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Gaussian quantum channels

This article provides an elementary introduction to Gaussian channels and their capacities. We review results on the classical, quantum, and entanglement assisted capacities and discuss related entropic quantities as well as additivity issues. Some of the known results are extended. In particular, it is shown that the quantum conditional entropy is maximized by Gaussian states and that some implications for additivity problems can be extended to the Gaussian setting.


Introduction
Any physical operation that reflects the time evolution of the state of a quantum system can be regarded as a channel. In particular, quantum channels grasp the way how quantum states are modified when subjected to noisy quantum communication lines. Couplings to other external degrees of freedom, often beyond detailed control, will typically lead to losses and decoherence, effects that are modelled by appropriate non-unitary quantum channels.
Gaussian quantum channels play a quite central role indeed. After all, good models for the transmission of light through fibers are provided by Gaussian channels. This is no accident: linear couplings of bosonic systems to other bosonic systems with quadratic Hamiltonians can in fact appropriately be said to be ubiquitous in physics. In this optical context then, the time evolution of the modes of interest, disregarding the modes beyond control, is then reflected by a Gaussian bosonic channel. Random classical noise, introduced by Gaussian random displacements in phase space, gives also rise to a Gaussian quantum channel, as well as losses that can be modelled as a beam splitter like interaction with the vacuum or a thermal mode.
This article provides a brief introduction into the theory of Gaussian quantum channels. 1 After setting the notation and introducing to the elementary concepts, we provide a number of practically relevant examples. Emphasis will later be put on questions concerning capacities: Capacities come in several flavours, and essentially quantify the usefulness of a quantum channel for the transmission of classical or quantum information. We will briefly highlight several major results that have been achieved in this field. Finally, we discuss a number of open questions, notably related to the intriguing but interesting and fundamental questions of additivities of quantum channel capacities.

Gaussian channels
In mathematical terms a quantum channel is a completely positive trace-preserving map ρ −→ T (ρ) that takes states, i.e., density operators ρ acting on some Hilbert space H, into states 2 . For simplicity we will always assume that output and input Hilbert spaces are identical. Every channel can be conceived as reduction of a unitary evolution in a larger quantum system. So for any channel T there exists a state ρ E on a Hilbert space H E , and a unitary U such that The system labeled E serves as an environment, embodying degrees of freedom of which elude the actual observation, inducing a decoherence process. The channel is then a local manifestation of the unitary evolution of the joint system. A Gaussian channel [1,2,3,4,5] is now a channel of the form as in Eq. (1), where U is a Gaussian unitary, determined by a quadratic bosonic Hamiltonian, and ρ E is a Gaussian state [6]. In many cases, of which the lossy optical fiber is the most prominent one, this restriction to quadratic Hamiltonians gives a pretty good description of the physical system. Note that although the channel is assumed to be Gausssian in the entire article, the input states are not necessarily taken to be Gaussian.

Preliminaries
It seems appropriate for the following purposes to briefly fix the notation concerning Gaussian states and their transformations [5,6,7,8]. For a quantum system with n modes, i.e., n canonical degrees of freedom, the canonical coordinates will be denoted as R = (x 1 , p 1 , ..., x n , p n ). Most naturally, these operators can be conceived as corresponding to field quadratures. Although all statements in this article hold true for any physical system having canonical coordinates, we will often refer to the optical context when intuitively describing the action of a channel. The creation and annihilation operators are related to these canonical coordinates according to The coordinates satisfy the canonical commutation relations, which can be expressed in terms of the Weyl operators or displacement operators W ξ = e iξ T σR with ξ ∈ Ê 2n : where we have seth = 1. The matrix σ defines the symplectic scalar product, simply indicating that position and momentum of the same mode do not commute.
The Fourier transform of the ordinary Wigner function in phase space Ê 2n is the charac- from which the state can be reobtained as ρ = d 2n ξ χ ρ (ξ)W † ξ /(2π) n . Gaussian states are exactly those having a Gaussian characteristic function, and therefore a Gaussian Wigner function in phase space: Here, the 2n × 2n-matrix Γ and the vector D ∈ Ê 2n are essentially the first and second moments: they are related to the covariance matrix γ and the displacements d as Γ = σ T γσ and D = σd. This choice is then consistent with the definition of the covariance matrix as having entries States always satisfy the Heisenberg uncertainty principle, which can be expressed as γ + iσ ≥ 0. This is a simple semi-definite constraint onto any matrix of second moments, also obeyed by every non-Gaussian state.

General Gaussian channels
The simplest Gaussian channel is a lossless unitary evolution, governed by a quadratic bosonic Hamiltonian: with H being a real and symmetric 2n × 2n matrix. Such unitaries correspond to a representation of the real symplectic group Sp(2n, Ê), formed by those real matrices for which SσS T = σ [7,8,9]. These are exactly the linear transformations which preserve the commutation relations. The relation between such a canonical transformation in phase space and the corresponding unitary in Hilbert space is given by S = e Hσ . Needless to say, Gaussian unitaries are ubiquitous in physics, in particular in optics, and this is the reason why Gaussian channels play such an important role. Notably, the action of ideal beam splitters, phase shifters, and squeezers correspond to symplectic transformations. 3 It is often instructive to consider transformations on the level of Weyl operators in the Heisenberg picture. For a symplectic transformation we have W ξ −→ W S −1 ξ . The action of a general Gaussian channel ρ −→ T (ρ) can be phrased as where X, Y are real 2n × 2n-matrices [1,5,7]. Additional linear terms in the quadratic form are omitted since they merely result in displacements in phase space, which are not interesting for later purpose. Not any transformation of the above form is possible: complete positivity of the channel dictates that 4 Depending on the context it may be more appropriate or transparent to formulate a Gaussian channel in the Schrödinger picture ρ −→ T X,Y (ρ) or to define it as a transformation of covariance matrices γ −→ X T γX + Y.
This is the most general form of a Gaussian channel. Roughly speaking X serves the purpose of amplification or attenuation and rotation in phase space, whereas the Y contribution is a noise term which may consist of quantum (required to make the map physical) as well as classical noise. Interestingly, X may be any real matrix, and hence, any map γ −→ X T γX can be approximately realized, as long as 'sufficient noise' is added. In this language, it also becomes immediately apparent how much noise will be introduced by any physical device approximating amplification or time reversal, meaning phase conjugation in an optical context. For second moments far away from minimal uncertainty, this additional noise may hardly have an impact (so classical fields can be phase conjugated after all), whereas close to minimal uncertainty this is not so any longer.

Important examples of Gaussian channels
The practically most important Gaussian channel is probably an idealized action of a fiber. Moreover, as mentioned earlier, any situation where a quadratic coupling to a Gaussian environment provides a good description can be cast into the form of a Gaussian channel. We will in the following consider a number of important special cases of Gaussian channels for single modes: 3 Any such S can be decomposed into a squeezing component, and a passive operation [9]. So one may write S = V ZW , with V, W ∈ K(n) = Sp(2n, Ê) ∩ SO(2n) are orthogonal symplectic transformations, forming the subgroup of passive, i.e., number-preserving, operations. In turn, Z = diag (z 1 , 1/z 1 , . . . , zn, 1/zn) with z 1 , ..., zn ∈ Ê\{0} are local single-mode squeezings.
1. The classical noise channel merely adds classical Gaussian noise to a quantum state, i.e., X = ½, Y ≥ 0 [3,10,11,12]. In Schrödinger picture this channel can be represented by a random displacement according to a classical Gaussian probability distribution: 2. In the thermal noise channel [3,11] a mode passively interacts with another mode in a thermal state, The result is as if the mode had been coupled in with a beam splitter of some transmittivity η. 5 For the second moments, we have that where c½ 2 , c ≥ 1, is the covariance matrix of a thermal Gibbs state with mean photon number (c − 1)/2.
[.] E denotes the leading 2 × 2 submatrix. The passive symplectic transformation S η is given by So we obtain 3. The lossy channel is obtained by setting c = 1 in Eq (13). It reflects photon loss with probability 1 − η. This channel is the prototype for optical communication through a lossy fiber, since thermal photons (leading to a contribution c > 1) are negligible at room temperature. When using an optical fiber of length l and absorbtion length l A we may set η = e −l/lA . The lossy channel with X = √ η½ 2 , Y = (1 − η)½ 2 is also called attenuation channel [3]. 4. The amplification channel [3] is of the form Here, the term Y is a consequence of the noise that is added due to Heisenberg uncertainty. Note that a classical noise channel can be recovered as a concatenation of a lossy channel, followed by an amplification.
All these examples correspond to a single mode characterized by a fixed frequency ω. This is often referred to as the narrowband case as opposed to broadband channels [13,14,15], which consist out of many uncoupled single-mode channels, each of which corresponds to a certain frequency ω i , i = 1, 2, ... . Best studied is the simple homogeneous case of a lossy broadband channel (equally spaced frequencies ω i , with equal transmittivity η in all the modes). 5 In the Heisenberg picture this means that the annihilation operator transforms as a → √ η a+ b is the annihilation operator of the ancillary mode.
It shall finally be mentioned that the very extensive literature on harmonic open quantum systems is essentially concerned with Gaussian channels of a specific kind, yet one where the environment consists of infinitely many modes, where the linear coupling is characterized by some spectral density.

Output entropies
Channels describing the physical transmission of quantum states typically introduce noise to the states as a consequence of a decoherence process. Pure inputs are generally transformed into mixed outputs, so into states ρ having a positive von-Neumann entropy The entropy of the output will clearly depend on the input and the channel itself, and the minimal such entropy can be taken as a characteristic feature of the quantum channel. Introducing more generally the α Renyi entropies for α ≥ 0 as this minimal output entropy [16] is then defined as 6 The Renyi entropies [17] are derived from the α-norms of the state, ρ α = tr[ρ α ] 1/α . In case of the limit lim αց1 one retains the von-Neumann entropy, i.e., lim αց1 S α (ρ) = S(ρ); for α = 2, this is the purity in the closer sense. Roughly speaking, the smaller the minimal output entropy, the less decohering is the channel (see, e.g., Ref. [18]). The actual significance of this quantity yet originates from its intimate relationship concerning questions of capacities. This will be elaborated on in the subsequent section.

Mutual information and coherent information
In Shannon's seminal channel coding theorem the capacity of a classical channel is expressed in terms of the classical mutual information [20]. In fact, as we will see below, the quantum analogue of this quantity plays a similar role in quantum information theory. For any quantum channel T and any quantum state ρ acting on a Hilbert space H, one defines the quantum mutual information I(ρ, T ) as where S(ρ, T ) = (½ ⊗ T )(|ψ ψ|) and |ψ ∈ H D ⊗ H is any purification of the state ρ = tr D [|ψ ψ|] [3,19]. It is not difficult to see that I(ρ, T ) does not depend on the chosen purification. The quantum mutual information has many desirable properties: it is positive, concave with respect to ρ, and additive with respect to quantum channels of the form T ⊗n .
The latter property comes in very handy when relating this quantity to the entanglementassisted classical capacity. An important part of the quantum mutual information is the coherent information given by J(ρ, T ) can be positive as well as negative, it is convex with respect to T but its convexity properties with respect to ρ are unclear.

Entropies of Gaussian states and extremal properties
When maximizing the rate at which information can be sent through a Gaussian channel, Gaussian states play an important role. In fact, in many cases it turns out that encoding the information into Gaussian states leads to the highest transmission rates. This is mainly due to the fact that for a given covariance matrix many entropic quantities take on their extremal values for Gaussian states. These entropic quantities, and in fact any unitarily invariant functional, can for Gaussian states immediately be read off the symplectic spectrum of the covariance matrix: any covariance matrix γ of n modes can be brought to the Williamson normal and {c i : i = 1, ..., n} being the positive part of the spectrum of iσγ. This is nothing but the familiar normal mode decomposition with the c i corresponding to the normal mode frequencies. Then, the problem of evaluating any of the above quantities is reduced to a single-mode problem. For example, the von-Neumann entropy is given by [2] where g(N ) = (N + 1) log(N + 1) − N log N is the entropy of a thermal Gaussian state with average photon number N . Similar expressions can be found for the other entropic quantities. Consider now any stateρ which has the same first and second moments as its Gaussian counterpart ρ. Then where the first term is the nonnegative relative entropy, and the second term vanishes since the expectation value of the operator ln ρ depends only on the first and second moments. Hence, the Gaussian state has the largest entropy among all states with a given covariance matrix 7 [2]. A more sophisticated argument, using ideas of convex optimization and the theorem of Kuhn and Tucker, shows that the same holds true for the quantum mutual information [3]: For any Gaussian channel T and fixed first and second moments of ρ, the respective Gaussian state maximizes I(ρ, T ). Whether a similar statement also holds for the coherent information is not known. Another very useful quantity that takes its extremal values on Gaussian states, we would like to mention at this point, is the quantum conditional entropy [22], defined as in a bi-partite system with parts A and B. Here ρ A is the reduction with respect to system A. It can be shown in a very similar fashion as before that this quantity is maximized on Gaussian states for fixed second moments, although we now encounter a difference between two von-Neumann entropies. Letρ again be a state with the same first and second moments as its Gaussian counterpart ρ, then In the last inequality, it is used that the relative entropy can only decrease under joint application of completely positive maps. This extremal property is helpful when assessing for example achievable rates in state merging [22], for which the quantum conditional entropy is an upper bound. More importantly in the Gaussian setting, the negative of the conditional entropy is a lower bound [23] to the distillable entanglement [24], which can be used to detect distillable entanglement in quantum states by measuring second moments only. Whenever one performs a measurement of second moments (estimation of the variances of the quadratures) of an unknown stateρ and finds that its Gaussian counterpart satisfies then one can argue that this is in turn a lower bound for the distillable entanglement E D (ρ) ofρ. In this way, one can infer about the distillable entanglement of an unknown quantum state without having to assume that the quantum state is Gaussian. This is relevant as any knowledge whether a state is Gaussian is typically not accessible without complete state tomography. Moreover, this bound is robust against small perturbations, which is also practically important since even complete state tomography will determine the state only up to some error.

Constrained quantities
There are essentially two subtleties [25,26] that arise in the infinite-dimensional context as we encounter it here for Gaussian quantum channels: on the one hand, there is the necessity of natural input constraints, such as one of finite mean energy. Otherwise, the capacities diverge. On the other hand, there is the possibility of continuous state ensembles 8 . The need for a constraint is already obvious when considering the von-Neumann entropy: On a state space over an infinite dimensional Hilbert space, the von-Neumann entropy is not (trace-norm) continuous, but only lower semi-continuous 9 , and almost everywhere infinite. This problem can be tamed by introducing an appropriate constraint. For our purposes, we may take the Hamiltonian H = n i=1 (x 2 i + p 2 i )/2. Then, instead of taking all states into account, one may consider the subset introducing for some h > 0 a constraint on the mean energy 10 or mean photon number N = h − 1/2. Similarly, for tensor products we consider K ⊗n = {ρ : tr[ρH ⊗n ] < nh}. On this very natural subset K the von-Neumann entropy and the classical information capacity retain their continuity. In fact, many entanglement measures also retain the continuity properties familiar in the finite-dimensional context, such that, e.g., the entropy of a subsystem for pure states can indeed be interpreted as the distillable entanglement [27].

Capacities
In classical information theory a single number describes how much information can reliably be sent through a channel: its capacity. In quantum information theory the situation is more complicated and each channel is characterized by a number of different capacities [28]. More precisely, which capacity is the relevant one depends on whether we want to transmit classical or quantum information, and on the resources and protocols we allow for. An important resource that we must consider is entanglement shared between sender and receiver. The presence or absence of this resource together with the question about sending classical or quantum information leads to four basic capacities, which we will discuss in the following.

Classical information capacity
The classical information capacity is the asymptotically achievable number of classical bits that can be reliably transmitted from a sender to a receiver per use of the channel. Here, it is assumed that the parties may coherently encode and decode the information in the sense that they may use entangled states as codewords at the input and joint measurements over arbitrary channel uses at the output. This answers essentially the question of how useful a quantum channel is for the transmission of classical information. This capacity is derived from the single-shot expression [29,30], appropriately constrained as above, where the supremum is taken over all probability distributions and sets of states satisfying ρ = i p i ρ i under the constraint ρ ∈ K [25,26] 11 . By the Holevo-Schumacher-Westmoreland (HSW) theorem [29,30], this single-shot expression gives the capacity if the encoding is restricted to product states. Hence, the full classical information capacity can formally be expressed as the regularization of C 1 , 10 More general constraints than this one can be considered, leading to compact subsets of state space on which one retains continuity properties in particular for the von-Neumann entropy and the classical information capacity [17,25,26,27]. Essentially, any unbounded positive operator H with a spectrum without limiting points would also be appropriate, such that tr exp[−βH] < ∞ for all β > 0. 11 The above constraint also ensures that (S • T )(ρ) < ∞. The convex hull function of S • T , given by ρ −→ S(ρ, T ) = inf i p i (S • T )(ρ i ) in Eq. (27), with the infimum being taken over all ensembles with i p i ρ i = ρ, is still convex in the unconstrained case, but no longer continuous, however, lower semi-continuous in the above sense.
Clearly, C(T, K) ≥ C 1 (T, K) since the latter does not allow for inputs which are entangled over several instances of the channel. Yet, it is in general not known whether this possibility comes along with any advantage at all, so whether entangled inputs facilitate a better information transfer. This will be remarked on later.
Note that in this infinite-dimensional setting, the constraint is required to obtain a meaningful expression for the capacity: for all non-trivial Gaussian channels the optimization over all input ensembles in Eq. (27) would lead to an infinite capacity. This can simply be achieved by encoding the information into phase space translates of any signal state. Then no matter how much noise is induced by the channel, we can always choose the spacing between the different signal states sufficiently large such that they can be distinguished nearly perfectly at the output.
Let us now follow the lines of Ref. [31] and sketch the derivation of the classical capacity for lossy channels. First of all, a lower bound on C(T, K) can be obtained by choosing an explicit input ensemble for Eq. (27). Random coding over coherent states according to a classical Gaussian probability distribution leads to an average input state of the form with covariance matrix γ = ½ + V . Hence, if we choose V = 2N ½, the average number of photons in the input state will be tr [ρa † a] = N . The constraint set K hence corresponds to the choice of h = N + 1/2. After passing a lossy channel with transmittivity η this changes to tr [T (ρ)a † a] = ηN , and since T (ρ) is a thermal state, its entropy is given by (S • T )(ρ) = g(ηN ). The action of a lossy channel on a coherent input state is to shift the state by a factor η towards the origin in phase space. In other words, the channel maps coherent states onto coherent states and since the latter have zero entropy, we have [3] C 1 (T, K) ≥ (S • T )(ρ) = g(ηN ).
Assume now thatρ is the average input state optimizing C 1 (T ⊗n , K ⊗n ) under a given constraint for the mean energy as described above. Then whereρ i is the reduction ofρ to the i-th mode and the second inequality is due to the subadditivity of the von-Neumann entropy. Since for a fixed average photon number tr [ρ i a † a] = N i the entropy is maximized by a Gaussian state, we have in addition that (S • T )(ρ i ) ≤ g(ηN i ).
Together with the lower bound this implies that the classical capacity of a lossy channel is indeed given by C(T, K) = g(ηN ) [31], if the average number of input photons per channel use is restricted to be not larger than N , corresponding to the constrained associated with K. Hence random coding over coherent states turns out to be optimal and neither non-classical signal states nor entanglement is required in the encoding step. 12 An immediate consequence of this result is that the classical capacity of the homogeneous broadband channel T is given by where P is the average input power and t is the transmission time related to the frequency spacing δω = 2π/t. For the lossless case η = 1 this capacity was derived in Ref. [15,32].

Quantum capacities
The quantum capacity is the rate at which qubits can be reliably transmitted through the channel from a sender to a receiver. This transmission is done again employing appropriate encodings and decodings before envoking instances of the quantum channel [3]. This capacity can be made precise using the norm of complete boundedness 13 . The question is how well the identity channel can be approximated in this norm. More specifically [33], the quantum capacity Q is the supremum of c ≥ 0 such that for all ε, δ > 0 there exist n, m ∈ AE, One may also consider a weaker instance, allowing for ε-errors, and then look at a Q εcapacity [3]. It is known that the quantum capacity does not increase if we allow for additional classical forward communication [34]. In Ref. [35] it was proven that the quantum capacity Q(T ) can be expressed in terms of the coherent information as Unfortunately, the asymptotic regularization is required in general, since the supremum over the coherent information is known to be not additive 14 . However, the single-shot quantity sup ρ J(ρ, T ) already gives a useful lower bound on Q(T ). For the classical noise Gaussian channel and Gaussian ρ this was first shown to be attainable in Ref. [10], based on earlier work [37], using methods of quantum stabilizer codes that embed a finite-dimensional protected code space in an infinite-dimensional one. For more general thermal noise channels, this is given by [3] J(ρ, where N ′ = ηN + (1 − η)(c − 1)/2 is the average photon number at the channel output. In fact, the same bound holds for the amplification channel, for which η > 1 and N ′ = ηN + (η − 1)(c + 1)/2. For broadband channels, lower bounds of this kind on the quantum capacity were discussed in Ref. [13].
A computable upper bound on the quantum capacity of any channel is given by Q(T ) ≤ log ||T θ|| cb [3]. For finite-dimensional systems θ is the matrix transposition, which corresponds to the momentum-reversal operation in the continuous variables case. This bound is zero for entanglement breaking channels 15 and additive for tensor products of channels. For attenuation and amplification channels with classical noise, i.e., channels acting as γ → ηγ + |1 − η|c, this leads to [3] Note that this bound is finite for all η = 1. This is remarkable since it does not depend on the input energy. That is, unlike the classical capacity, the unconstrained quantum capacity does typically not diverge. Moreover, it is even zero in the case η ≤ 1/2, since then the no-cloning theorem forbids an asymptotic error-free transmission of quantum information.

Entanglement-assisted capacities
Needless to say, in a quantum information context, it is meaningful to see what rates can be achieved for the transfer of classical information when entanglement is present. This is the kind of information transfer considered in the entanglement-assisted classical capacity C E [38,36]. It is defined as the rate at which bits that can be transmitted in a reliable manner in the presence of an unlimited amount of prior entanglement shared between the sender and the receiver. In just the same manner, the entanglement-assisted quantum capacity Q E may be defined [36,13,14]. Similarly, this quantifies the rate at which qubits can asymptotically be reliably transmitted per channel use, again in the presence of unlimited entanglement. Exploiting teleportation and dense coding is not difficult to see that 2Q E = C E . Now, the entanglement-assisted capacity C E is intimately related to the quantum mutual information, as just the supremum of this quantity with respect to all states ρ ∈ K as in Eq. (26) Again, with this constraint [36], the quantity regains the appropriate continuity properties 16 .
Note that in this case, no asymptotic version has to be considered, and due to the additivity of the quantum mutual information the single-shot expression already provides the capacity. In a sense Eq. (38) is the direct analogue of Shannon's classical coding theorem. The latter states that the classical capacity of a classical channel is given by the maximum mutual information. The main difference is, however, that in the classical case shared randomness does not increase the capacity, whereas for quantum channels shared entanglement typically increases the capacity, Again, similar to the classical case C E is conjectured to characterize equivalence classes of channels within which all channels can efficiently simulate one another [38]. For Gaussian channels the extremal property of Gaussian states with respect to the quantum mutual information allows us to calculate C E (T, K) by only maximizing over constrained Gaussian states ρ. For attenuation channels with classical noise, i.e., γ → ηγ + (1 − η)c with 0 ≤ η ≤ 1, it was shown in Ref. [3] that with the coherent information J(ρ, T ) taken from Eq. (35). For the homogeneous broadband lossy channel, extensively discussed in Ref. [13,14], it holds again that C E (T, K) ∝ t √ P .

Additivity issues
In the previous sections, we have encountered additivity problems of several quantities related to quantum channels. Such questions are at the core of quantum information theory: essentially, the questions is whether for product channels one can potentially gain from utilizing entangled inputs. This applies in particular to the additivity of the single-shot expression C 1 and the minimal output entropy 17 [16]. A number of partial results on additivity problems have been found. Yet, a conclusive answer to the most central additivity questions is still lacking. In particular, it is one of the indeed intriguing open questions of quantum information science whether the single-shot expression C 1 in Eq. (27) is already identical to the classical information capacity as it is true for the case of the lossy channel [31].

Equivalence of additivity problems
Interestingly, a number of additivity questions are related in the sense that they are either all true or all false. This connection is particularly well-established in the finite-dimensional context [39,40,41]: then, the equivalence of the (i) additivity of the minimum output 1entropy, the von-Neumann entropy, (ii) the additivity of the single-shot expression C 1 , (iii) the additivity of the entanglement of formation, and (iv) the strong superadditivity of the entanglement of formation have been shown to be equivalent [39,40,41]. This equivalence, besides being an interesting result in its own right, provides convenient starting points for general studies on additivity, as in particular the minimal output entropies appear much more accessible than the classical information capacity.
In the infinite-dimensional context, the argument concerning equivalence is somewhat burdened with technicalities. We will here state the main part of an equivalence theorem of additivity questions concerning any pair T 1 , T 2 of Gaussian channels [26]. The following properties (1.) and (2.) are equivalent and imply (3.): (1.) For any state ρ on the product Hilbert space and for all appropriately constraint sets K 1 and K 2 we have that 18 where for a channel T and i p i ρ i = ρ (3.) For the minimal output entropies where the bar indicates that in order to evaluate the minimal output entropy of T 1 ⊗ T 2 , the infimum is taken only over all pure states ρ such that S(tr 2 [ρ]) = S(tr 1 [ρ]) < ∞ and (S • (T 1 ⊗ T 2 ))(ρ) < ∞.
In particular, this means that once a general answer to (1.) or (2.) was known for Gaussian channels, a general single-shot expression for the classical information capacity of such channels would be available, solving a long-standing open question. Moreover, it was proven that the above implications hold true if one of the additivity conjectures is proven for the general finite dimensional case [26].

Integer output entropies
For specific channels, the unconstrained minimal output α-entropies for tensor products can be identified for integer α. These integer instances of output purities are not immediately related to the question of the classical information capacity, for which the limit α ց 1 is needed. However, they provide a strong indication of additivity also in the general case. Notably, for the single-mode classical and thermal noise channels T , has been established for integer α [11]. The concept of entrywise positive maps also provides a general framework for assessing integer minimal output entropies for Gaussian channels [45], generalizing previous results. It is worth mentioning that in the above cases the minimal output entropy ν α (T ), 2 ≤ α ∈ AE is attained for Gaussian input states [11].

Output entropies for Gaussian inputs
In all known cases Gaussian input states achieve the minimal output entropy or attain the capacity of Gaussian channels. Hence, one may be tempted to believe that this could be true in general and thus consider only Gaussian input states from the very beginning. In this restricted settings, quite far-reaching statements concerning additivity can yet be made. For example, if one requires that the encoding is done entirely in Gaussian terms, the additivity for minimal output entropies can be proven in quite some generality [42]. The Gaussian minimal output entropy is defined as where the infimum is taken over all Gaussian states. Then one finds that the minimal output α-entropy for single-mode Gaussian channels T 1 , ..., T n , as in Eq. (8) characterized by X 1 , ..., X n and Y 1 , ..., Y n , Y i ≥ 0, and det[X i ] = det[X j ] for all i, j is additive for all α ∈ (1, ∞). This includes the important case of identical Gaussian channels T , for all n and all α ∈ (1, ∞). Moreover, for α = 2 this kind of additivity was proven for arbitrary multi-mode Gaussian channels for which det[X i ] = 0 [42].

Equivalence of Gaussian additivity problems
The aim of this section is to make a first step towards proving that the equivalence of additivity problems [39,40,41] holds within the Gaussian world, where all resources and quantities are appropriately replaced by their Gaussian counterparts. Since all states in this section are Gaussian and thus essentially characterized by their covariance matrices (except from the first moments, which are not relevant, e.g., for their entanglement content and their entropy), we will for simplicity of notation use the covariance matrix γ as the argument of functions which are supposed to act on density operators. That is, we will write T (γ) and S(γ) meaning T (ρ γ ) and S(ρ γ ), with ρ γ being the centered Gaussian state with covariance matrix γ. Let us first define the quantities under consideration: I. Gaussian entanglement of formation: The Gaussian version [43] of the entanglement of formation (EoF) [24] restricts to decompositions into Gaussian states -the probability distribution in the decomposition is however not restricted to be Gaussian (although there exists always an optimal decomposition with this property [43]). As proven in Ref. [43], we have where E(Γ) is the entropy of entanglement of the pure Gaussian state with covariance matrix Γ, i.e., the von-Neumann entropy of its reduced state. The corresponding decomposition contains only phase-space displaced versions of this state. Obviously this is an upper bound for the (unconstrained) entanglement of formation [24], i.e., E G (γ) ≥ E F (γ) where equality holds at least for the case of symmetric two-mode states (for which the reduced states are unitarily equivalent) [44]. For these states E G was proven to be additive [43] and convex on the level of covariance matrices, i.e., for all λ ∈ [0, 1].

II. Gaussian capacity:
We introduce the Gaussian counterpart C 1,G (T, K) of the singleshot expression C 1 (T, K) in Eq. (27) by restricting the input ensemble to be a set of phase-space translates of a Gaussian state, distributed according to a Gaussian distribution. Evidently, C 1,G (T, K) ≤ C 1 (T, K) and the question of additivity is, whether for all Gaussian channels equality holds in As mentioned above, for the lossy channel we have indeed that [31] C 1,G (T, K) = C 1 (T, K) = C(T, K).
Gaussian MSW correspondence: Following Matsumoto, Shimono, and Winter (MSW) in Ref. [40], one can easily establish a relation between E G , C 1,G , and ν 1,G . Let T : γ → XγX T +Y be a Gaussian channel acting on systems of n-modes. Then there exists a dilation, i.e., a pure state of m ≤ 2n modes with covariance matrix γ 0 and a symplectic transformation S, such that where A and B refer to an n-mode and m-mode subsystem, and as before [γ ′ ] A denotes the covariance matrix corresponding to subsystem A. This is nothing but the corresponding principal submatrix. Replacing K by the singleton set of a fixed average Gaussian input state with covariance matrix γ leads to a new quantity C 1,G (T, γ), which is defined as This relates the Gaussian EoF to the capacity C 1,G . In fact, if G(T, K) is the set of all covariance matrices γ ′ in Eq. (53) for which γ ∈ K, then which is the Gaussian analogue of the MSW correspondence [40]. Moreover, the simplicity of the Gaussian EoF in Eq. (48) leads to a relation between E G and the minimum output entropy: if γ, γ ′ , and T are again related via Eq. (53), then Implications for Gaussian additivity problems: Using the above Gaussian analogue of the MSW correspondence and following the argumentation in Ref. [39,40], one can easily prove that Gaussian versions of all the above additivity statements would be implied by the super-additivity of the Gaussian entanglement of formation. Let γ be the covariance matrix of a bi-partite Gaussian state consisting of n bi-partite sub-systems 19 with respective reduced covariance matrices [γ] i . Then E G is said to be super-additive if Note that γ is not assumed to be of direct sum structure. If Eq. (58) holds for all covariance matrices γ, then (I.) E G is additive, i.e., E G (γ 1 ⊕ γ 2 ) = E G (γ 1 ) + E G (γ 2 ), (II.) the constrained Gaussian classical capacity is additive, i.e., equality holds in Eq. (51), (III.) the minimal output entropy restricted to Gaussian inputs is additive, i.e., ν 1,G (T 1 ⊗ T 2 ) = ν 1,G (T 1 ) + ν 1,G (T 1 ), (IV.) E G is convex on the level of covariance matrices.
Remarkably, this implication has a simple converse: if E G is additive and convex on the level of covariance matrices, then it is super-additive. To see this, we introduce a local symplectic transformation θ = ½ ⊕ (−½) with block structure as in Eq. (59). Then, for every we have where the inequality is due to the assumed convexity and the last equation reflects additivity of E G . Note that by the above result, if E G is not convex on covariance matrices, then either E F = E G or E F is not additive.

Outlook
This article was concerned with the theory of Gaussian quantum communication channels. Such channels arise in several practical contexts, most importantly as models for lossy fibers. Emphasis was put on questions related to capacities, which give the best possible bounds on the rates that can be achieved when using channels for the communication of quantum or classical information. Though many basic questions have been solved over the last few years, many interesting questions in the theory of bosonic Gaussian channels are essentially open. This applies in particular to additivity issues: general formulae for the classical information capacity are simply not available before a resolution of these issues. For specific channels, a number of methods can yet be applied to find additivity of output purities. It may be interesting to see how far the idea of relating minimal 1-entropies to 2-entropies as in Ref. [46] could be extended in the infinite-dimensional context.
Then, there is the old conjecture that to take Gaussian ensembles does not constitute a restriction of generality anyway when transmitting information through a Gaussian quantum channel. In the light of this conjecture, it would be interesting whether a complete theory 20 E G S(γ 1 ⊕ γ 2 )S T = E G (γ 1 ⊕ γ 2 ) since S is a local unitary. of quantum communication can be formulated, restricting both to Gaussian ensembles and Gaussian channels.
Finally, all what has been stated on capacities in this article refers to the case of memoryless channels. For Gaussian channels with memory, the situation can be quite different. For example, notably, the classical information capacity can be enhanced using entangled instead of product inputs [47,48]. It would in this context also be interesting to see the program of Ref. [49] implemented in the practically important case of Gaussian quantum channels.