.
• Obtain examples of channels which satisfy the condition stated in Theorem 3.8 for all positive integers n ≤ m for some fixed m ∈ N, but not for n > m.Such channels, if they exist, would have positive n-shot coherent information for all n > m but at the same time, would satisfy the necessary condition for having zero n-shot coherent information for all n ≤ m, and hence could potentially shed new light on the superadditivity of coherent information.• Investigate whether Theorem 3.1 can be used to detect positive quantum capacity of a (C)DUC channel when Theorem 4.11 is inapplicable.Further work on applying the techniques developed in this paper to random quantum channels and quantum channels acting on infinite dimensional Hilbert spaces is currently in progress.
Proof.This proof exploits some basic properties of real analytic functions, see [KP02, Chapter 1].Firstly, recall that since the Taylor series of f converges in the interval [0, R), the derived series f also converges in the same interval.Hence, for every r ∈ (0, R) we can uniformly bound both f and f on the domain [0, r].Moreover, since the zeros of analytic functions are isolated, it is easy to find an r ∈ (0, R) such that no zeros of f lie within the range (0, r], i.e. ∀ ∈ (0, r] : f ( ) > 0. Hence, there exist constants c 1 , c 2 > 0 such that ∀ ∈ (0, r) : c 1 ≤ f ( ) ≤ c 2 , except possibly when f 0 = f (0) = 0, in which case f is still strictly positive and bounded on (0, r) but f ( ) → 0 as → 0.
Now, since log is real analytic on (0, ∞) and f maps (0, r) within (0, ∞), it is clear that g is real analytic on (0, r).Then, g is also analytic on (0, r) and admits an expression of the form where we have used the symbol O( k ) as a placeholder for functions h for which there exists 0 < r 0 ≤ r and C > 0 such that ∀ ∈ [0, r 0 ) : |h( )| ≤ C k .It is then straightforward to infer from our previous discussion that the terms in the square brackets above are always bounded in the interval (0, r).Now, if either f 0 = 0 or f 1 = 0, the first term is also bounded.However, if f 0 = 0 and f 1 = 0, the first term splits up into f 1 log + f 1 log(f 1 + O( )), which blows up as → 0. Hence, we can write the derivative as follows where both K 1 and K 2 are bounded on (0, r).Observe that for our purposes, the spectrum of a d × d matrix is just the (unordered) sequence of all its eigenvalues counted with multiplicities.Now, once we turn on the perturbation and is sufficiently small (say 0 ≤ < R), we get the following analytic eigenvalue functions (see Theorem A.1): }, where i = 1, 2, . . ., n, j = 1, 2, . . ., d out − n and k = 1, 2, . . ., d env − n.Assume, without loss of generality, that all the above functions are not identically zero (otherwise, we can just restrict ourselves to those which are non-zero).We are interested in analyzing the coherent information of ρ( ) with respect to Φ, which has the following form (see Section 2.2) Observe that we can immediately apply Lemma A.2 to each term in the above sum, so that we obtain 0 < r < R such that I( ) is analytic on (0, r) and its derivative can be written as where K is bounded on (0, r), see Eq (36).Now, Theorem A.1 tells us that the term in the brackets above is nothing but , we can find a small enough 0 < δ < r such that the first term above dominates the other when < δ and consequently, ∀ ∈ (0, δ) : I ( ) > 0. A simple application of the mean value theorem then tells us that I( ) is strictly increasing on the interval [0, δ].Moreover, since I(0) = 0, we obtain ∀ ∈ (0, δ] : I( ) > 0 =⇒ Q (1) (Φ) > 0. The other case can be tackled similarly.
where S d denotes the permutation group of the set [d] and X π = i∈[d] X iπ i .
Finally, let us prove a lemma regarding an intriguing matrix determinant.
Lemma C.1.Let X ∈ M d (d ≥ 2) be such that it has exactly two non-zero entries in each row, i.e., ∀i ∈ [d], there exist unique i 1 , i 2 ∈ [d] such that X ij = 0 ⇐⇒ j ∈ {i 1 , i 2 }.For |ψ ∈ C d , define where n i denotes the number of non-zero entries in the i th column of X.
Proof.Let us begin by observing that for every π ∈ S d , we must have ψ(X) π = ( i∈[d] ψ )X π , where {k i (π)} i∈[d] is a set of positive integers.This is because the matrix ψ(X) has the same entries as X upto multiplication by the entries of |ψ .Hence, our task is to show that where n i denotes the number of non-zero entries in the i th column.Firstly, note that if π is such that X π = 0, then ψ(X) π = 0 as well, so there's nothing to prove here.Hence, let π ∈ S d be an arbitrary permutation with X π = 0. Now, suppose on the contrary that the desired claim does not hold and there exists j ∈ [d] such that k j (π) = n j − 1.Let us collect all the rows i ∈ [d] with X ij = 0 in the set R j , so that n j = |R j |.Then, for each row i ∈ R j , one of the two non-zero entries is present in the column i 1 = j.Since X π = 0, it must be the case that π i = i 2 for all i ∈ R j such that π i = j = i 1 , which implies that Since ψ j cannot arise from any other term in the product i∈[d] ψ(X) iπ i , we arrive at a contradiction.
where the vectors |λ i ∈ span{|i } ⊆ C r (for i ∈ I) and |λ k ij ∈ span{|i , |j } ⊆ C r (for i < j, i, j ∈ I, and k ∈ {1, 2}) form the rank one decompositions of the corresponding positive semidefinite matrices from the previous sum, and Λ ∈ M n×r stores these vectors in its rows.Notice that the conditions A 0, B = B † and A ij A ji ≥ |B ij | 2 ensure that all the matrices in the two sums above are indeed positive semi-definite.Similar decomposition can be obtained for an arbitrary pure input state |ψ ψ| ∈ S d : where the vectors |ψ(λ) i ∈ span{|i } ⊆ C r (for i ∈ I), |ψ(λ) k ij ∈ span{|i , |j } ⊆ C r (for i < j, i, j ∈ I, k ∈ {1, 2}) and the matrix ψ(Λ) ∈ M n×r are defined as before.Notice that for all i, j, k: If det Λ (r) = 0 because ∀π ∈ S r : Λ (r) π = 0, then since Λ (r) ij = 0 =⇒ ψ (r) (Λ) ij = 0 for all i, j, the desired claim is trivial to prove.Otherwise, there must exist at least two distinct permutations π 1 , π 2 ∈ S r such that Λ (r) π 1 and Λ (r) π 2 are non-zero.In this case, after suitable elementary row and column operations, Λ (r) and ψ (r) (Λ) can be brought, respectively, into the block diagonal forms X 0 0 Y and ψ(X) where X, ψ(X) ∈ M k are such that det X = X π = 0 and det ψ(X) = ψ(X) π for a unique π ∈ S k and Y, ψ(Y ) ∈ M r−k are of the form described in Lemma C.1, see Eq. (40).In the notation of Lemma C.1, the vector ψ ∈ C r−k which implements the transformation Y → ψ(Y ) can be obtained by choosing some r − k entries from ψ ∈ C d .Its exact form is irrelevant for our purposes.When written in the above form, it should be evident that det Λ (r) = 0 ⇐⇒ det Y = 0, in which case Lemma C.1 informs us that det ψ(Y ) = 0 =⇒ det ψ (r) (Λ) = 0, and the proof is complete.
(D p : M d → M d and D q : M d → M d , respectively) have positive complementary quantum capacities for all p > 0 and q < d d−1 ; see Theorem 4.1.Moreover, the Werner-Holevo channel Φ WH ≡ D d d−1 : M d → M d is shown to have positive complementary quantum capacity for all d ≥ 4; see Theorem 4.3.

4. 1 .
Depolarizing and transpose depolarizing channels.The depolarizing channels D p : M d → M d and the transpose depolarizing channels D q : M d → M d form one-parameter families of completely positive and trace-preserving linear maps within their respective parameter ranges

Theorem 4. 1 .
Let D p : M d → M d and D q : M d → M d be the qudit depolarizing and transposedepolarizing channels with p ∈ [0, d 2 d 2 −1 ] and q ∈ [ d d+1 , d d−1 ].Then, • if p > 0, any complementary channel D c p ∈ C Dp has positive quantum capacity.• if q < d d−1 , any complementary channel (D p ) c ∈ C D p has positive quantum capacity.

)
Theorem 4.3.Any complementary channel to the Werner-Holevo channel Φ WH : M d → M d has positive quantum capacity when d ≥ 4. Proof.Let us denote the Werner-Holevo channel simply by Φ : M d → M d .We already know that d(d − 1)/2 = d * env (Φ) > d * out (Φ) = d whenever d ≥ 4.However, Corollary 3.2 cannot be applied to this channel, since it maps every pure state to a state with rank = d − 1, see Eq. (21).Thus, our aim is to exploit Theorem 3.1 instead.Let us consider the pure state |0 0| ∈ S d , (recall that {|k } d−1 k=0 denotes the standard basis of C d ), so that the orthogonal projection onto ker Φ(|0 0|) becomes K 0 = |0 0|.Now, it was shown in [Hol07] that Φ admits a complementary channel Φ c : M d → M d ⊗ M d of the following form: the flip operator whose action on the canonical basis {|i ⊗ |j } d−1 i,j=0 of C d ⊗ C d is defined as F |i ⊗ |j = |j ⊗ |i .Let us now see how Φ c acts on |0 0|:

4. 4 .
(Conjugate) diagonal unitary covariant channels.Let us denote the group of diagonal unitary matrices in M d by DU d .A channel Φ : M d → M d is said to be diagonal unitary covariant (DUC) (resp.conjugate diagonal unitary covariant (CDUC)) if for all U ∈ DU d and X ∈ M d , Proposition 4.10.Let Φ : M d → M d be a DUC or a CDUC channel and |e = i |i ∈ C d .Then, max |ψ ψ|∈S d rank Φ(|ψ ψ|) = rank Φ(|e e|).
jj| has the same rank as B, so that according to Lemma 2.2, d * env (Φ B ) = rank B. Let us split the remaining proof into two cases.Case 1. rank B < d.Since d * env (Φ B ) < d * out (Φ B ), the result follows by noting that Φ B (|e e|) = B, see Theorem 4.11.Case 2. rank B = d.In this case, we have d * env (Φ B ) = d * out (Φ B ) = d, so Corollary 3.2 can no longer be applied.Let {|φ i } d−1 i=0 ⊆ C d be a set of d linearly independent unit vectors such that B = gram{|φ i } d−1 i=0 , i.e., ∀i, j : B ij = φ j |φ i .Then, we can define an isometry V : C d → C d ⊗ C d by setting V |i = |i ⊗ |φ i as its action on the standard basis of C d , so that ∀X ∈ M d : Φ B (X) = Tr 2 (V XV † ) and Φ c B (X) = Tr 1 (V XV † ), where Tr 1(2) denote the partial trace over the first (second) system and Φ c B ∈ C Φ B .Now, since Φ is not entanglement breaking and hence B = 1 d , there exist indices i = j such that B ij = 0. Choose |ψ = |i , so that in the notation of Theorem 3.8, R ψ = |i i| and R c ψ = |φ i φ i |.Let us now analyse the action of the adjoint maps Φ * B and (Φ c B ) * on the projectors R ψ and R c ψ

A
Proof of Proposition 4.10 Recall that in order to prove the proposition, it suffices to show that Φ(|e e|)[I] is singular =⇒ Φ(|ψ ψ|)[I] is singular for all |ψ ψ| ∈ S d , where I ⊆ [d] is an arbitrary index set of size |I| = r.Case 1. Φ = Φ A,B is a CDUC channel with A 0 and B ≥ 0. Let us express Φ A,B (|e e|)[I] = i∈I k =i A ik |i i| + B[I] as a sum of two r × r positive semidefinite matrices, so that Φ A,B (|e e|)[I] is singular if and only if the two matrices in the above sum have a non-trivial common kernel.Sinceker i∈I k =i A ik |i i| = span{|j } j∈I\J ⊆ C r with J = {j ∈ I : ∃k ∈ [d], k = j, A jk = 0 }, we obtain Φ A,B (|e e|)[I] is singular ⇐⇒ {|B[I] j } j∈I\J ⊆ C r is linearly dependent, (38)where |B[I] j := B[I] |j is the j th column of B[I].Now, for |ψ ∈ C d , we have Φ A,B (|ψ ψ|)[I] = i∈I k =i A ik |ψ k | 2 |i i| + (B |ψ ψ|)[I].(39) Note that span{|j } j∈I\J ⊆ ker i∈I k =i A ik |ψ k | 2 |i i| and equality holds whenever ψ i = 0 for all i ∈ [d].Clearly, since the j th column of (B |ψ ψ|)[I] is nothing but ψ j |B[I] j |ψ[I] , the column set {|(B |ψ ψ|)[I] j } j∈I\J ⊆ C r is linearly dependent as well, so that the two matrices in the sum in Eq. (39) again have a non-trivial common kernel.Hence, Φ A,B (|ψ ψ|)[I] is singular.Case 2. Φ = Ψ A,B is a DUC channel with A 0, B = B † and A ij A ji ≥ |B ij | 2 .In this case, since B is not positive semi-definite, the previous approach would not work.Even so, let us still try to express Ψ A,B (|e e|)[I] as a sum of positive semi-definite matrices: Ψ A,B (|e e|)[I] = ij |i i| + B ij |i j| + B ji |j i| + A ji |j j| = i∈I |λ i λ i | + i<j i,j∈I where denotes the entrywise vector product with respect to the standard basis in C r .Now, if n < r, we have rank Φ A,B (|e e|)[I] = rank Λ < r and rank Φ A,B (|ψ ψ|)[I] = rank ψ(Λ) < r, so that both Φ A,B (|e e|)[I] and Φ A,B (|ψ ψ|)[I] are always singular.Otherwise, if n ≥ r, we need to show that for r × r submatrices Λ[J|r] := Λ (r) and ψ(Λ)[J|r] := ψ (r) (Λ) constructed by arbitrarily choosing r rows according to a random index set J ⊆ [n] with |J| = r, det Λ (r) = 0 =⇒ det ψ (r) (Λ) = 0.
[DS05]nce for a given channel Φ, all complementary channels Φ c ∈ C Φ are isometrically related (in the sense of Remark 2.1), the stated capacity expressions do not depend on the choice of Φ c .More generally, the quantum capacities of any two isometrically related channels are identical.Hence, while computing quantum capacities, one can choose to work with channels Φ : M d → M dout and Φ c : M d → M denv that are minimally defined (i.e.d env = d * env (Φ) andd out = d * out (Φ)).Degradable, anti-degradable, more capable and PPT channels.Degradable channels are the quintessential examples of channels which have additive coherent information[DS05].A quantum channel Φ : M d → M dout is said to be degradable if there exists a channel N :