Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels

We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the ‘data-processing bound,’ is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the ‘ε-degradable bound’ and the ‘ε-close-degradable bound,’ are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint. What remains open for several interesting channel divergences, such as the diamond norm or the Rényi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.


Introduction
One of the main aims of quantum information theory is to characterize the capacities of quantum communication channels [Hol12,Hay06,Wil16].A quantum channel is a model for a communication link between two parties.The properties of a quantum channel and its coupling to environment govern the evolution of a quantum state that is sent through the channel.
The quantum capacity Q(N ) of a quantum channel N is the maximum rate at which quantum information (qubits) can be reliably transmitted from a sender to a receiver by using the channel many times.The private capacity P (N ) of a quantum channel N is defined to be the maximum rate at which a sender can reliably communicate classical messages to a receiver by using the channel many times, such that the environment of the channel gets negligible information about the transmitted message.In general, the best known characterization of quantum or private capacity of a quantum channel is given by the optimization of regularized information quantities over an unbounded number of uses of the channel [Llo97,Sho02,CWY04,Dev05]. Since these information quantities are additive for a special class of channels called degradable channels [DS05,Smi08], the capacities of these channels can be calculated without any regularization.However, for the channels that are not degradable, these information quantities can be superadditive [DSS98, SS07, SRS08, CEM + 15, ES15], and quantum capacities can be superactivated for some of these channels [SY08,SSY11].Hence, it is difficult to determine the quantum or private capacity of channels that are not degradable, and the natural way to characterize such channels is to bound these capacities from above and below.
An important class of channels called bosonic Gaussian channels act as a good model for the transmission of light through optical fibers or free space (see, e.g., [Ser17] for a review).Within the past two decades, there have been advances in finding quantum and private capacities of bosonic channels.In particular, when there is no constraint on the energy available at the transmitter, the quantum and private capacities of single-mode quantum-limited attenuator and amplifier channels were given in [HW01, WPGG07, WHG12, QW17, WQ16].However, the availability of infinite energy at the transmitter is not practically feasible, and it is thus natural to place energy constraints on any communication protocol.Recently, a general theory of energy-constrained quantum and private communication has been developed in [WQ16], by building on notions developed in the context of other energy-constrained information-processing tasks [Hol04].For the particular case of bosonic Gaussian channels, formulas for the energy-constrained quantum and private capacities of the single-mode pure-loss channel were conjectured in [GSE08] and proven in [WHG12,WQ16].Also, for a single-mode quantum-limited amplifier channel, the energyconstrained quantum and private capacities have been established in [QW17,WQ16].
What remains a pressing open question in the theory of Gaussian quantum information [Ser17] is to determine formulas for or bounds on the quantum and private capacities of non-degradable bosonic Gaussian channels.Of particular interest is the thermal channel, which serves as a variant of the pure-loss channel, incorporating environmental imperfections.In this article, we address this query by providing several bounds on the energy-constrained quantum and private capacities of the thermal channel.
To motivate the thermal channel model, consider that almost all communication systems are affected by thermal noise [Cav82].Even though the pure-loss channel has relevance in free-space communication [YS78,Sha09], it represents an ideal situation in which the environment of the channel is prepared in a vacuum state.Instead, consideration of a thermal state with a fixed mean photon number N B as the state of the environment is more realistic, and such a channel is called a bosonic Gaussian thermal channel [Sha09, RGR + 17].Hence, quantum thermal channels model free-space communication with background thermal radiation affecting the input state in addition to transmission loss.In the context of private communication, a typical conservative model is to allow an eavesdropper access to the environment of a channel, and in particular, tampering by an eavesdropper can be modeled as the excess noise realized by a thermal channel [NH04,LDTBG05].

Summary of results
Some of our main contributions in this paper are upper bounds on the energy-constrained quantum capacity of thermal channels.A first upper bound is established by decomposing a thermal channel as a pure-loss channel followed by a quantum-limited amplifier channel [CGH06, GPNBL + 12] and using a data-processing argument.We note that the same method was employed in [KS13], in order to establish an upper bound on the classical capacity of the thermal channel.Throughout, we call this first upper bound the "data-processing bound."We also prove that this upper bound can be at most 1.45 bits larger than a known lower bound [HW01,WHG12] on the energy-constrained quantum and private capacity of a thermal channel.Moreover, the dataprocessing bound is very near to a known lower bound for the case of low thermal noise and both low and high transmissivity.
Recently, the notion of approximate degradability of quantum channels was developed in [SSWR14], and upper bounds on the quantum and private capacities of approximately degradable channels were established for quantum channels with finite-dimensional input and output systems.In our paper, we establish general upper bounds on the energy-constrained quantum and private capacities of approximately degradable channels for infinite-dimensional systems.These general upper bounds can be applied to any quantum channel that is approximately degradable with energy constraints on the input and output states of the channels.In particular, we apply these general upper bounds to bosonic Gaussian thermal channels.
Our second upper bound is based on the notion of ε-degradability of thermal channels, and we call this bound the "ε-degradable bound."In this method, we first construct a degrading channel, such that a complementary channel of the thermal channel is close in diamond distance [Kit97] to the serial concatenation of the thermal channel followed by this degrading channel.In general, it seems to be computationally hard to determine the diamond distance between two quantum channels if the optimization is over input density operators acting on an infinite-dimensional Hilbert space.However, in our setup, we address this difficulty by constructing a simulating channel, which simulates the serial concatenation of the thermal channel and the aforementioned degrading channel.Using this technique, an upper bound on the diamond distance reduces to the calculation of the quantum fidelity between the environmental states of the thermal channel and the simulating channel.Based on the fact that, for certain parameter regimes, the resulting capacity upper bound is better than all other upper bounds reported here, we believe that our aforementioned choice of a degrading channel is a good choice.
A third upper bound on the energy-constrained quantum capacity of thermal channels is established using the concept of ε-close-degradability of a thermal channel, and we call this bound the "ε-close-degradable bound."In particular, we show that a low-noise thermal channel is ε-close degradable, given that it is close in diamond distance to a pure-loss channel.We find that the εclose-degradable bound is very near to the data-processing bound for the case of low thermal noise.
We compare these three different upper bounds with a known lower bound on the quantum capacity of a thermal channel [HW01,WHG12].We find that the data-processing bound is very near to a known capacity lower bound for low thermal noise and for both medium and high transmissivity.Moreover, we show that the maximum difference between the data-processing bound and a known lower bound never exceeds 1/ ln 2 ≈ 1.45 bits for all possible values of parameters, and this maximum difference is attained in the limit of infinite input mean photon number.This result places a strong limitation on any possible superadditivity of coherent information of the thermal channel.We note here that this kind of result was suggested without proof by the heuristic developments in [SS13].Next, we plot these three upper bounds as well as a known lower bound versus input mean photon number for different values of the channel transmissivity η and thermal noise N B .In particular, we find that the ε-close-degradable bound is very near to the dataprocessing bound for low thermal noise and for both medium and high transmissivity.Moreover, all three upper bounds are very near to a known lower bound for low thermal noise and high transmissivity.We also examine different parameter regimes where the ε-close-degradable bound is tighter than the ε-degradable bound and vice versa.In particular, we find that the ε-degradable bound is tighter than the ε-close degradable bound for the case of high thermal noise.
We find an interesting parameter regime where the ε-degradable bound is tighter than all other upper bounds, as it becomes closest to a known lower bound for the case of high noise and high input mean photon number.However, for the same parameter regime, if the input mean photon number is low, then the data-processing bound is tighter than the ε-degradable bound.This suggests that the upper bounds based on the notion of approximate degradability are good for the case of high input mean photon number.We suspect that these bounds could be further improved for the case of low input mean photon number if it were possible to compute or tightly bound the energy-constrained diamond norm [Shi17,We17].
As one of the last technical developments of our paper, we address this latter question in a very broad sense, by considering the energy-constrained, generalized channel divergence of two quantum channels, as an extension of the generalized channel divergence developed in [LKDW17].In particular, we prove that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint.It is an interesting open question to determine whether the two-mode squeezed vacuum is optimal among all input states, but we leave this for future work, simply noting for now that an answer would lead to improved upper bounds on the energyconstrained quantum and private capacities of the thermal channel.
Similar to our bounds on the energy-constrained quantum capacity, we establish three different upper bounds on the energy-constrained private capacity of bosonic thermal channels.We also develop an improved lower bound on the energy-constrained private capacity of a bosonic thermal channel.In particular, we find that for certain values of the channel transmissivity, a higher private communication rate can be achieved by using displaced thermal states as information carriers instead of coherent states.
The rest of the paper is structured as follows.In Section 3, we summarize definitions and prior results relevant to our paper.We provide general upper bounds on the energy-constrained quantum and private capacities of approximately degradable channels in Section 4. We use these tools to establish three different upper bounds on the energy-constrained quantum and private capacities of a thermal channel in Sections 5 and 7, respectively.A comparision of these different upper bounds on energy-constrained quantum capacity of a thermal channel is discussed in Section 6.We present an improvement on the achievable rate of private communication through thermal channels, in Section 8. We discuss the optimization of the Gaussian energy-constrained generalized channel divergence in Section 9. Finally, we summarize our results and conclude in Section 10.

Preliminaries
Background on quantum information in infinite-dimensional systems is available in [Hol12] (see also [Hol04,SH08,HS10,HZ11,Shi15,Shi16]).In this section, we explain our notations and discuss prior results relevant for our paper.
Quantum states and channels.Let H denote a separable Hilbert space, let B(H) denote the set of bounded operators acting on H, and let P(H) denote the subset of B(H) that consists of positive semi-definite operators.Let T (H) denote the set of trace-class operators, defined such that their trace norm is finite: Let D(H) denote the set of density operators (positive semi-definite with unit trace) acting on H.A quantum channel N : T (H A ) → T (H B ) is a completely positive, trace-preserving linear map.Using the Stinespring dilation theorem [Sti55], a quantum channel can be expressed in terms of a linear isometry: i.e., there exists another Hilbert space H E and a linear isometry U : Quantum entropies and information.The quantum entropy of a state ρ ∈ D(H) is defined as H(ρ) ≡ − Tr{ρ log 2 ρ}.It is a non-negative, concave, lower semicontinuous function [Weh76] and not necessarily finite [BV13].The binary entropy function is defined for x ∈ [0, 1] as (3.1) Throughout the paper we use a function g(x), which is the entropy of a bosonic thermal state with mean photon number x ≥ 0: By continuity, we have that h 2 (0) = lim x→0 h 2 (x) = 0 and g(0 where {|i } ∞ i=1 is an orthonormal basis of eigenvectors of the state ρ, if supp(ρ) ⊆ supp(σ) and D(ρ σ) = ∞ otherwise.The quantum relative entropy D(ρ σ) is non-negative for ρ, σ ∈ D(H) and is monotone with respect to a quantum channel [Lin75] (3.4) The quantum mutual information The coherent information I(A B) ρ of ρ AB is defined as [SN96, HS10, Kuz11] when H(A) ρ < ∞.This expression reduces to Quantum fidelity, trace distance, and diamond distance.The fidelity of two quantum states The trace distance between two density operators ρ, σ ∈ D(H) is equal to ρ − σ 1 .The operational interpretation of trace distance is that it is linearly related to the maximum success probability in distinguishing two quantum states.The diamond norm of a Hermiticity preserving linear map S is defined as , where id R is the identity map acting on Hilbert space H R of the reference system [Kit97].It suffices to optimize with ρ being a pure quantum state.The diamond-norm distance N − M is a measure of distinguishability of two quantum channels N and M.

Approximate degradability. The concept of approximate degradability was introduced in [SSWR14].
The following two definitions of approximate degradability will be useful in our paper.

Definition 3 (Energy observable)
Let G be a positive semi-definite operator.We assume that it has discrete spectrum and that it is bounded from below.In particular, let {|e k } k be an orthonormal basis for a Hilbert space H, and let {g k } k be a sequence of non-negative real numbers bounded from below.Then is a self-adjoint operator that we call an energy observable.

Definition 4 (Extension of energy observable)
The nth extension G n of an energy observable G is defined as where n is the number of factors in each tensor product above.
For a Gibbs observable G, let us consider a quantum state ρ such that Tr{Gρ} ≤ W .There exists a unique state that maximizes the entropy H(ρ), and this unique maximizer has the Gibbs form γ(W ) = exp(−β(W )G)/Z(β(W )), where β(W ) is the solution of the equation: Tr{exp(−βG)(G − W )} = 0.In particular, for the Gibbs observable G = ωn, where n = â † â is the photon number operator, a thermal state (mean photon number n) that saturates the energy constrained inequality Tr{Gρ} ≤ W , gives the maximum value of the entropy: Here, we have fixed the ground-state energy to be equal to zero.In some parts of our paper, we take the Gibbs observable to be the number operator, and we use the terminology "mean photon number" and "energy" interchangeably.
The following lemma is a uniform continuity bound for the conditional quantum entropy with energy constraints [Win16]: (3.11) Throughout the paper, we consider only those quantum channels that satisfy the following finite output entropy condition: Condition 6 (Finite output entropy) Let G be a Gibbs observable and W ∈ [0, ∞).A quantum channel N satisfies the finite-output entropy condition with respect to G and W if (3.12) Gaussian states and channels.We now deliver a brief review of Gaussian states and channels, and we point to [Ser17] for more details.Gaussian channels model natural physical processes such as photon loss, photon amplification, thermalizing noise, or random kicks in phase space.They satisfy Condition 6 when the Gibbs observable for m modes is taken to be where ω j > 0 is the frequency of the jth mode and âj is the photon annihilation operator for the jth mode, so that â † j âj is the photon number operator for the jth mode.Let x ≡ [q 1 , . . ., qm , p1 , . . ., pm ] ≡ [x 1 , . . ., x2m ] (3.14) denote a vector of position-and momentum-quadrature operators, satisfying the canonical commutation relations: and I m denotes the m × m identity matrix.We take the annihilation operator for the jth mode as âj = (q j + ip j )/ √ 2. For ξ ∈ R 2m , we define the unitary displacement operator D(ξ) ≡ exp(iξ T Ωx).Displacement operators satisfy the following relation: (3.16) Every state ρ ∈ D(H) has a corresponding Wigner characteristic function, defined as and from which we can obtain the state ρ as (3.18) A quantum state ρ is Gaussian if its Wigner characteristic function has a Gaussian form as where µ ρ is the 2m × 1 mean vector of ρ, whose entries are defined by µ ρ j ≡ xj ρ and V ρ is the 2m × 2m covariance matrix of ρ, whose entries are defined as The following condition holds for a valid covariance matrix: V + iΩ ≥ 0, which is a manifestation of the uncertainty principle [SMD94].
A thermal Gaussian state θ β of m modes with respect to Êm from (3.13) and having inverse temperature β > 0 thus has the following form: and has a mean vector equal to zero and a diagonal 2m × 2m covariance matrix.One can calculate that the photon number in this state is equal to j 1 e βω j − 1 . (3.22) A single-mode thermal state with mean photon number n = 1/(e βω − 1) has the following representation in the photon number basis: It is also well known that thermal states can be written as a Gaussian mixture of displacement operators acting on the vacuum state: where p(ξ) is a zero-mean, circularly symmetric Gaussian distribution.From this, it also follows that randomly displacing a thermal state in such a way leads to another thermal state of higher temperature: where β ≥ β and q(ξ) is a particular circularly symmetric Gaussian distribution.
In our paper, we employ the two-mode squeezed vacuum state with parameter n, which is equivalent to a purification of the thermal state in (3.23) and is defined as (3.26) A 2m × 2m matrix S is symplectic if it preserves the symplectic form: SΩS T = Ω.According to Williamson's theorem [Wil36], there is a diagonalization of the covariance matrix V ρ of the form, where S ρ is a symplectic matrix and D ρ ≡ diag(ν 1 , . . ., ν m ) is a diagonal matrix of symplectic eigenvalues such that ν i ≥ 1 for all i ∈ {1, . . ., m}.Computing this decomposition is equivalent to diagonalizing the matrix The entropy H(ρ) of a quantum Gaussian state ρ is a direct function of the symplectic eigenvalues of its covariance matrix V ρ [Ser17]: where g(•) is defined in (3.2).The Hilbert-Schmidt adjoint of a Gaussian quantum channel N X,Y from l modes to m modes has the following effect on a displacement operator D(ξ) [Ser17]: where X is a real 2m × 2l matrix, Y is a real 2m × 2m positive semi-definite matrix, and d ∈ R 2m , such that they satisfy The effect of the channel on the mean vector µ ρ and the covariance matrix V ρ is thus as follows: All Gaussian channels are covariant with respect to displacement operators.That is, the following relation holds and note that D(Xξ) is a tensor product of local displacement operators.
Just as every quantum channel can be implemented as a unitary transformation on a larger space followed by a partial trace, so can Gaussian channels be implemented as a Gaussian unitary on a larger space with some extra modes prepared in the vacuum state, followed by a partial trace [CEGH08].Given a Gaussian channel N X,Y with Z such that Y = ZZ T we can find two other matrices X E and Z E such that there is a symplectic matrix which corresponds to the Gaussian unitary transformation on a larger space.The complementary channel NX E ,Y E from input to the environment then effects the following transformation on mean vectors and covariance matrices: where

Quantum thermal channel.
A quantum thermal channel is a Gaussian channel that can be characterized by a beamsplitter of transmissivity η, coupling the signal input state with a thermal state with mean photon number N B .In the Heisenberg picture, the beamsplitter transformation is given by the following Bogoliubov transformation: where â, b, ê, and ê are the annihilation operators representing the sender's input mode, the receiver's output mode, an environmental input mode, and an environmental output mode of the channel, respectively.Throughout the paper, we represent the thermal channel by L η,N B .If the mean photon number at the input of a thermal channel is no larger than N S , then the total number of photons that make it through the channel to the receiver is no larger than ηN Continuity of output entropy.The following theorem on continuity of output entropy for infinitedimensional systems with finite average energy constraints is a direct consequence of [LS09, Theorem 11] and Lemma 1.
Theorem 7 Let N A→B and M A→B be quantum channels, G ∈ P(H B ) be a Gibbs observable, such that and consider the following chain of inequalities: The first inequality follows from the triangle inequality.The second equality follows from the fact that the states ρ j and ρ j−1 are the same except for the jth output system.Let W j denote an energy constraint on the jth output state of both the channels N and M, i.e., Tr{GN (ρ A j )}, Tr{GM(ρ A j )} ≤ W j and 1 n j W j ≤ W . Then the second inequality follows because 1 2 ρ j − ρ j−1 1 ≤ ε for the given channels, and we use Lemma 1 for the jth output system.The third inequality follows from concavity of entropy.The final inequality follows because and γ(W/δ) is the Gibbs state that maximizes the entropy corresponding to the energy W/δ.

Continuity of capacities for channels.
The continuity of various capacities of quantum channels has been discussed in [LS09, Lemma 12].The general form for the classical, quantum, or private capacity of a channel N can be defined as , where {f n } n denotes a family of functions, and P (n) represents states or parameters over which an optimization is performed.Then the following lemma holds [LS09].
Energy-constrained quantum and private capacities.The energy-constrained quantum and private capacities of quantum channels have been defined in [WQ16,Section III].In what follows, we review the definition of quantum communication and private communication codes, achievable rates, and regularized formulas for energy-constrained quantum and private capacities.

Energy-constrained quantum capacity. An (n, M, G, W, ε) code for energy-constrained quantum communication consists of an encoding channel E n : T (H S ) → T (H ⊗n
A ) and a decoding channel , where M = dim(H S ).The energy constraint is such that the following bound holds for all states resulting from the output of the encoding channel E n : where where (3.51) due to the i.i.d.nature of the observable G n .Furthermore, the quantum communication code satisfies the following reliability condition such that for all pure states φ RS ∈ D(H R ⊗ H S ), where H R is isomorphic to H S .A rate R is achievable for quantum communication over N subject to the energy constraint W if for all ε ∈ (0, 1), δ > 0, and sufficiently large n, there exists an of N is equal to the supremum of all achievable rates.If the channel N satisfies Condition 6 and G is a Gibbs observable, then the quantum capacity Q(N , G, W ) is equal to the regularized energy-constrained coherent information of the channel N where the energy-constrained coherent information of the channel is defined as [WQ16] and N denotes a complementary channel of N .Note that another definition of energy-constrained quantum communication is possible, but it leads to the same value for the capacity in the asymptotic limit of many channel uses [WQ16].
for all m ∈ {1, . . ., M }, with ω E n some fixed state in D(H ⊗n E ).In the above, N is a channel complementary to N .A rate R is achievable for private communication over N subject to energy constraint W if for all ε ∈ (0, 1), δ > 0, and sufficiently large n, there exists an (n, 2 n[R−δ] , G, W, ε) private communication code.The energy-constrained private capacity P (N , G, W ) of N is equal to the supremum of all achievable rates.
An upper bound on the energy-constrained private capacity of a channel has been established in [WQ16], but the lower bound still needs a detailed proof.However, the results in [WQ16] suggest the validity of the following form.If the channel N satisfies Condition 6 and G is a Gibbs observable, then the energy-constrained private capacity P (N , G, W ) is given by the regularized energy-constrained private information of the channel: where the energy-constrained private information is defined as and ρE A ≡ dx p X (x)ρ x A is an average state of the ensemble and N denotes a complementary channel of N .Note that another definition of energy-constrained private communication is possible, but it leads to the same value for the capacity in the asymptotic limit of many channel uses [WQ16].

Bounds on energy-constrained quantum and private capacities of approximately degradable channels
In this section, we derive upper bounds on the energy-constrained quantum and private capacities of approximately degradable channels.We derive these bounds for both ε-degradable (Definition 1) and ε-close-degradable (Definition 2) channels.This general form for the upper bounds on the energy-constrained quantum and private capacities of approximately degradable channels will be directly used in establishing bounds on the capacities of quantum thermal channels.We begin by defining the conditional entropy of degradation, which will be useful for finding upper bounds on the energy-constrained quantum and private capacities of an ε-degradable channel.A similar quantity has been defined for the finite-dimensional case in [SSWR14].

Definition 8 (Conditional entropy of degradation)
Let N A→B and D B→E be quantum channels, and let G ∈ P(H A ) be a Gibbs observable.We define the conditional entropy of degradation as follows: where We note that the conditional entropy of degradation can be understood as the negative entropy gain of the channel D B→E [Ali04, Hol10, Hol11a, Hol11b], with the optimization over input states N (ρ) restricted to being in the image of N and obeying the energy constraint Tr{Gρ} ≤ W . Next, we show that the conditional entropy of degradation in (4.2) is additive.
Lemma 3 Let N A→B and D B→E be quantum channels, let G ∈ P(H A ) be a Gibbs observable, and let W ∈ [0, ∞).Then for all integer n ≥ 1 Proof.The following inequality follows trivially because a product input state is a particular state of the form required in the optimization of U D ⊗n (N ⊗n , G n , W ). We now prove the less trivial inequality Consider the following chain of inequalities: where ρn = 1 n n i=1 ρ A i .The first inequality follows from several applications of strong subadditivity [LR73b,LR73a].The second inequality follows from concavity of conditional entropy [LR73b,LR73a].The last inequality follows because Tr{G n ρ A n } = Tr{Gρ n } ≤ W and the conditional entropy of degradation U D (N , G, W ) involves an optimization over all input states obeying this energy constraint.Since the chain of inequalities is true for all input states ρ A n satisfying the input energy constraint, the desired result follows.

Bound on the energy-constrained quantum capacity of an ε-degradable channel
An upper bound on the quantum capacity of an ε-degradable channel was established as [SSWR14, Theorem 3.1(ii)] for the finite-dimensional case.Here, we prove a related bound for the infinitedimensional case with finite average energy constraints on the input and output states of the channels.
Theorem 9 Let N A→B be an ε-degradable channel with a degrading channel D B→E , and let G ∈ P(H A ) and G ∈ P(H E ) be Gibbs observables, such that for all input states ρ A n ∈ D(H ⊗n A ) satisfying input average energy constraints Tr{G n ρ A n } ≤ W , the following output average energy constraints are satisfied: where NA→E is a complementary channel of N and E E. Then the energy-constrained quantum capacity Q(N , G, W ) is bounded from above as and consider the following chain of inequalities: The first inequality follows from the definition in (4.1).The second equality follows from Lemma 3 and the telescoping technique.Let W j denote the energy constraint on the jth output state of both the channels D • N and N , i.e., Tr{G (D • N )(ρ A j )}, Tr{G N (ρ A j )} ≤ W j where 1 n j W j ≤ W . Then the second inequality holds because 1 2 ρ j − ρ j−1 1 ≤ ε for the given channels, and we use Lemma 1 for the jth output system.The third inequality follows from concavity of entropy.The last inequality follows because Tr{ 1 n n j=1 Gγ(W j /δ)} = 1 n n j=1 Tr{Gγ(W j /δ)} ≤ W /δ, and γ(W /δ) is the Gibbs state that maximizes the entropy corresponding to the energy W /δ. Since the chain of inequalities is true for all ρ A n satisfying the input average energy constraint, from (3.54) and the above, we get that Since the last inequality holds for all n, we obtain the desired result by taking the limit n → ∞ and applying (3.53).

Bound on the energy-constrained quantum capacity of an ε-close-degradable channel
An upper bound on the quantum capacity of an ε-close-degradable channel was established as [SSWR14, Proposition A.2(i)] for the finite-dimensional case.Here, we provide a bound for the infinite-dimensional case with finite average energy constraints on the input and output states of the channels.
Theorem 10 Let N A→B be an ε-close-degradable channel, i.e., 1 2 N − M ≤ ε < ε ≤ 1, where M A→B is a degradable channel.Let G ∈ P(H A ), G ∈ P(H B ) be Gibbs observables, such that for all input states ρ RA n ∈ D(H R ⊗ H ⊗n A ) satisfying the input average energy constraint Tr{G n ρ A n } ≤ W , the following output average energy constraints are satisfied: where W, W ∈ [0, ∞).Then the energy-constrained quantum capacity Q(N , G, W ) is bounded from above as and consider the following chain of inequalities: The first inequality follows from applying Theorem 7 twice.Then from Lemma 2, The desired result follows from the fact that the energy-constrained quantum capacity of a degradable channel is equal to the energy-constrained coherent information of the channel [WQ16].

Bound on the energy-constrained private capacity of an ε-degradable channel
In this section, we first derive an upper bound on the private capacity of an ε-degradable channel for the finite-dimensional case, which is different from any of the bounds presented in [SSWR14].
Then, we generalize this bound to the infinite-dimensional case with finite average energy constraints on the input and output states of the channels.
Theorem 11 Let N A→B be a finite-dimensional ε-degradable channel with a degrading channel D B→E , and let N : then the private capacity P (N ) of N is bounded from above as Proof.Consider Stinespring dilations U : T (A) → T (B) ⊗ T (E) and V : T (B) → T (E ) ⊗ T (F ) of the channel N and the degrading channel D, respectively.Let ρ XA n be a classical-quantum state in correspondence with an ensemble {p X (x), ρ x A n }: and let Consider the following extension of ω XE n E n F n : where ψ x,y A n is a pure state, and let Consider the following chain of inequalities: The first two equalities follow from entropy identities.The first inequality follows by applying the telescoping technique twice and using the continuity result of the conditional quantum entropy for finite-dimensional quantum systems [Win16].The second inequality follows from the quantum data processing inequality for conditional quantum mutual information.The last two equalities follow from entropy identities and by using that σ x,y E n E n F n is a pure state, so that H(F n E n ) σ x,y = H(E n ) σ x,y .The last inequality follows from the definition in (4.23), and additivity of U D (N ) [SSWR14].Also, we applied the telescoping technique for each σ x,y in the summation, and used the continuity result of the conditional quantum entropy for finite-dimensional systems [Win16].Since the chain of inequalities is true for any ensemble {p X (x), ρ x A n }, the final result follows from the definition of private information of the channel, dividing by n, taking the limit n → ∞, and noting that the regularized private information is equal to the private capacity of any channel.
Next, we derive an upper bound on the energy-constrained private capacity of an ε-degradable channel.
Theorem 12 Let N A→B be an ε-degradable channel with a degrading channel D B→E , and let G ∈ P(H A ), G ∈ P(H E ) be Gibbs observables, such that for all input states ρ A n ∈ D(H ⊗n A ) satisfying input average energy constraints Tr{G n ρ A n } ≤ W , the following output average energy constraints are satisfied: where NA→E is a complementary channel of N , and E E. Then the energy-constrained private capacity is bounded from above as Proof.Since the proof is similar to the above one and previous ones, we just summarize it briefly below.Consider Stinespring dilations U : T (A) → T (B) ⊗ T (E) and V : T (B) → T (E ) ⊗ T (F ) of the channel N and the degrading channel D, respectively.Then the action of U ⊗n followed by V ⊗n on the ensemble {p X (x), ρ x A n } leads to the following ensemble: (4.37) Similar to the above proof, from applying the telescoping technique three times and using Lemma 1, concavity of entropy, and Lemma 3, we get the following bound: The desired result follows from dividing by n, taking the limit n → ∞, the definition of the energyconstrained private information of the channel, and using the fact that the regularized energyconstrained private information is an upper bound on the energy-constrained private capacity of a quantum channel [WQ16].

Bound on the energy-constrained private capacity of an ε-close-degradable channel
An upper bound on the private capacity of an ε-close-degradable channel was established as [SSWR14, Proposition A.2(ii)] for the finite-dimensional case.Here, we provide a bound for the infinite-dimensional case with finite average energy constraints on the input and output states of the channels.
Theorem 13 Let N A→B be an ε-close-degradable channel, i.e., 1 2 N − M ≤ ε < ε ≤ 1, where M A→B is a degradable channel.Let G ∈ P(H A ), G ∈ P(H B ) be Gibbs observables, such that for all input states ρ A n ∈ D(H ⊗n A ) satisfying input average energy constraints Tr{G n ρ A n } ≤ W , the following output average energy constraints are satisfied: where W, W ∈ [0, ∞).Then Proof.We follow the proof of [LS09, Corollary 15] closely, but incorporate energy constraints.Consider Stinespring dilations U : T (A) → T (B) ⊗ T (E) and V : T (A) → T (B) ⊗ T (E) of the channels N and M, respectively.Consider an input ensemble {p X (x), ρ x A n }, which leads to the output ensembles Supposing at first that the index x is discrete, from four times applying Theorem 7 and employing the same expansions as in the proof of [LS09, Corollary 15] , we get The upper bound is uniform and has no dependence on the particular ensemble except via the energy constraints.Thus, by approximation, the same bound applies to ensembles for which the index x is continuous.Then from Lemma 2, we find that The equality in the last line follows from the fact that the energy-constrained private capacity of a degradable channel is equal to the energy-constrained coherent information of the channel [WQ16].

Upper bounds on energy-constrained quantum capacity of bosonic thermal channels
In this section, we establish three different upper bounds on the energy-constrained quantum capacity of a thermal channel: 1. We establish a first upper bound using the theorem that any thermal channel can be decomposed as the concatenation of a pure-loss channel followed by a quantum-limited amplifier channel [CGH06, GPNBL + 12].We call this bound the data-processing bound and denote it by Q U 1 .
2. Next, we show that a thermal channel is an ε-degradable channel for a particular choice of degrading channel.Then an upper bound on the energy-constrained quantum capacity of a thermal channel directly follows from Theorem 9. We call this bound the ε-degradable bound and denote it by Q U 2 .
3. We establish a third upper bound on the energy-constrained quantum capacity of a thermal channel using the idea of ε-close-degradability.We show that the thermal channel is ε-close to a pure-loss bosonic channel for a particular choice of ε.Since a pure-loss bosonic channel is a degradable channel [WPGG07], the bound on the energy-constrained quantum capacity of a thermal channel follows directly from Theorem 10.We call this bound the ε-close-degradable bound and denote it by Q U 3 .
In Section 6, we compare, for different parameter regimes, the closeness of these upper bounds with a known lower bound on the quantum capacity of thermal channels.

Data-processing bound on the energy-constrained quantum capacity of bosonic thermal channels
In this section, we provide an upper bound using the theorem that any thermal channel L η,N B can be decomposed as the concatenation of a pure-loss channel L η ,0 with transmissivity η followed by a quantum-limited amplifier channel A G with gain G [CGH06, GPNBL + 12], i.e., where G = (1 − η)N B + 1, and η = η/G.In Theorem 22, we prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound.
Theorem 14 An upper bound on the quantum capacity of a thermal channel L η,N B with transmissivity η ∈ [1/2, 1], environment photon number N B , and input mean photon number constraint N S is given by (5.3) Proof.An upper bound on quantum capacity can be established by using (5.1) and a dataprocessing argument.We find that The first inequality follows from definitions and data processing-the energy-constrained capacity of A G • L η ,0 cannot exceed that of L η ,0 .The second equality follows from the formula for the energy-constrained quantum capacity of a pure-loss bosonic channel with transmissivity η and input mean photon number N S [WHG12, WQ16].

ε-degradable bound on the energy-constrained quantum capacity of bosonic thermal channels
In this section, we provide an upper bound on the energy-constrained quantum capacity of a thermal channel using the idea of ε-degradability.In Theorem 9, we established a general upper bound on the energy-constrained quantum capacity of an ε-degradable channel.Hence, our first step is to construct the degrading channel D given in (5.14), such that the concatenation of a thermal channel L η,N B followed by D is close in diamond distance to the complementary channel Lη,N B of the thermal channel L η,N B .We start by motivating the reason for choosing the particular degrading channel in (5.14), which is depicted in Figure 1, and then we find an upper bound on the diamond distance between D • L η,N B and Lη,N B .In general, it is computationally hard to perform the optimization over an infinite dimensional space required in the calculation of the diamond distance between Gaussian channels.However, we address this problem in this particular case by introducing a channel that simulates the serial concatenation of the thermal channel and the degrading channel, and we call it the simulating channel, as given in (5.18).This allows us to bound the diamond distance between the channels from above by the trace distance between the environment states of the complementary channel and the simulating channel (Theorem 15).Next, we argue that, for a given input mean photon-number constraint N S , a thermal state with mean photon number N S maximizes the conditional entropy of degradation defined in (4.2), which also appears in the general upper bound established in Theorem 9. We finally provide an upper bound on the energy-constrained quantum capacity of a thermal channel by using all these tools and invoking Thereom 9.
We now establish an upper bound on the diamond distance between the complementary channel of the thermal channel and the concatenation of the thermal channel followed by a particular degrading channel.Let B and B represent beamsplitter transformations with transmissivity η and (1 − η)/η, respectively.In the Heisenberg picture, the beamsplitter transformation (5.8) Similarly, the beamsplitter transformation where R is a reference system and ψ TMS (N B ) E E 1 is a two-mode squeezed vacuum state with parameter N B , as defined in (3.26).
Here and what remains in the proof, we consider the action of various transformations on the covariance matrices of the states involved, and we furthermore track only the submatrices corresponding to the position-quadrature operators of the covariance matrices.It suffices to do so because all channels involved in our discussion are phase-insensitive Gaussian channels.
The submatrix corresponding to the position-quadrature operators of the covariance matrix of ψ TMS (N B ) E E 1 has the following form: (5.12) The action of a complementary channel Lη,N B on an input state φ RA is given by (5.13) It can be understood from Figure 1 that the system R is correlated with the input system A for the channel, and the system E is the environment's input.The beamsplitter transformation B then leads to systems B and E 2 .Hence, the output of the thermal channel L η,N B is system B, and the outputs of the complementary channel Lη,N B are systems E 1 and E 2 .Our aim is to introduce a degrading channel D, such that the combined state of R and the output of D • L η,N B emulate the combined state of R, E 1 , and E 2 , to an extent.This will then allow us to bound the diamond distance between D • L η,N B and Lη,N B from above.For the case when there Figure 1: The figure plots a thermal channel with transmissivity η ∈ [1/2, 1] and a degrading channel as described in (5.14).φ RA is an input state to the beamsplitter B with transmissivity η and ψ TMS (N B ) represents a two-mode squeezed vacuum state with parameter N B .System B is the output of the thermal channel, and systems E 1 E 2 are the outputs of the complementary channel.The second beamsplitter B has transmissivity (1 − η)/η, and system B acts as an input to B .Systems E 1 E 2 represent the output systems of the degrading channel, whose action is to tensor in the state ψ TMS (N B ) F E 1 , interact the input system B with F according to B , and then trace over system G. is no thermal noise, i.e., N B = 0, a thermal channel reduces to a pure-loss channel.Moreover, we know that a pure-loss channel is a degradable channel and the corresponding degrading channel can be realized by a beamsplitter with transmissivity (1 − η)/η [GSE08].Hence, we consider a degrading channel, such that it also satisfies the conditions for the above described special case.
Consider a beamsplitter with transmissivity (1 − η)/η and the beamsplitter transformation B from (5.9)-(5.10).As described in Figure 1, the output B of the thermal channel L η,N B becomes an input to the beamsplitter B .We consider one mode (F in Figure 1) of the two-mode squeezed vacuum state ψ TMS (N B ) F E 1 as an environmental input for B , so that the subsystem E 1 mimics E 1 .Hence, our choice of degrading channel seems reasonable, as the combined state of system R and output systems E 1 , E 2 of D • L η,N B emulates the combined state of R, E 1 , and E 2 , to an extent.We suspect that our choice of degrading channel is a good choice because an upper bound on the energy-constrained quantum capacity of a thermal channel using this technique outperforms all other upper bounds for certain parameter regimes.We denote our choice of degrading channel by D (1−η)/η,N B : T (B) → T (E 1 ) ⊗ T (E 2 ).More formally, D (1−η)/η,N B has the following action on the output state L η,N B (φ RA ): (5.14) Next, we provide a strategy to bound the diamond distance between D (1−η)/η,N B • L η,N B and Lη,N B .Consider the following submatrix corresponding to the position-quadrature operators of the covariance matrix of an input state φ RA : where a, b, c ∈ R are such that the above is the position-quadrature part of a legitimate covariance matrix.Let ξ RE 2 E 2 E 1 GE 1 denote the state after the beamsplitter transformations act on an input state φ RA : (5.16) Then the submatrix corresponding to the position-quadrature operators of the covariance matrix of the output state in (5.14) is given by [Not]: (5.17) Now, we introduce a particular channel that simulates the action of D (1−η)/η,N B • L η,N B on an input state φ RA .We denote this channel by Ξ, and it has the following action on an input state φ RA : where ω(N B ) E E 1 represents a noisy version of a two-mode squeezed vacuum state with parameter N B and has the following submatrix corresponding to the position-quadrature operators of the covariance matrix: Figure 2: The figure plots the simulating channel Ξ described in (5.18).φ RA is an input state to a beamsplitter B with transmissivity η and ω(N B ) represents a noisy version of a two-mode squeezed vacuum state with parameter N B (see (5.19)), one mode of which is an input to the environment mode of the beamsplitter.The simulating channel is such that system B is traced over, so that the channel outputs are E 1 and E 2 .Finally, the simulating channel is exactly the same as the channel from system A to systems E 1 E 2 in Figure 1.
The matrix V in (5.19) is a well defined submatrix of the covariance matrix for the noisy version of a two-mode squeezed vacuum state, because The submatrix of the covariance matrix corresponding to the state in (5.18) is the same as the submatrix in (5.17) [Not].In other words, the covariance matrix for the sytems R, E 1 , and E 2 in Figure 1 is exactly the same as the covariance matrix for the systems R, E 1 , and E 2 in Figure 2.This equality of covariance matrices is sufficient to conclude that the following equivalence holds for any quantum input state φ RA (see [Ser17, Chapter 5] for a proof): (5.20) Thus, the channels D (1−η)/η,N B • L η,N B and Ξ are indeed the same.From (5.13), (5.18), and (5.20), the action of both Lη,N B and Ξ can be understood as tensoring the state of the environment with the input state of the channel, performing the beamsplitter transformation B, and then tracing out the output of the channels.Using these techniques, we now establish an upper bound on the diamond distance between the complementary channel in (5.13) and the concatenation of the thermal channel followed by the degrading channel in (5.14).
Theorem 15 Fix η ∈ [1/2, 1].Let L η,N B be a thermal channel with transmissivity η, and let D (1−η)/η,N B be a degrading channel as defined in (5.14).Then (5.22) Proof.Consider the following chain of inequalities: The first equality follows from (5.20).The second equality follows from (5.13) and (5.18).The first inequality follows from monotonicity of the trace distance.The third equality follows from invariance of the trace distance under a unitary transformation (beamsplitter).The last inequality follows from the Powers-Stormer inequality [PS70].
Next, we compute the fidelity between ψ TMS (N B ) E E 1 and ω(N B ) E E 1 by using their respective covariance matrices in (5.12) and (5.19), in the Uhlmann fidelity formula for two-mode Gaussian states [MM12].We find [Not] (5.29) Since these inequalities hold for any input state φ RA , the final result follows from the definition of the diamond norm.
Theorem 16 An upper bound on the quantum capacity of a thermal channel L η,N B with transmissivity η ∈ [1/2, 1], environment photon number N B , and input mean photon-number constraint N S is given by (5.31) Proof.From Theorem 15, we have an upper bound on the diamond distance between the complementary channel of the thermal channel and the concatenation of the thermal channel followed the degrading channel, i.e., (5.36) Using Proposition 17, we find that the thermal state with mean photon number N S optimizes the conditional entropy of degradation U D (1−η)/η,N B (L η,N B , N S ).For the given thermal channel in (5.11) and the degrading channel in (5.14), we find the following analytical expression [Not]: (5.37) with ζ ± defined as in the theorem statement.
Proposition 17 Let L η,N B be a thermal channel with transmissivity η ∈ [1/2, 1], environment photon number N B , and input mean photon number constraint N S .Let D (1−η)/η,N B be the degrading channel from (5.14).Then the thermal state with mean photon number N S optimizes the conditional entropy of degradation U D (1−η)/η,N B (L η,N B , N S ), defined from (4.2).
Proof.Consider the Stinespring dilation in (5.16) of the degrading channel D (1−η)/η,N B from (5.14), and denote it by W. Then according to (4.2), (5.38) Our aim is to find an input state ρ with a certain photon number N t ≤ N S , such that it maximizes the conditional entropy in (5.38).From the extremality of Gaussian states applied to the conditional entropy [EW07], it suffices to perform the optimization in (5.38) over only Gaussian states.Now, we argue that for a given input mean photon number N t , a thermal state is the optimal state for the conditional output entropy in (5.38).For a thermal channel and our choice of a degrading channel, a phase rotation on the input state is equivalent to a product of local phase rotations on the outputs.Let us denote the state after the local phase rotations on the outputs by (5.39) and let (5.40) Note that the phase covariance property mentioned above is the statement that the following equality holds for all φ ∈ [0, 2π) [Not]: (5.41) Consider the following chain of inequalities for a Gaussian input state ρ: The first equality follows from invariance of the conditional entropy under local unitaries.The second equality follows from the phase covariance property of the channel.The inequality follows from concavity of conditional entropy.The last equality follows from linearity of the channel, and the following identity: dφ e iφn ρe −iφn . (5.46) In (5.46), the state after the phase averaging is diagonal in the number basis, and furthermore, the resulting state has the same photon number N t as the Gaussian state ρ.The thermal state θ(N t ) is the only Gaussian state of a single mode that is diagonal in the number basis with photon number equal to N t .
Next, we argue that, for a given photon number constraint, a thermal state that saturates the constraint is the optimal state for the conditional output entropy.Let (5.47) Consider the following chain of inequalities: where q N (α) = exp{−|α| 2 /N }/πN is a complex-centered Gaussian distribution with variance N ≥ 0. The first equality follows by placing a probability distribution in front, and the second follows from invariance of the conditional entropy under local unitaries.The third equality follows because the channel is covariant with respect to displacement operators, as reviewed in (3.33).
The last inequality follows from concavity of conditional entropy, and from the fact that a thermal state with a higher mean photon number can be realized by random Gaussian displacements of a thermal state with a lower mean photon number, as reviewed in (3.25).Hence, for a given input mean photon number constraint N S , a thermal state with mean photon number N S optimizes the conditional entropy of degradation defined from (4.2).

Remark 18
The arguments used in the proof of Proposition 17 can be employed in more general situations beyond that which is discussed there.The main properties that we need are the following, when the channel involved takes a single-mode input to a multi-mode output: • The channel should be phase covariant, such that a phase rotation on the input state is equivalent to a product of local phase rotations on the output.
• The channel should be covariant with respect to displacement operators, such that a displacement operator acting on the input state is equivalent to a product of local displacement operators on the output.
• The function being optimized should be invariant with respect to local unitaries and concave in the input state.
If all of the above hold, then we can conclude that the thermal-state input saturating the energy constraint is an optimal input state.We employ this reasoning again in the proof of Theorem 22.

ε-close-degradable bound on the energy-constrained quantum capacity of bosonic thermal channels
In this section, we first establish an upper bound on the diamond distance between a thermal channel and a pure-loss channel.Since a pure-loss channel is a degradable channel, an upper For simplicity, we denote (1 − η)N B as Y , employ the natural logarithm for g(x), and omit the prefactor 1/ ln 2 from all instances of g(x).We use the following property of the function g(x): For large x, g(x) = ln(x + 1) + 1 + O(1/x), (6.4) so that as x → ∞, the approximation g(x) ≈ ln(x + 1) + 1 holds.Using (6.4), the data-processing bound in (5.3) can be expressed as follows for large N S : Similarly, the lower bound Q L in (6.1) can be expressed as Then the difference simplifies as The second equality follows from the definition of D 2 .Next, we show that as N S → ∞, and hence we get the desired result.Consider the following expression and take the limit N S → ∞: After incorporating the 1/ ln 2 factor, which was omitted earlier for simplicity, we find that the difference between the upper and lower bounds approaches 1/ ln 2 (≈ 1.45 bits) as N S → ∞.Now, we show that the difference ∆(L η,N B , N S ) is a monotone increasing function with respect to input mean photon number N S ≥ 0. Let U η A→B 1 E 1 and V G B 1 →B 2 E 2 denote Stinespring dilations of a pure-loss channel L η ,0 : A → B 1 and a quantum limited amplifier channel A G : B 1 → B 2 , respectively.For the energy-constrained quantum capacity of a pure-loss channel, the thermal state as an input is optimal for any fixed energy or input mean photon number constraint N S [WHG12].Moreover, the lower bound in (6.1) is obtained for a thermal state with mean photon number N S as input to the channel.Then the action of a thermal channel L η,N B on an input state θ(N S ) can be expressed as Consider the following state: Since the data-processing bound Q U 1 (L η,N B , N S ) is equal to the quantum capacity of a pure-loss channel with transmissivity η , which in turn is equal to coherent information for this case, (5.3) can also be represented as Similarly, the lower bound can be expressed as Hence the difference between (6.17) and (6.18) is given by Now, our aim is to show that the conditional entropies in (6.19) are monotone increasing functions of N S .We employ displacement covariance of the channels, and note that this argument is similar to that used in the proof of Proposition 17.Let Let N S − N S ≥ 0, and consider the following chain of inequalities: The first equality follows by placing a probability distribution in front, and the second follows from invariance of the conditional entropy under local unitaries.The third equality follows because the channel is covariant with respect to displacement operators, as reviewed in (3.33).The last inequality follows from concavity of conditional entropy, and from the fact that a thermal Figure 3: The figures plot the data-processing bound (Q U 1 ), the ε-degradable bound (Q U 2 ), the ε-close-degradable bound (Q U 3 ) and the lower bound (Q L ) on energy-constrained quantum capacity of thermal channels.In each figure, we select certain values of η and N B , with the choices indicated above each figure.In all the cases, the data-processing bound Q U 1 is close to the lower bound Q U L .In (a), for medium transmissivity and low thermal noise, the ε-close-degradable bound is close to the data-processing bound, and they are tighter than the ε-degradable bound.In (b), for medium transmissivity and high thermal noise, only the data-processing bound is close to the lower bound.Also the ε-degradable bound is tighter than the ε-close-degradable bound.In (c), for high transmissivity and low thermal noise, all upper bounds are very near to the lower bound.In (d), for high transmissivity and high noise, the ε-degradable bound is tighter than the ε-close-degradable bound.
state with a higher mean photon number can be realized by random Gaussian displacements of a thermal state with a lower mean photon number, as reviewed in (3.25).Hence, the difference between the data-processing bound in (5.3) and the lower bound in (6.1) attains its maximum value in the limit N S → ∞.Next, we perform numerical evaluations to see how close the three different upper bounds are to the lower bound Q L in (6.1).Since there is a free parameter ε in both the ε-degradable bound in (5.30) and the ε-close-degradable bound in (5.61), we optimize these bounds with respect to ε [Not].In Figure 3, we plot the data-processing bound Q U 1 , the ε-degradable bound Q U 2 , the ε-close-degradable bound Q U 3 and the lower bound Q L versus N S for certain values of the transmissivity η and thermal noise N B .In particular, we find that the data-processing bound is close to the lower bound Q L for both low and high thermal noise.This is related to Theorem 22, as the data-processing bound can be at most 1.45 bits larger than the lower bound Q L .In Figure 3(a), we plot for medium transmissivity and low thermal noise.We find that the ε-close-degradable bound is very near to the data-processing bound and is tighter than the ε-degradable bound.In Figure 3(b), we plot for medium transmissivity and high thermal noise.We find that the εdegradable bound is tighter than the ε-close degradable bound.In Figure 3(d), we plot for high transmissivity and high thermal noise.In Figure 3(c), we plot for high transmissivity and low thermal noise.We find that all upper bounds are very near to the lower bound Q L .From Figures 3(a) and 3(c), it is evident that in the low-noise regime, there is a strong limitation on any potential super-additivity of coherent information of a thermal channel.Similar results were obtained on quantum and private capacities of low-noise quantum channels in [LLS17].It is important to stress that the upper bound Q U 3 can serve as a good bound only for low values of the thermal noise N B , as the technique to calculate this bound requires the closeness of a thermal channel with a pureloss channel (discussed in Theorem 19), and the closeness parameter is equal to N B /(N B + 1).
In Figure 4, we plot all the upper bounds and the lower bound Q L versus N S , for high transmissivity and high thermal noise.In Figure 4(a), we find that the ε-degradable bound is tighter than all other bounds for high values of N S .In Figure 4(b), we plot for the same parameter values, but for low values of N S .It is evident that for low input mean photon number, the data-processing bound is tighter than the ε-degradable bound.
The plots suggest that our upper bounds based on the notion of approximate degradability are good for the case of high input mean photon number.We suspect that these bounds can be further improved for the case of low input mean photon number by considering the energy-constrained diamond norm [Shi17,We17].To address this question, we consider the generalized channel divergences of quantum Gaussian channels in Section 9 and argue about their optimization.(a) Figure 4: The figures plot the data-processing bound (Q U 1 ), the ε-degradable bound (Q U 2 ), the ε-close-degradable bound (Q U 3 ) and the lower bound (Q L ) on energy-constrained quantum capacity of thermal channels.In each figure, we select η = 0.99 and N B = 0.1.In (a), the εdegradable upper bound is tighter than all other upper bounds.In (b), for low values of N S , the data-processing bound is tighter than the ε-degradable bound.

Data-processing bound on the energy-constrained private capacity of bosonic thermal channels
Theorem 23 An upper bound on the private capacity of a thermal channel L η,N B with transmissivity η ∈ [1/2, 1], environment photon number N B , and input mean photon number constraint N S is given by P (L η,N B , N S ) ≤ max{0, P U 1 (L η,N B , N S )} (7.1) with η = η/((1 − η)N B + 1).
Proof.A proof follows from arguments similar to those in the proof of Theorem 14.Since a pureloss channel is a degradable channel [WPGG07, GSE08], its energy-constrained private capacity is the same as its energy-constrained quantum capacity [WQ16].
Proof.A proof follows from arguments similar to those in the proof of Theorem 16.The final result is obtained using Theorem 12.
Proof.A proof follows from arguments similar to those in the proof of Theorem 21.The final result is obtained using Theorem 13.
8 Lower bound on energy-constrained private capacity of bosonic thermal channels In this section, we establish an improvement on the best known lower bound [WHG12] on the energy-constrained private capacity of bosonic thermal channels, by using displaced thermal states as input to the channel.We note that a similar effect has been observed in [RGK05] for the finite-dimensional case.
The energy-constrained private information of a channel N , as defined in (3.59), can also be written as (8.1) where ρE A ≡ dx p X (x)ρ x A is an average state of the ensemble E A ≡ {p X (x), ρ x A } and N denotes a complementary channel of N .If the energy-constrained private information is calculated for coherent-state inputs, then for each element of the ensemble, the following equality holds ) is an achievable rate, which is the same as the energy-constrained coherent information.
However, we show that displaced thermal-state inputs provide an improved lower bound for certain values of the transmissivity η, low thermal noise N B , and both low and high input mean photon number N S .We start with the following ensemble of displaced thermal states, chosen according to the Gaussian probability distribution where D(α) denotes the displacement operator, θ(N 2 S ) denotes the thermal state with mean photon number N 2 S , and N 1 S and N 2 S are chosen such that N 1 S + N 2 S = N S , which is the mean number of photons input to the channel.By employing (3.25), the average of this ensemble is a thermal state with mean photon number N S , i.e., Hence, this ensemble meets the constraint that the average number of photons input to the channel is equal to N S .After the action of the channel on one of the states in the ensemble, the entropy of the output state is given by where the first equality follows because thermal channel is covariant with respect to displacement operators, as reviewed in (3.33).The second equality follows because D( √ ηα) is a unitary operator and entropy is invariant under the action of a unitary operator.Since H(L η,N B (θ(N 2 S ))) is independent of the Gaussian probability distribution in (8.3), Similar arguments can be made for the output states at the environment mode.
Hence, a lower bound on the energy-constrained private information in (8.1) for the bosonic thermal channel is as follows: where Lη,N B denotes the complementary channel of L η,N B , and we denote the lower bound in (8.9) on the private information by P L (L η,N B , N S ).The first inequality follows from (3.59).Here, I c (L η,N B , N S ) denotes the coherent information of the channel for the thermal state with mean photon number N S as input to the channel.I c (L η,N B , N S ) has the same form as (6.1), i.e., where We optimize the lower bound in (8.9) on the private information P L (L η,N B , N S ) with respect to N 2 S for a fixed value of N S [Not].In Figure 5, we plot the optimized value of the lower bound in (8.9) on the private information P L (L η,N B , N S ) (dashed line) and the coherent information in (8.10)I c (L η,N B , N S ) (solid line) of the thermal channel versus the transmissivity parameter η, for low thermal noise N B and for both low and high input mean number of photons N S .We find that a larger rate for private communication can be achieved by using displaced thermal states as input to the channel instead of coherent states, for certain values of the transmissivity η.

On the optimization of generalized channel divergences of quantum Gaussian channels
In this section, we address the question of computing the energy-constrained diamond norm of several channels of interest that have appeared in our paper.We provide a very general argument, based on some definitions and results in [LKDW17] and phrased in terms of the "generalized channel divergence" as a measure of the distingishability of quantum channels.We find that, among all Gaussian input states with a fixed energy constraint, the two-mode squeezed vacuum state saturating the energy constraint is the optimal state for the energy-constrained generalized channel divergence of two particular Gaussian channels.We describe these results in more detail in what follows.We begin by recalling some developments from [LKDW17]: where N is a quantum channel.
Particular examples of a generalized divergence are the trace distance, quantum relative entropy, and the negative root fidelity.
We say that a generalized channel divergence possesses the direct-sum property on classicalquantum states if the following equality holds: where p X is a probability distribution, {|x } x is an orthonormal basis, and {ρ x } x and {σ x } x are sets of states.We note that this property holds for trace distance, quantum relative entropy, and the negative root fidelity.

Definition 27 (Generalized channel divergence [LKDW17]
) Given quantum channels N A→B and M A→B , we define the generalized channel divergence as In the above definition, the supremum is with respect to all mixed states and the reference system R is allowed to be arbitrarily large.However, as a consequence of purification, data processing, and the Schmidt decomposition, it follows that such that the supremum can be restricted to be with respect to pure states and the reference system R isomorphic to the channel input system A.
Particular cases of the generalized channel divergence are the diamond norm of the difference of N A→B and M A→B as well as the Rényi channel divergence from [CMW16].
Covariant quantum channels have symmetries that allow us to simplify the set of states over which we need to optimize their generalized channel divergence [Hol02].Let G be a finite group, and for every g ∈ G, let g → U A (g) and g → V B (g) be unitary representations acting on the input and output spaces of the channel, respectively.Then a quantum channel N A→B is covariant with respect to {(U A (g), V B (g))} g if the following relation holds for all input density operators ρ A and group elements g ∈ G: where We say that channels N A→B and M A→B are jointly covariant with respect to {(U A (g), V B (g))} g∈G if each of them is covariant with respect to {(U A (g), V B (g))} g [TW16, DW17].
The following lemma was established in [LKDW17]: Lemma 4 ([LKDW17]) Let N A→B and M A→B be quantum channels, and let {(U A (g), V B (g))} g∈G denote unitary representations of a group G. Let ρ A be a density operator, and let φ ρ RA be a purification of ρ A .Let ρA denote the group average of ρ A according to a distribution p G , i.e., and let φ ρ RA be a purification of ρA .If the generalized divergence possesses the direct-sum property on classical-quantum states, then the following inequality holds By approximation, the above lemma can be extended to continuous groups for several generalized channel divergences of interest: Lemma 5 Let N A→B and M A→B be quantum channels, and let {(U A (g), V B (g))} g∈G denote unitary representations of a continuous group G. Let ρ A be a density operator, and let φ ρ RA be a purification of ρ A .Let ρA denote the group average of ρ A according to a measure µ(g), i.e., ρA = dµ(g) U g A (ρ A ), (9.10) and let φ ρ RA be a purification of ρA .If the generalized divergence possesses the direct-sum property on classical-quantum states and is a Borel function, then the following inequality holds We can apply this lemma effectively in the context of quantum Gaussian channels.To this end, we consider an energy-constrained generalized channel divergence for W ∈ [0, ∞) and an energy observable G as follows: In what follows, we specialize this measure even further to the Gaussian energy-constrained generalized channel divergence, meaning that the optimization is constrained to be with respect to Gaussian input states: where G denotes the set of Gaussian states.We then establish the following proposition: Proposition 28 Suppose that channels N A→B and M A→B are Gaussian, they each take one input mode to m output modes, and they have the following action on a single-mode, input covariance matrix V : V → XV X T + Y N , (9.14) where X is an m × 1 matrix, Y N and Y M are m × m matrices such that N A→B and M A→B are legitimate Gaussian channels.Suppose furthermore they these channels are jointly phase covariant (phase-insensitive), in the sense that for all φ ∈ [0, 2π) and input density operators ρ, the following equality holds i.e., the state after phase averaging is diagonal in the number basis, and furthermore, the resulting state has the same photon number N 1 as ψ A .The thermal state θ(N 1 ) is the only Gaussian state of a single mode that is diagonal in the number basis with photon number equal to N 1 .A purification of the thermal state θ(N 1 ) is the two-mode squeezed vacuum ψ TMS (N 1 ) with parameter N 1 .By applying Lemma 5 and the joint phase covariance relations in (9.16)-(9.17),we find that the following inequality holds where f i for i ∈ {1, . . ., m} are functions depending on the entries of the matrix X.We can then exploit the joint covariance of the channels with respect to displacements, the observation in (9.21), and Lemma 5 to conclude that D(N A→B (ψ TMS (N S )) M A→B (ψ TMS (N S ))) ≥ D(N A→B (ψ TMS (N 1 )) M A→B (ψ TMS (N 1 ))), (9.24) for all N 1 ≤ N S .This concludes the proof.
The above proposition applies to the various settings and channels that we have considered in this paper for ε-degradable and ε-close degradable bosonic thermal channels.Thus, we can conclude in these situations that the Gaussian energy-constrained generalized channel divergence is achieved by the two-mode squeezed vacuum state.where F denotes the quantum fidelity.Proposition 28 implies that the Gaussian-constrained versions of these quantities reduce to the following for channels satisfying the assumptions stated there: (9.28) We note that the latter quantity is readibly expressed as a closed formula in terms of the Gaussian specification of the channels N A→B and M A→B in (9.14)-(9.15)and the parameter N S by employing the general formula for the fidelity of zero-mean Gaussian states from [PS00].One could also employ the formulas from [SLW17] or [Che05,Kru06] to compute Gaussian, energy-constrained channel divergences based on Rényi relative entropy or quantum relative entropy, respectively.It is a very interesting open question to determine whether, under the conditions given in the above proposition, the energy-constrained generalized channel divergence is always achieved by the two-mode squeezed vacuum state (if the restriction to Gaussian input states is lifted).Divergences of interest are the trace distance, fidelity, quantum relative entropy, and Rényi relative entropies.All of these measures lead to a very interesting suite of Gaussian optimizer questions, which we leave for future work.If there is a positive answer to this question, then we would expect to see, in the low-photon-number regime, significant improvements of the ε-degradable and ε-close degradable upper bounds on the capacities of the thermal channel.

Conclusion
In this paper, we established three different upper bounds on the energy-constrained quantum capacity of thermal channels.We discussed the closeness of these three upper bounds with a known lower bound.In particular, we have shown that the ε-close degradable bound works well only in the low-noise regime and that the data-processing upper bound is close to a lower bound for both low and high thermal noise.Moreover, we found that the data-processing bound can be at most 1.45 bits larger than a known lower bound.We also discussed an interesting case where the ε-degradable bound is tighter than all other upper bounds.Also, our results establish strong limitations on any potential superadditivity of coherent information of a thermal channel in the low-noise regime.
Similarly, we established three different upper bounds on the energy-constrained private capacity of thermal channels.We have also shown an improvement in the achievable rates of private communication through quantum thermal channels by using displaced thermal states as inputs to the channel.
Since thermal noise is present in almost all communication and optical systems, our results have implications for quantum computing and quantum cryptography.The knowledge of bounds on quantum capacity can be useful to quantify the performance of distributed quantum computation between remote locations, and private communication rates are connected to the ability to generate secret key.
We finally used the generalized channel divergence from [LKDW17] to address the question of optimal input states for the energy-bounded diamond norm and other related divergences.In particular, we showed that for two Gaussian channels that are jointly phase covariant, the Gaussian energy-constrained generalized channel divergence is achieved by a two-mode squeezed vacuum state that saturates the energy constraint.It is an interesting open question to determine whether, among all input states, the two-mode squeezed vacuum is the optimal input state for several energy-constrained, generalized channel divergences of interest.
As another task for future work, it would be good to extend the results of our paper to quantum amplifier channels and additive-noise Gaussian channels, and we note that many of the methods used and developed in our paper can be applied.
.35) Due to the input mean photon number constraint N S , and environment photon number N B for both L η,N B and D (1−η)/η,N B , there is a total photon number constraint (1 − η)N S + (1 + η)N B for the average output of n channel uses of both Lη,N B and D (1−η)/η,N B • L η,N B .Using these results in Theorem 9, we find the following upper bound on the energy-constrained quantum capacity of a thermal channel: 99, N B = 0.1

Figure 5 :
Figure 5: The figures plot the optimized value of the lower bound on the private information P L (L η,N B , N S ) (dashed line) and coherent information I c (L η,N B , N S ) (solid line) of a thermal channel versus transmissivity parameter η.In each figure, we select certain values of thermal noise N B and input mean photon number N S , with the choices indicated above each figure.In all the cases, there is an improvement in the achievable rate of private communication for certain values of the transmissivity η.

N
A→B (e inφ ρe −inφ ) = m i=1 e in i (−1) a i φ N A→B (ρ) m i=1 e −in i (−1) a i φ , (9.16) M A→B (e inφ ρe −inφ ) = m i=1 e in i (−1) a i φ M A→B (ρ) m i=1e −in i (−1) a i φ , (9.17)where a i ∈ {0, 1} for i ∈ {1, . . ., m} and ni is the photon number operator for the ith mode.Then the Gaussian energy-constrained generalized channel divergence is achieved by the two-mode squeezed vacuum state with parameter N S , i.e.,D G n,N S (N M) = D((id R ⊗N A→B ) (ψ TMS (N S )) (id R ⊗M A→B ) (ψ TMS (N S ))).(9.18) Proof.This result is an application of Lemma 5 and previous developments in our paper.Let ψ RA denote an arbitrary pure Gaussian state of two modes such that Tr{nψ A } = N 1 ≤ N S .Consider that ψ A e −inφ = ∞ n=0 |n n|ψ A |n n|, (9.19)