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Quantifying the noise of a quantum channel by noise addition

In this paper we introduce a way to quantify the noise level associated to a given quantum transformation. The key mechanism lying at the heart of the proposal is"noise addition": in other words we compute the amount of extra noise we need to add to the system, through convex combination with a reference noisy map or by reiterative applications of the original map, before the resulting transformation becomes entanglement-breaking. We also introduce the notion of entanglement-breaking channels of order n (i.e. maps which become entanglement-breaking after n iterations), and the associated notion of amendable channels (i.e. maps which can be prevented from becoming entanglement-breaking after iterations by interposing proper quantum transformations). Explicit examples are analyzed in the context of qubit and one-mode Guassian channels.


I. INTRODUCTION
In quantum information several entropic functionals (the so called quantum capacities) have been introduced that provide a sort of "inverse measures" of the noise level associated with a given process, see e.g. Refs. [1][2][3][4]. In this approach the evolution of the system of interest S is described as a linear, completely positive, trace preserving mapping (CPT), the quantum channel Φ, which associates a final state Φ(ρ) to each possible initial density matrix ρ of S. The quantum capacities of Φ have a clear operational meaning as they gauge the optimal communication transmission rates achievable when operating in parallel on multiple copies of S: consequently the noisier the channel is, the lower are its associated quantum capacities. Unfortunately however, even for small systems, these quantities are also extremely difficult to evaluate since require optimization over large coding spaces, e.g. see Ref. [5,6].
In this paper we introduce an alternative way to determine how disruptive a channel might be which, while still having a simple operational interpretation, it is easier to compute than the quantum capacities. The starting point of our analysis is to use Entanglement-Breaking (EB) channels [7,8] as the fundamental benchmarks for evaluating the noise level of a transformation. We remind that a map operating on a system S, is said to be entanglement-breaking if for all its extensions to an ancillary system A it annihilates the entanglement between the system and the ancilla [7,8]. From the point of view of quantum information, the action of these maps represents hence the most disruptive form of noise a quantum mechanical system can undergo. A reasonable way to quantify the noise level of a generic map Φ can then be introduced by computing how much extra noise we need to "add" to it before the resulting transformation becomes entanglement-breaking. The intuitive idea behind this approach is that channels which are less disruptive on S should require larger amount of extra noise to behave like an entanglement-breaking map.
In the following we analyze two different mechanisms of noise addition. The first one assumes to form convex combinations (1 − µ)Φ + µΦ ρ0 DEP of the input channel Φ with generalized completely depolarizing channels Φ ρ0 DEP (these are the most drastic examples of entanglementbreaking channels which bring every state of S into a unique output configuration ρ 0 -the fixed point of the map). In this approach the level of noise associated with the original map Φ is gauged by the minimum value µ c of the mixing parameter µ which transforms (1 − µ)Φ + µΦ ρ0 DEP into an entanglement-breaking map (as we will see a proper characterization of this measure requires an optimization upon ρ 0 ).
The second mechanisms of noise addition we consider assumes instead the reiterative application of Φ on S. In this case the noise level is determined by the minimum value n c of iterations needed to transform Φ in an entanglement-breaking map (if such minimum exists). As we shall see, due to the lack of monotonicity under concatenation with other maps, this second functional cannot be considered a proper measure of the noise level introduced by Φ (regularized version of n c do however retain this property). Nonetheless n c captures some important aspects of the dynamics associated with Φ: namely it counts the number of discrete time evolutions induced by the map that a system can sustain before its entanglement with an external ancilla is completely destroyed. The definition of n c gives us also the opportunity of introducing the set EB n of the entanglement-breaking channels of order n, and the notion of amendable channels. The former is composed by all CPT maps Φ which, when applied n times, are entanglement-breaking. Vice-versa a channel Φ is amendable if it can be prevented from becoming entanglement-breaking after n c iterations via a proper application of intermediate quantum channels -a similar problem was also discussed in Ref. [9].
In the following we will discuss some general features of the functional µ c and n c computing their exact values for some special class of channels. Specifically in Sec. II we introduce the functionals and characterize some general properties. In Sec. III we will restrict our attention to qubit channels computing the value of µ c and n c for the arXiv:1204.5589v3 [quant-ph] 18 Jul 2013 set of unital maps and for the generalized amplitudedamping channels. Also using examples from these sets we will show that the set EB n is not convex for n ≥ 2 and provide evidence of the existence of amendable maps. In Sec. IV we will consider the analogous of qubit maps for continuous variable systems, that is one-mode gaussian channels, and in particular we will evaluate the functional n c for attenuation and amplification channels. Conclusions and final remarks are given in Sec. V.

II. DEFINITIONS AND BASIC PROPERTIES
It is a well known [7,8] fact that any entanglementbreaking channel Φ EB can be described as measure and re-prepare scheme according to which the system S is initialized in some state ρ depending on the outcome of a measurement performed on the input state ρ, i.e.
(here {F } are the elements of the positive operator valued measure (POVM) which defines the measuring process). These maps cannot be used to convey quantum information even in the presence of two-way classical side channel (of course they might allow to achieve this goal when the communicating parties are given some prior shared entanglement via teleportation). In this respect they can thus be considered as "classical" communication lines -notice however that they might be used to create quantum discord [10]. One can easily verify that the set EB formed by the entanglement-breaking channels is closed under convex combinations and under sequential concatenation, i.e. given p ∈ [0, 1] and Φ EB , Φ EB ∈ EB we have pΦ EB + (1 − p)Φ EB ∈ EB and Φ EB • Φ EB ∈ EB [here "•" stands for the composition of super-operators defined in Sec. II A 1]. More generally, it also true that the set EB is stable under concatenation with other (non necessarily entanglement-breaking) maps, i.e.
A. Noise addition via convex convolution A proper subset of the EB channels is provided by the completely depolarizing maps, which transform any input state ρ of S into an assigned fixed point ρ 0 ∈ S(H S ), i.e.
(we indicate with S(H S ) the set of density matrices of S).
In a sense, these maps form the hard core of the EB set which prevents any sort of communication (not even classical). We can thus use them as a fundamental "unity" of added noise. Let then Φ be a generic map acting on S. For each completely depolarizing channel Φ ρ0 DEP we define µ(Φ; ρ 0 ) to be the minimum value of the mixing probability parameter µ ∈ [0, 1] that transforms the convex convolution (1 − µ)Φ + µΦ ρ0 DEP into an element of EB, i.e. µ(Φ; ρ 0 ) := min Thanks to the Choi-Jamiolkowski isomorphism [11], computing µ(Φ; ρ 0 ) corresponds to determining the minimum µ for which the state is separable. Here we used the symbol ψ + to represent the density matrix |ψ + ψ + | associated with the maximally entangled state |ψ + = 1 √ d d j=1 |j ⊗ |j ∈ H ⊗2 S , {|j } being an orthonormal basis of the Hilbert space of the system S, and d = dim H S being its dimension. We remind that the specific choice of such basis is completely irrelevant when constructing the Choi-Jamiolkowski state.
The quantity (4) measures the amount of Φ ρ0 DEP we need to mix to Φ via a classical stochastic process in order to make the resulting map entanglement-breaking. Clearly µ(Φ; ρ 0 ) nullifies if Φ is already an element of EB. Vice-versa for non entanglement-breaking channels Φ the value of µ(Φ; ρ 0 ) is always non null and in general might depend on the selected Φ ρ0 DEP (some of the properties of µ(Φ; ρ 0 ) are discussed in Appendix A). To get a functional of Φ alone we need hence to optimize with respect to the possible choices of the fix point ρ 0 . This brings us to define the function which constitutes our first (inverse) measure of the noise level associated with Φ. Solving the minimization in Eq. (6) is not simple in general: still in the next section we shall see that for some classes of channels which possess special symmetries this is feasible. Here we present some basic properties of the functional µ c (Φ).
An almost trivial consequence of the above inequalities is the fact that µ c is constant under unitary transformations, i.e.
where U and V stands for unitary channels, i.e. U(ρ) := U ρU † with U being unitary.

Extremal values
By definition µ c (Φ) is always positive (it nullifies if and only if Φ ∈ EB) and smaller than 1. As a matter of fact, it is possible to show that the upper bound being achieved for noiseless (unitary) channels. To see this we first use the fact that the generalized Werner states are separable iff µ ≥ d/(1+d) [12]. Therefore since Γ I 1 1/d,µ is the Choi-Jamiolkowski state of the identity channel Φ = I this implies, Now the bound (11) can be easily established by noticing that where the first inequality is just a consequence of the definition (6) while the second one follows from the fact that the Choi-Jamiolkowski state Γ Thanks to the property (10) we can just focus on the case U = I for which we have already established the condition (13). To prove the relation (15) we need hence only to verify that µ(I; 1 1/d) ≤ µ(I; ρ 0 ) for all ρ 0 . This can be proven by observing that Γ I 1 1/d,µ is obtained from Γ I ρ0,µ via the application of the twirling isotropic map [13,14] (16) where the integral is evaluated over the elements V of the unitary group on H S , dm(V ) is the Haar measure, while V * represents the unitary transformation obtained from V by complex conjugation with respect to the canonical basis {|j }. Indeed, on one hand |ψ + is invariant under the application of V ⊗ V * (i.e. V ⊗ V * |ψ + = |ψ + ) while on the other hand dm(V )V ρ 0 V † = 1 1/d, so that Take then µ = µ(I; ρ 0 ). With this choice the state Γ I ρ0,µ is separable. Since P is LOCC [13], also Γ I 1 1/d,µ is separable. Therefore we must have µ(I; 1 1/d) ≤ µ concluding the derivation.

Convexity rules
Consider {p j ; Φ j } a statistical ensemble of CPT channels Φ j distributed according to the probabilities p j . Then given µ j := µ c (Φ j ) the value assumed by the functional µ c on the j-th element of the ensemble, it results that The proof of the first inequality trivially follows from the fact that convex combinations of separable states are still separable. To verify the second one instead it is convenient to introduce the probabilities q j := ( 1−µ k ) and the states ρ j associated to the optimal depolarizing channel which saturates the minimization (6) for the map Φ j . Then the result simply follows by noticing that the density matrix is separable (here ρ 0 is the state j qj µj µ ρ j ). Equations (18) and (19) state that the value of µ c associated with the average map j p j Φ j of {p j ; Φ j } always lays between the extremal values it takes on the elements of the ensemble.

B. Noise addition via concatenation
As evident from Sec. II A 1 a simple method for increasing the noise level of a given map Φ is via concatenation. In particular Eq. (7) implies that where for each n integer we have set This allows us to introduce a new criterion for classifying noisy quantum channels. For this purpose, given n integer we define EB n to be the set formed by the CPT maps Φ acting on S such that Φ n is entanglement-breaking, i.e. explicitly [Note: The set EB 2 was introduced in [15] in connection to the set of 2-local entanglement-annihilating channels]. Form Eq. (21) it follows that for all m ≥ n we have the inclusion being strict (explicit examples of maps belonging to EB m but not to EB n will be provided in the next sections). Clearly a channel Φ which is an element of EB n+1 but which does not belong to EB n can be considered "less noisy" than the elements of the latter set as they break the entanglement shared between the system and an arbitrary ancilla just after n reiterations. This motivates the introduction of a new functional n c (Φ) to "measure" the noise level of Φ. In particular if Φ ∈ EB we set n c (Φ) = 1. Vice-versa, if for an integer number n ≥ 1 we have Φ ∈ EB n /EB n−1 , we say that the map Φ is an entanglement-breaking channel of order n, and we set n c (Φ) = n. Finally if for all n integers, Φ does not belong to EB n /EB n−1 , then we set n c (Φ) = ∞. Trivial examples of channels with n c (Φ) = ∞ are the unitary transformations. Other maps which do not become entanglement-breaking after any number of iterations are the amplitude-damping channels -see Sec. III B 2 for details.

Amendable channels
As anticipated in Sec. I, differently from µ c of Sec. II A, the functional n c is not monotonic under concatenation, i.e. it is not true that for an arbitrary CPT channels Ψ one has n c (Ψ • Φ) ≤ n c (Φ). Notice however that from Eq. (24) it trivially follows that for each Φ ∈ CPT and n integer we still have Furthermore exploiting the fact that the entanglementbreaking property of a map is invariant under unitary redefinition of the input and output spaces, and the fact that U † being the inverse of U.
The absence of monotonicity under concatenation implies that n c will be typically not invariant under unitary equivalence built upon uncorrelated unitary redefinitions of the input and output subspaces, i.e. exists Φ and U, V unitaries such that (compare this with the identity (26)). These facts are strictly related with the notion of amendable channels.
Specifically we say that a map Φ is amendable if there exists a filtering CPT mapΦ such that [16]. In particular we will focus on unitary filtersΦ. Indeed explicit examples can be found (see next section) of transformations Φ ∈ EB 2 /EB for which there exists V unitary such that Φ • V • Φ (and therefore its unitarily conjugate V • Φ • V • Φ) is not entanglementbreaking. Accordingly, the following inequality holds, which explicitly disproves the monotonicity of n c under concatenation.
The notion of amendable channels will be clarified in Sec. III where, focusing on unital and generalized amplitude-damping channels for qubits, we will provide explicit examples of the inequality (28). In particular in Sec. III B 2 we will show that, given an arbitrary integer m ≥ 2, it is possible to find Φ and V such that (28) holds with n c (V • Φ) = m (maps Φ for which this is possible will be called amendable channel of order m). Such examples show that there are cases where even though two successive uses of the same channel Φ are sufficient to destroy completely the entanglement present in the system, one can delay such detrimental effect by m − 2 steps by simply acting with the same intermediate filtering rotation V after each channel application. In other words, for these special maps it is possible to make the sequences not entanglement-breaking for all m < m.

Regularization of nc
Property (27) implies that the functional n c cannot be considered a proper measure of noise for quantum maps but rather a criterion for classifying them. More stable versions of n c can be obtained by optimizing with respect to all filtering transformations. For instance a possibility is to consider the regularization where the maximization is performed over all CPT maps Ψ. By construction (30) is explicitly monotonic under concatenation, i.e.
for all Φ 1 , Φ 2 CPT. Notice however that it is still possible that not uniform filtering CPT transformations not EB for m ≥n c (Φ). Removing this last instability is possible (e.g. adopting the quantum comb formalism of Ref. [17]) but clearly the resulting expression becomes extremely complex. Furthermore any optimization with respect to the filtering transformations tends to hide some of the structural properties of Φ. For this reason, in what follows we will restrict our attention to compute the non regularized version of n c introduced in Sec. II B.

III. QUBIT CHANNELS
In what follows we analyze the behavior of the noisequantifiers introduced in the previous sections by focusing on two important classes of qubit channels which, due to their special symmetries, allow us to provide closed expressions for both µ c and n c . We start in Sec. III A by studying the set of qubit unital maps [18]. Here we will construct explicit examples of amendable channels of order 2 and prove that the sets EB n are not convex for n ≥ 2. In Sec. III B instead we will focus on generalized amplitude-damping channels which will allow us to show that exist examples of amendable channels of arbitrary order.

A. Unital channels
We recall that unital CPT channels transform the identity operator into itself, i.e. Φ[1 1] = 1 1. This set includes unitary transformations as a proper subset, and it is closed under closed convex combination and channel multiplication [1]. When operating on a single qubit, unital channels admit a simple one-to-one parameterization in terms of the 3 × 3 real matrices T which fulfills the necessary and sufficient constraint [18] (the inequality being saturated by the T which correspond to unitary transformations -in other words unitary qubit channels can be uniquely associated with 3 × 3 real matrices O which are orthogonal, i.e. O † O = 1 1).
The correspondence (33) can be easily established by expanding the system density operators in the canonical operator basis {1 1, σ 1 , σ 2 , σ 3 } given by the unit 1 1 and the Pauli matrices σ i , 1 ≤ i ≤ 3. Accordingly, given a generic qubit density matrix ρ = (1 1 with T being 3 × 3 real matrix fulfilling (34). It also follows that given two unital maps Φ 1 and Φ 2 , characterized by the 3 × 3 real matrices T 1 and T 2 respectively, Within the correspondence (33) it was shown [18] that the entanglement-breaking property of a unital qubit channel Φ only depends on the singular values of T . More precisely, a unital map is EB if and only if where ||·|| 1 is the trace norm and corresponds to the sum of the singular values of the matrix it is applied to, while the real non-negative matrix Λ is related to T via polar decomposition, with O being a 3 × 3 orthogonal matrix [19]. Observe that Λ satisfies the condition (34) and can be hence associated with a unital qubit CPT channel via the correspondence (33). As a matter of fact this new map Φ p is the polar form of the original channel Φ which is related to the latter through a unitary transformation according to the identity which mimics (38) at the super-operator level via the identity (36) (here U is the unitary channel which is associated to the orthogonal matrix O of Eq. (38) through the correspondence (33)). The above construction in particular shows also that, without loss of generality, when characterizing the entanglement-breaking property of a map Φ we can just focus on its polar form Φ p . At the level of super-operators this follows simply from (39) and by the fact that channels which differ by unitary transformations share the same entanglement-breacking property (alternatively this can also verified through Eq. (37) and noticing that the singular values of Λ coincides with those of T so that Λ 1 = T 1 ). The decomposition (38) can be further simplified by expressing T as where O 1,2 are orthogonal matrices, while D = diag{λ 1 , λ 2 , λ 3 } is a diagonal (not necessarily positive) matrix with elements λ j , which in modulus correspond to the singular values of T (i.e. to the eigenvalues of Λ) so that In terms of the equivalence (33), the matrix D defines a unital channel Φ λ which gives the canonical representation of the map Φ [18], and which is related to the latter through the identity

Noise addition via concatenation and amendable unital channels
For what concerns the mechanism of noise addition via concatenation, we start noting that differently from the previous section, a unital qubit channel Φ and its polar Φ p or canonical Φ λ forms might not share the same properties: indeed, as anticipated in Sec. II B 1, the functional n c is not invariant under unitary equivalence (an explicit counterexample will be provided in the following). Still it turns out that Φ p and Φ λ have same the value of n c [21] and that a sufficient (but not necessary) condition for a generic unital channel Φ to be EB n is Φ p ∈ EB n , implying To prove this relation we first notice that from Eqs. (36) and (37) it follows that Φ n is entanglement-breaking if and only if where T is the matrix associated to Φ via the correspondence (33). Hence, reminding Eq. (25) we can write where the minimization is performed over all n integers (notice that for entanglement-breaking channels this yields correctly n c (Φ) = 1). Equation (52) can then be derived by noticing that the matrices T and Λ which represent Φ and Φ p , respectively, are related as in (38) and by using the Hölder inequality [22]. Indeed where · n is the operator n-norm, and where we used the fact that O is orthogonal and Λ non-negative. From this we can hence conclude that if n c (Φ p ) = n (that is ||Λ n || 1 ≤ 1), then Φ = U •Φ p ∈ EB n too (||(OΛ) n || 1 ≤ 1), concluding the proof. On the contrary, for n ≥ 2 there exist Φ ∈ EB n such that their polar form Φ p is not in EB n [23]. As an example consider for instance the case in which Φ p is characterized by the matrix which satisfies the condition (34) and for which one has that imply n c (Φ p ) = 3 and hence Φ p ∈ EB 3 /EB 2 ⊂ EB 3 . Now construct Φ as the unital map associated with the matrix T defined as in Eq. (38) where O is the orthogonal matrix that is With this choice we have which gives n c (Φ) = 2 and thus Φ ∈ EB 2 /EB ⊂ EB 2 . The same example proves the existence of unital qubit maps that are amendable. To see this explicitly take V of Eq. (28) as the inverse U † of the unitary U of Eq. (39) and notice that while the map Φ introduced above is an element of EB 2 (i.e. Φ 2 ∈ EB), the channel Φ • U † • Φ = U • Φ 2 p is not entanglement-breaking (it is indeed unitarily equivalent to Φ 2 p which is not in EB according to Eq. (57)): this implies that, by interposing a unitary transformation, we can prevent two consecutive applications of Φ of becoming entanglement-breaking. For the sake of completeness is worth mentioning that this fact is in contradiction with one of the claims of Ref. [9] where it was stated (without presenting a formal proof) that unital qubit channels are no rectifiable (and hence no amendable). At the origin of this incompatibility between such claim and our findings, we believe is the fact that the Authors of [9] assumed the possibility of always putting the channel in the canonical representation (42), without considering the fact that the properties of noise addition via concatenation are not invariant under unitary equivalence.

Non convexity of the sets EB n
The set of EB channels is known to be convex. The same property however does not apply for the set EB n with n ≥ 2. In what follows we will show this fact by presenting counter-examples taken from the unital qubit maps for n = 2, 3 (even though we did not check for larger n we believe that the similar counter-examples can be found also for those cases). Specifically we will prove that even though convex combinations of untial channels are still unital, the unital qubit channels which are in EB 2 and in EB 3 are not closed under convex convolution.
a. Non convexity of EB 2 : Via the correspondence (33) the fact that convex combinations of qubit unital channels in EB 2 are not necessarily elements of the same set, can be traced back to the fact that the set formed by operators T satisfying the condition is not convex. To show this thesis it is sufficient to provide a counter-example. For instance take T as in (59) and its transpose T . Since both these matrices satisfy the condition (34) they properly define two unital qubit channels Φ and Ψ via the correspondence (33). Furthermore since these maps belong to EB 2 . Take then the channelΦ = (Φ + Ψ)/2: it is clearly unital and it is described by a matrix (33) formed by the convex convolution Computing its T 2 1 we notice however that it does not fulfill (62), indeed T 2 implying thatΦ is not in EB 2 , completing hence the proof. b. Non convexity of EB 3 : A counter-example can be obtained by taking the unital channels Φ and Ψ, and Φ = (Φ + Ψ)/2 associated with matrices T , T , and T = (T + T )/2 where now however T is expressed as OΛ with O is agin as in Eq. (58) but with Λ being With this choice in fact we have T 3 1 = (T ) 3 1 0.991 < 1 (hence Φ and Ψ are in EB 3 ) but T 3 1 1.03 > 1 which instead implies thatΦ / ∈ EB 3 .
The example (63) seems to contradict one of the claims of [15] which states that the set of unital qubit EB 2 is closed under convex convolution. As for the incompatibility with [9] underlined at the end of Sec. III A 2, this originates from the fact that Ref. [15] always assumes the canonical structure (42) when analyzing the untial qubit channels. Indeed, even though, in general the set of unital qubit channels which are EB n is not convex, it is true that the subset formed by qubit unital maps of EB n which are in polar (or canonical) forms is convex. For n = 2 this was explicitly proved in [15]. Here we generalize this fact to arbitrary n. Let then Φ p1 and Φ p2 be unital qubit maps in polar form which are EB n , that is ||Λ 1 n || 1 ≤ 1 and ||Λ 2 n || 1 ≤ 1 (Λ 1 and Λ 2 being the non-negative matrices that define the two maps within the correspondence (33)). According to the Hölder inequality [22] we have that given an arbitrary product Λ i1 Λ i2 . . . Λ in with {i α } 1≤α≤n = 1, 2, one has Thus, exploiting the sub-additivity of the trace norm, we get that for arbitrary probabilities q 1 , q 2 we have which proves that the mixed channel q 1 Φ p1 + q 2 Φ p2 is an element of EB n . The convexity of the set of unital qubit maps in canonical form can be proved in an similar way.
Similarly from Eqs. (37) and (41) we have that the set of unital entanglement-breaking maps EB for qubits in canonical form is restricted to the octahedron of vertices (±1, 0, 0) and permutations, which correspond to the intersections between the tetrahedron and its inversion through the origin [18]. The generalization to the case EB n is straightforward. Indeed from Eqs. (40) and (53) it immediately follows that the canonical forms Φ λ which are EB n are identified by the intersection between the tetrahedron and the region (for instance, the set EB 2 corresponds to the intersection between the tetrahedron and the unit ball in the origin, see also Fig. 2 of Ref. [15]). For sake of simplicity, in Fig. 1 we have reduced the tridimensional (λ 1 , λ 2 , λ 3 )'s space to the space (λ 1 , λ 2 ) by fixing λ 3 = 1/2, and studied the domain associated to the sets EB n , n ≥ 1. The shaded region (orange in online version), refers to the untial maps -i.e. it is a section of the tetrahedron (68). Notice that while the EB square is contained in the CPT set, this in not the case for higher order entanglementbreaking maps. The corners of the CPT rectangle, λ 1 = ±λ 3 with λ 2 = ±1 and λ 2 = ±λ 3 with λ 1 = ±1, refer to unital maps which can never belong to EB n for finite λ 3 , as the EB n condition reduces to 2|λ 3 | n ≤ 0, n ≥ 1. Finally, observe that EB n ⊆ EB n+1 , n ≥ 1 (see Eq. (24)).

B. Generalized amplitude-damping channels
A generalized qubit amplitude-damping channel Ψ p,γ describes the physical process through which the system approaches equilibrium with its environment by spontaneous emission at finite temperature [4]. This is done by modifying the statistics of the populations associated to the ground state |0 and the excited state |1 which, in the following we identify with the eigenvectors of the Pauli matrix σ z . We can represent them as  (42), identify the entanglementbreaking qubit maps of order n (here we set λ3 = 1/2). The shaded region (orange in the online version) refers to the tetrahedron (68) which identifies all CPT maps Φ λ . The domain associated to the EB set is the square at the center of the figure and corresponds to the section of the octahedron [18]. It is the only region (69) which is completely contained in the tetrahedron. The circle of radius √ 3/2 identifies instead the condition for Φ λ to be EB 2 .
where E i are the Kraus operators [24] which, in the canonical basis {|0 , |1 }, are expressed by the matrices with 0 ≤ p ≤ 1 and 0 ≤ γ ≤ 1. For these channels we will provide an explicit expression for the functional µ c (Ψ p,γ ) and compare it with µ c (Ψ 2 p,γ ) verifying Eq. (21). We will also compute explicitly the value of n c (Ψ p,γ ) and construct examples of amendable channels of arbitrary order. Before entering in the details of these calculations however we observe that from Eq. (71) it follows where S 1 is the Pauli rotation (48). This implies that we can restrict the analysis to the case γ ∈ [0, 1/2] since the following identities hold (the first deriving from Eq. (10) while the second from (26)).

Noise addition via convex convolution
By definition, in order to determine µ c we have to perform two minimizations: the first leads to µ(Ψ p,γ ; ρ 0 ), and the second consists of a minimization over all possible choices of ρ 0 (see Eqs. (4) and (6)). Regarding the second step it is useful to observe that for the maps (70) we can focus on ρ 0 lying on the z axis, i.e. on states with the following Bloch form This can be proved following the same line of reasoning that led us to the identy (46) via Eq. (47). Indeed we notice that the channel Ψ p,γ commutes with all the unitary rotations U z = exp[iθσ z ] around the z axis, i.e.
being U z [ρ] := U z ρU † z . We can thus conclude that if there is a state ρ 0 = (1 1 + v · σ)/2 with arbitrary v ∈ R 3 such that Γ Ψp,γ ρ0,µ is separable for some µ, then also the state dm(U z )(U z ⊗ U z )[Γ Ψp,γ ρ0,µ ] obtained by mixing over all the local rotations U z ⊗ U z will be separable (dm(U z ) being the measure over the set of unitary rotations U z ). The latter however is given by which coincides with Γ Ψp,γ ρ z 0 ,µ with ρ z 0 being the density matrix of the form (75) defined by Thus if Γ Ψp,γ ρ0,µ ∈ SEP, then Γ Ψp,γ ρ z 0 ,µ ∈ SEP which implies µ(Φ; ρ z 0 ) ≤ µ(Φ; ρ 0 ) as anticipated. To determine µ(Ψ p,γ ; ρ z 0 ), we recall that for two qubits the state Γ Ψp,γ ρ z 0 ,µ is separable iff, when partially transposed, it shows a positive determinant [25]. Exploiting this we can then write (see also [15] for n = 1, 2). Starting from this observation we can now construct examples of amendable channels, by looking for maps of the form with U being a unitary, such that Ψ U p,γ is entanglementbreaking of oder 2 with n c (Ψ U p,γ ) = 2, while Ψ p,γ is not. Accordingly, some simple algebra allows us to write which provides an instance of Eq. (28) with the filtering transformation V being now U † (indeed for such examples we will have Ψ U p,γ • U † • Ψ U p,γ / ∈ EB proving that we have amended the action of the mapping (Ψ U p,γ ) 2 ). The region of the parameter space {p, γ} for which the inequality (91) holds for some U is shown in gray (red in online version) in Fig. 3. A numerical investigation shows that it is given by the union of two areas. The first one contains all points for which (28) holds with U being the Pauli rotation (48) S 1 : this region can be determined analytically and it is delimited from below by the condition  88) and (89)). Notice that for all γ's we have pn(γ) = pn(1 − γ) in agreement with Eq. (74). Furthermore, amplitude-damping channels, corresponding to γ = 0, 1 do not become EB after any number of iterations. In the inset figure, is shown the dependence of µc(Ψp,γ) and µc(Ψ 2 p,γ ) on p, for γ = 1/3. This functional vanishes as soon as the map becomes entanglement-breaking. As expected, for all p's we find µc(Ψ 2 p,γ ) ≤ µc(Ψp,γ) (see Eq. (21)).
only numerically and it is delimited from below by the dotted line of the figure).
Notice that for any given m integer the shaded area of Fig. 3 has a not-null overlap with the region (88) associated with channels Φ p,γ having n c (Φ p,γ ) = m. In Fig. 4 we finally compare the values of µ c (Ψ 2 p,γ ) with that of µ c (Ψ p,γ • U • Ψ p,γ ) as functions of p having fixed U = S 1 and γ = 1/10 (part (a)), and U = R 2 (π/2) • R 1 (π/2) and γ = 2/5 (part (b)). In both cases the amended channels Ψ p,γ present an higher value of the functional µ c .

IV. GAUSSIAN CHANNELS
In what follows we will show that behaviors similar to those seen in the previous section for finite dimensional systems, still hold also for continuous variables. In particular we will discuss how n c can be used to classify the set of Gaussian maps [26] in a way which mimics what obtained for qubit channels.
Gaussian maps Φ can be conveniently described in terms of the action of their duals Φ * on the Weyl operator: where l ∈ R 2f , and K and β are 2f × 2f matrices such that ∆ being the canonical symplectic form on R 2f . The above inequality is the necessary and sufficient condition which garantees the complete positivity of Φ. A Gaussian channel is therefore characterized by the triplet (K, l, β). The composition of two Gaussian maps, Φ = Φ 2 • Φ 1 , described by (K 1 , l 1 , β 1 ) and (K 2 , l 2 , β 2 ) respectively, is still a Gaussian map whose parameters are given by Finally, a Gaussian map Φ is entanglement-breaking [27] if and only if For the case of continuous variables systems, one-mode Gaussian mappings (f = 1) can be seen as the analogous of the qubit channels for finite dimensional systems. In this section we will quantify the noise level of two representative examples of this class of maps, i. e. the attenuation and the amplification channels. By exploiting the conservation of the Gaussian character of these transformations under concatenation, and the entanglementbreaking conditions (96)-(97), in what follows we will focus on the classification of attenuation and amplification channels according to n c , postposing the analysis of the addition via mixing to future works.

Attenuation channels
One-mode attenuation channels Φ At are Gaussian mappings such that: where 1 1 = 1 0 0 1 , k ∈ (0, 1) and N 0 ≥ 0, the latter condition deriving from the CPT requirement (94) on Φ At . From Eq. (96) it follows that Φ At ∈ EB and n c (Φ At ) = 1 iff In order to generalize the condition above to n c (Φ At ) = n with n ≥ 2, let us start by studying the composite map Φ At 2 := Φ At • Φ At . It is given by Notice that if we define we get which plays the same role of β for Φ At . It immediately follows that Φ At 2 ∈ EB if and only if (see Eq. (99)), that is n c (Φ At ) = 2 for We can now generalize these results to the case n c (Φ At ) = n ≥ 2. From Eq. (95) we get where k n := k n ≤ 1 and (N 0 ) n := N 0 It follows that Φ At n ∈ EB iff (N 0 ) n ≥ k 2 n , and n c (Φ At ) = n if and only if k 2n

Amplification channels
One-mode amplification maps are very similar to the attenuation channels, where now k ≥ 1 and with k n and (N 0 ) n given by Eq. (106). It follows that the composite map Φ Am n is entaglement-breaking iff (N 0 ) n ≥ 1, and n c (Φ Am ) = n ≥ 2 for ( Analogously to what done in Fig. 2 for the case of generalized amplitude-damping channels, in Fig. 5 we propose a scan of the parameter space {k, N 0 } for attenuation and amplification maps according the n c criterion by plotting the boundaries of the regions such that n c (Φ At ) = n (dashed red lines), and n c (Φ Am ) = n (solid blue lines) for n = 1, . . . , 6. Notice that, the region such that Φ At/Am ∈ EB n grows with n, as it is limited from below by the curves

V. CONCLUSIONS
In our analysis we have introduced two new functionals (µ c and n c ) that can be used to quantify the noise level of a map. These have been characterized in terms of general properties and have been explicitly evaluated for some class of channels. Along the way we have also introduced the notion of amendable channels showing how there exist maps which can be prevented from becoming entanglement-breaking after subsequent applications by interposing some extra transformations. For the sake of simplicity in our study we have only considered the case in which the same unitary transformation is interposed between successive channel uses. This however is not the only possibility and it is an interesting question of quantum control to determine what are the optimal operations one has to perform in order to guarantee that after certain number of reiterations entanglement will not be destroyed in the system.
We finally notice that when introducing the functionals n c and µ c we started from the identification of the entanglement-breaking channels as a benchmark set to evaluate the noise level of other quantum transformations (the choice being operationally motivated by the extreme deteriorating effects that EB maps have on the system). Again this however is not the only possibility. For instance another reasonable choice is to replace the entanglement-breaking channels with the set formed by the Positive Partial Transpose (or binding) maps [28] producing new noise measures µ (P P T ) c and n (P P T ) c for Φ. We remind that a channel is said to be PPT if when extended on an ancilla A the only entanglement one can find in the output of S + A is non-distillable [13]. Similarly to the case of EB maps, also the PPT set is stable under convex convolution and iteration: therefore µ (P P T ) c and n (P P T ) c will again assume the minimum allowed values on the benchmark set guarantying that they are well defined quantities for all Φ. Now, since for qubits the entanglement is always distillable [29], when S has dimension 2 we have µ