One-mode Bosonic Gaussian channels: a full weak-degradability classification

A complete degradability analysis of one-mode Gaussian Bosonic channels is presented. We show that apart from the class of channels which are unitarily equivalent to the channels with additive classical noise, these maps can be characterized in terms of weak- and/or anti-degradability. Furthermore a new set of channels which have null quantum capacity is identified. This is done by exploiting the composition rules of one-mode Gaussian maps and the fact that anti-degradable channels can not be used to transfer quantum information.

Within the context of quantum information theory [1] Bosonic Gaussian channels [2,3,4] play a fundamental role. They include all the physical transformations which preserve "Gaussian character" of the transmitted signals and can be seen are the quantum counterpart of the Gaussian channels in the classical information theory [5]. Bosonic Gaussian channels describe most of the noise sources which are routinely encountered in optics, including those responsible for the attenuation and/or the amplification of signals channels which are weakly degradable with pure ancillas (i.e. those which are degradable in the sense of Ref. [22]) have quantum capacity which can be expressed in terms of a single-letter expression. Here we will focus mostly on the anti-degradability property, and, additionally, we will show that by exploiting the composition rules of one-mode Bosonic Gaussian channels, one can extend the set the maps with null quantum capacity well beyond the set of anti-degradable maps.
The paper is organized as follows. In Sec. 1 we introduce the notion of weakly complementarity and weak-degradability in a rather general context. In Sec. 2 we give a detailed description of the canonical decomposition of one-mode Bosonic Gaussian channels. In Sec. 3 we discuss the weak-degradability properties of one-mode channels. Finally, in Sec 4 we determine the new set of channels with null quantum capacity.

Weakly complementary and weakly degradable channels
In quantum mechanics, quantum channels describe evolution of an open system A interacting with external degrees of freedom. In the Schödinger picture these transformations are described by completely positive trace preserving (CPT) linear maps Φ acting on the set D(H a ) of the density matrices ρ a of the system. It is a well known (see e.g. [2], [24]) that Φ can be described by a unitary coupling between the system A in input state ρ a with an external ancillary system B (describing the environment) prepared in some fixed pure state. This follows from Stinespring dilation [23] of the map which is unique up to a partial isometry. More generally, one can describe Φ as a coupling with environment prepared in some mixed state ρ b , i.e.
where Tr b [...] is the partial trace over the environment B, U ab is a unitary operator in the composite Hilbert space H a ⊗ H b . We call Eq. (1.1) a "physical representation" of Φ to distinguish it from the Stinespring dilation, and to stress its connection with the physical picture of the noisy evolution represented by Φ. Any Stinespring dilation gives rise to a physical representation. Moreover from any physical representation (1.1) one can construct a Stinespring dilation by purifying ρ b with an external ancillary system C, and by replacing U ab with the unitary coupling U abc = U ab ⊗ 1 1 c . Equation (1.1) motivates the following [19] Definition 1 For any physical representation (1.1) of the quantum channel Φ we define its weakly complementary as the mapΦ : D(H a ) → D(H b ) which takes the input state ρ a into the state of the environment B after the interaction with A, i.e. where T is the partial trace over the purifying system (here " • " denotes the composition of channels). As we will see, the properties of weakly complementary and complementary maps in general differ.
Definition 2 Let Φ,Φ be a pair of mutually weakly-complementary channels such that
Similarly if for some channel Ψ : D(H b ) → D(H a ) and all density matrix ρ a ∈ D(H a ), then Φ is anti-degradable whileΦ is weakly-degradable. In Ref. [22] the channel Φ is called degradable if in Eq. (1.4) we replaceΦ with a complementary map Φ com of Φ. Clearly any degradable channel [22] is weakly degradable but the opposite is not necessarily true. Notice, however, that due to Eq. (1.3), in the definition of anti-degradable channel we can always replace weakly complementary with complementary (for this reason there is no point in introducing the notion of weakly anti-degradable channel). This allows us to verify that if Φ is anti-degradable (1.5) then its complementary channel Φ com is degradable [22] and vice-versa. It is also worth pointing out that channels which are unitarily equivalent to a channel Φ which is weakly degradable (anti-degradable) are also weakly degradable (anti-degradable).
Finally an important property of anti-degradable channels is the fact that their quantum capacity [13] is null. As discussed in [19] this is a consequence of the no-cloning theorem [27] (more precisely, of the impossibility of cloning with arbitrary high fidelity [28]).
It is useful also to reformulate our definitions in the Heisenberg picture. Here the states of the system are kept fixed and the transformation induced on the system by the channel is described by means of a linear map Φ H acting on the algebra B(H a ) of all bounded operators of A so that for all ρ a ∈ D(H a ) and for all Θ a ∈ B(H a ). From this it follows that the Heisenberg picture counterpart of the physical representation (1.1) is given by the unital channel Similarly, from (1.2) it follows that in the Heisenberg picture the weakly complementary of the channel is described by the completely positive unital map for all Θ a ∈ B(H a ).

One-mode Bosonic Gaussian channels
Gaussian channels arise from linear dynamics of open Bosonic system interacting with Gaussian environment via quadratic Hamiltonians. Loosely speaking, they can be characterized as CPT maps that transform Gaussian states into Gaussian states [3,4,29].
Here we focus on one-mode Bosonic Gaussian channels which act on the density matrices of single Bosonic mode A. A classification of such maps obtained recently in the paper [20] allows us to simplify the analysis of the weak-degradability property. In the following we start by reviewing the result of Ref. [20], clarifying the connection with the analysis of Ref. [19] (cf. also Ref. [18]). Then we pass to the weak-degradability analysis of these channels, showing that with some important exception, they are either weakly degradable or anti-degradable.

General properties
Consider a single Bosonic mode characterized by canonical observables Q a , P a obeying the canonical commutation relation [Q a , P a ] = i. A consistent description of the system can be given in terms of the unitary Weyl operators V a (z) = exp [i(Q a , P a ) · z], with z = (x, y) T being a column vector of R 2 . In this framework the canonical commutation relation is written as with σ 2 being the second Pauli matrix. Moreover the density operators ρ a of the system can be expressed in terms of an integral over z of the V a (z)'s, i.e.
being the characteristic function of ρ a 1 . Consequently a complete description of a quantum channel on A is obtained by specifying its action on the operators V a (z), or, equivalently, by specifying how to construct the characteristic function φ(Φ(ρ a ); z) of the evolved states. In the case of Gaussian channels Φ this is done by assigning a mapping of the Weyl operators in the Heisenberg picture, or the transformation of the characteristic functions in the Schrödinger picture. Here m is a vector, while K and α are real matrices (the latter being symmetric and positive). Equation (2.5) guarantees that any input Gaussian characteristic function will remain Gaussian under the action of the map. A useful property of Gaussian channels is the fact that the composition of two of them (say Φ ′ and Φ ′′ ) is still a Gaussian channel. Indeed one can easily verify that the composite map Φ ′′ • Φ ′ is of the form (2.5) with m, K and α given by Here m ′ , K ′ , and α ′ belongs to Φ ′ while m ′′ , K ′′ , and α ′′ belongs to Φ ′′ .
Not all possible choices of K, α correspond to transformations Φ which are completely positive. A necessary and sufficient condition for this last property (adapted to the case of one mode) is provided by the nonnegative definiteness of the following 2 × 2 Hermitian matrix [3,20] 2 This matrix reduces to 2α + (Det[K] − 1) σ 2 and its nonnegative definiteness to the inequality (2.8) Within the limit imposed by Eq. (2.8) we can use Eq. (2.5) to describe the whole set of the one-mode Gaussian channels.

Channels with single-mode physical representation
An important subset of one-mode Gaussian channels is given by the maps Φ which possess a physical representation (1.1) with ρ b being a Gaussian state of a single external Bosonic mode B and with U ab being a canonical transformation of Q a , P a , Q b and P b (the latter being the canonical observables of the mode B). In particular let ρ b be a thermal state of average photon number N, i.e.
and let U ab be such that This yields the following evolution for the characteristic function φ(ρ a ; z), which is of the form (2.5) by choosing m = 0, K = m 11 and α = (N + 1/2) m T 12 · m 12 . It is worth stressing that in the case of Eq. (2.12) the inequality (2.8) is guaranteed by the symplectic nature of the matrix M, i.e. by the fact that Eq. (2.10) preserves the commutation relations among the canonical operators. Indeed we have where in the second identity the condition (2.21) was used.
As we shall see, with certain important exception one-mode Gaussian channels (2.4) are unitarily equivalent to transformations which admit physical representation with ρ b and U ab as in Eqs. (2.9) and (2.10).

Canonical form
Following Ref. [20] any Gaussian channel (2.5) can be transformed (through unitarily equivalence) into a simple canonical form. Namely, given a channel Φ characterized by the vector m and the matrices K, α of Eq. (2.5), one can find unitary operators U a and W a such that the channel defined by the mapping is of the form (2.5) with m = 0 and with K, α replaced, respectively, by the matrices K can , α can of Table 1 An important consequence of Eq. (2.15) is that to analyze the weak-degradability properties of a one-mode Gaussian channel it is sufficient to focus on the canonical map Φ (can) which is unitarily equivalent to it (see remark at the end of Sec. 1). Here we will not enter into the details of the derivation of Eqs. (2.14) and (2.15), see Ref. [20]. The dependence on the matrix K can of Φ (can) upon the parameters of Φ can be summarized as follows, with σ 3 being the third Pauli matrix. Analogously for α can we have  Table 2 for details. Notice that the weakly complementary channel of A 1 belongs to B 2 . However, not all the channels of B 2 have weakly complementary channels which are in A 1 -see Sec. 2.5.
The quantity N c is a free parameter which can set to any positive value upon properly calibrating the unitaries U a and W a of Eq. (2.14). Following Ref. [20] we will assume N c = 1/2. Notice also that from Eq. (2.8), rank[α] = 1 is only possible for Det[K] = 1.
Equations (2.16) and (2.17) show that only the determinant and the rank of K and α are relevant for defining K can and α can . Indeed one can verify that K can and α can maintain the same determinant and rank of the original matrices K and α, respectively. This is a consequence of the fact the Φ and Φ (can) are connected through a symplectic transformation for which Det[K], Det[α], rank[K], and rank[α] are invariant quantities. [In particular Det[K] is directly related with the invariant quantity q analyzed in Ref. [19].] The six inequivalent canonical forms of Table 1 follow by parametrizing the value of Det[α] to account for the constraints imposed by the inequality (2.8). It should be noticed that to determine which class a certain channel belongs to, it is only necessary to know if Det[K] is null, equal to 1, negative or positive ( = 1). If Det[K] = 0 the class is determined by the rank of the matrix. If Det[K] = 1 the class is determined by the rank of α (see Fig. 1). Within the various classes, the specific expression of the canonical form depends then upon the effective values of Det[K] and Det [α]. We observe also that the class A 1 can be obtained as a limiting case (for κ → 0) of the maps of class C or D. Analogously the class B 2 can be obtained as a limiting case of the maps of class C. Indeed consider the channel with K can = κ1 1 and α can = |κ 2 − 1|(N ′ 0 + 1/2)1 1 with N ′ 0 = N 0 /(|κ 2 − 1|) − 1/2, with N 0 and κ positive (κ = 0, 1). For κ sufficiently close to 1, N ′ 0 is positive and the maps belongs to the class C of Table 1. Moreover in the limit of κ → 1 this channel yields the map B 2 .
Finally it is interesting to study how the canonical forms of Table 1 compose under the product (2.6). A simple calculation shows that the following rules apply In this table, for instance, the element on the row 2 and column 3 represents class (i.e. A 2 ) associated to the product Φ ′′ • Φ ′ between a channel Φ ′ of B 1 and a channel Φ ′′ of A 2 . Notice that the canonical form of the products B 1 • B 2 , B 2 • B 1 and C • C is not uniquely defined. In the first case in fact, even though the determinant of the matrix K of Eq. (2.6) is one, the rank of the corresponding α might be one or different from one depending on the parameters of the two "factor" channels: consequently the B 1 • B 2 and B 2 • B 1 might belong either to B 1 or to B 2 . In the case of C • C instead it is possible that the resulting channel will have Det[K] = 1 making it a B 2 map. Typically however C • C will be a map of C. Composition rules analogous to those reported here have been extensively analyzed in Refs. [19,16,17].

Single-mode physical representation of the canonical forms
Apart from the case B 2 that will be treated separately (see next section), all canonical transformations of Table 1  which guarantees that U † ab Q a U ab and U † ab P a U ab satisfy canonical commutation relations. It is worth noticing that once m 11 and m 12 are determined within the constraint (2.21) the remaining blocks (i.e. m 21 and m 22 ) can always be found in order to satisfy the remaining symplectic conditions of M. An explicit example will be provided in few paragraphs. Class To complete the definition of the unitary operators U ab we need to provide also the transformations of Q b and P b . This corresponds to fixing the blocks m 21 and m 22 of M (see next section and Ref. [4]).
and cannot be done uniquely: one possible choice is presented in the following table The above definitions make explicit the fact that the canonical form C represents attenuator (κ < 1) and amplifier (κ > 1) channel [3]. We will see in the following sections that the class D is formed by the weakly complementary of the amplifier channels of the class C. For the sake of clarity the explicit expression for the matrices M of the various classes has been reported in App. A. Finally it is important to notice that the above physical representations are equivalent to Stinespring representations only when the average photon number N of ρ b nullifies. In this case the environment B is represented by a pure input state (i.e. the vacuum). According to our definitions this is always the case for the canonical form B 1 while for the canonical forms A 1 , A 2 , C and D it happens for N 0 = 0.

The class B 2 : additive classical noise channel
As mentioned in the previous section the class B 2 of Table 1 must be treated separately. The map B 2 corresponds 3 to the additive classical noise channel [3] defined by Φ(ρ a ) = d 2 z p(z) V a (z) ρ a V a (−z) , (2.22) with p(z) = (2πN 0 ) −1 exp[−|z| 2 /(2N 0 )] which, in Heisenberg picture, can be seen as a random shift of the annihilation operator a. These channels admit a natural physical representation which involve two environmental modes in a pure state (see Ref. [20] [20]. A pictorial representation of the above weak-degradability connections is given in Fig. 1. This would correspond to representing the channel B 2 in terms of a linear coupling with a single-mode thermal state ρ b of "infinite" temperature. Unfortunately this is not a well defined object. However we can use the "asymptotic" representation described at the end of Sec. 2.3 where it was shown how to obtain B 2 as limiting case of C class maps, to claim at least that there exists a one-parameter family of one-mode Gaussian channels which admits single-mode physical representation and which converges to B 2 .

Weak-degradability of one-mode Gaussian channels
In the previous section we have seen that all one-mode Gaussian channels are unitarily equivalent to one of the canonical forms of Table 1. Moreover we verified that, with the exception of the class B 2 , all the canonical forms admits a physical representation (1.1) with ρ b being a thermal state of a single environmental mode and U ab being a linear coupling. Here we will use such representations to construct the weakly complementary (1.2) of these channels and to study their weak-degradability properties.

Weakly complementary channels
In this section we construct the weakly complementary channelsΦ of the class A 1 , A 2 , B 1 , C and D starting from their single-mode physical representations (1.1) of Sec. 2.4. Because of the linearity of U ab and the fact that ρ b is Gaussian, the channelsΦ are Gaussian. This can be seen for instance by computing the characteristic function (2.3) of the output stateΦ(ρ a ) 1) has the same structure (2.5) of the one-mode Gaussian channel of A. Therefore by cascadingΦ with an isometry which exchanges A with B (see Refs. [31,19]) we can then treatΦ as an one-mode Gaussian channel operating on A (this is possible because both A and B are Bosonic one-mode systems). With the help of Table 1 we can then determine which classes can be associated with the transformation (3.1). This is summarized in Table 2.

Weak-degradability properties
Using the compositions rules of Eqs. (2.6) and (2.18) it is easy to verify that the canonical forms A 1 , A 2 , D and C with κ 1/2 are anti-degradable (1.10). Vice-versa one can verify that the canonical forms B 1 and C with κ 1/2 are weakly degradable (1.9)for C, D and A 1 these results have been proved in Ref. [19]. Through unitary equivalence Moreover B 1 is also degradable in the sense of Ref. [22]. The same holds for channels of canonical form C with N 0 = 0: the exact expression for the quantum capacity of these channels has been given in Ref. [21]. this can be summarized by saying that all one-mode Gaussian channels (2.5) having Det[K] 1/2 are anti-degradable, while the others (with the exception of the channels belonging to B 2 ) are weakly degradable (see Fig. 2). In the following we verify the above relations by explicitly constructing the connecting channels Ψ and Ψ of Eqs. (1.9) and (1.10) for each of the mentioned canonical forms. Indeed one has: • For a channel Φ of standard form A 1 or A 2 , anti-degradability can be shown by simply taking Ψ of Eq. (1.10) coincident with the channel Φ. The result immediately follows from the composition rule (2.6).
• For a channel Φ of B 1 , weak-degradability comes by assuming the map Ψ to be equal to the weakly complementary channelΦ of Φ (see Table 2). As pointed out in Ref. [20] this also implies the degradability of Φ in the sense of Ref. [22]. Let us remind that for B 1 the physical representation given in Sec. 2.4 was constructed with an environmental state ρ b initially prepared in the vacuum state, which is pure. Therefore in this case our representation gives rise to a Stinespring dilation.
-If κ ∈ [ 1/2, 1[ the channel is weakly degradable and the connecting map Ψ is again a channel of C defined as in the previous case but with κ ′ = √ 1 − κ 2 /κ < 1. For N 0 = 0 the channel is also degradable [22] since our physical representation is equivalent to a Stinespring representation.
Concerning the case B 2 it was shown in Ref. [20] that the channel is neither antidegradable nor degradable in the sense of [22] (apart from the trivial case N 0 = 0 which corresponds to the identity map). On the other hand one can use the continuity argument given in Sec. 2.5 to claim that the channel B 2 can be arbitrarily approximated with maps which are weakly degradable (those belonging to C for instance).
4 One-mode Gaussian channels with Det[K] > 1/2 and having null quantum capacity In the previous section we saw that all channels (2.5) with Det[K] 1/2 are antidegradable. Consequently these channel must have null quantum capacity [19,31]. Here we go a little further showing that the set of the maps (2.5) which can be proved to have null quantum capacity include also some maps with Det[K] > 1/2. To do this we will use the following simple fact: Let be Φ 1 a quantum channel with null quantum capacity and let be Φ 2 some quantum channel. Then the composite channels Φ 1 • Φ 2 and Φ 2 • Φ 1 have null quantum capacity.
The proof of this property follows by interpreting Φ 2 as a quantum operation performed either at the decoding or at encoding stage of the channel Φ 1 . This shows that the quantum capacities of Φ 1 • Φ 2 and Φ 2 • Φ 1 cannot be greater than the capacity of Φ 1 (which is null). In the following we will present two cases where the above property turns out to provide some nontrivial results.

Composition of two class D channels
We observe that according to composition rule (2.18) the combination of any two channels Φ 1 and Φ 2 of D produces a map Φ 21 ≡ Φ 2 • Φ 1 which is in the class C. Since the class D is anti-degradable the resulting channel must have null quantum capacity. Let then κ j σ 3 and (κ 2 j + 1)(N j + 1/2)1 1 be the matrices K can and α can of the channels Φ j , for j = 1, 2. From Eq. (2.6) one can then verify that Φ 21 has the canonical form C with parameters κ = κ 1 κ 2 , (4.1) Equation (4.1) shows that by varying κ j , κ can take any positive values: in particular it can be greater than 1/2 transforming Φ 21 into a channel which does not belong to the anti-degradable area of Fig. 2. On the other hand, by varying the N j and κ 2 , but keeping the product κ 1 κ 2 fixed, the parameter N 0 can assume any value satisfying the inequality N 0 1 2 We can therefore conclude that all channels C with κ and N 0 as in Eq. (4.3) have null quantum capacity -see Fig. 3. A similar bound was found in a completely different way in Ref. [3].

Composition of two class C channels
Consider now the composition of two class C channels, i.e. Φ 1 and Φ 2 , with one of them (say Φ 2 ) being anti-degradable.
Here, the canonical form of Φ 1 and Φ 2 have matrices K can and α can given by K i = κ j 1 1 and α j = |κ 2 j − 1|(N j + 1/2)1 1, where for j = 1, 2, N j and κ j are positive numbers, with κ 1 = 0, 1 and with κ 2 ∈]0, 1/2] (to ensure anti-degradability). From Eq. (2.6) follows then that the composite map Φ 21 = Φ 2 • Φ 1 has still a C canonical form with parameters κ = κ 1 κ 2 ,  As in the previous example, κ can assume any positive value. Vice-versa keeping κ fixed, and varying κ 1 > 1 and N 1,2 it follows that N 0 can take any values which satisfy the inequality N 0 1 2 We can then conclude that all maps C with κ and N 0 as above must possess null quantum capacity. The result has been plotted in Fig. 3. Notice that the constraint (4.6) is an improvement with respect to the constraint of Eq. (4.3).

Conclusion
In this paper we provide a full weak-degradability classification of one-mode Gaussian channels by exploiting the canonical form decomposition of Ref. [20]. Within this context we identify those channels which are anti-degradable. By exploiting composition rules of Gaussian maps, this allows us to strengthen the bound for one-mode Gaussian channels which have not null quantum capacity.