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

Let ${ \mathcal H }$ denote a separable Hilbert space, let ${ \mathcal B }({ \mathcal H })$ denote the set of bounded operators acting on ${ \mathcal H }$, and let ${ \mathcal P }({ \mathcal H })$ denote the subset of ${ \mathcal B }({ \mathcal H })$ that consists of positive semi-definite operators. Let ${ \mathcal T }({ \mathcal H })$ denote the set of trace-class operators, defined such that their trace norm is finite: ${\parallel A\parallel }_{1}\equiv \mathrm{Tr}\{| A| \}\lt \infty$, where $| A| \equiv \sqrt{{A}^{\dagger }A}$. Let ${ \mathcal D }({ \mathcal H })$ denote the set of density operators (positive semi-definite with unit trace) acting on ${ \mathcal H }$. A quantum channel ${ \mathcal N }\,:{ \mathcal T }({{ \mathcal H }}_{A})\to { \mathcal T }({{ \mathcal H }}_{B})$ 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 ${{ \mathcal H }}_{E}$ and a linear isometry $U\,:{{ \mathcal H }}_{A}\to {{ \mathcal H }}_{B}\otimes {{ \mathcal H }}_{E}$ such that for all ${\omega }_{A}\in { \mathcal T }({{ \mathcal H }}_{A})$, the following equality holds : ${ \mathcal N }({\omega }_{A})={\mathrm{Tr}}_{E}\{U{\omega }_{A}{U}^{\dagger }\}$. A complementary channel ${\hat{{ \mathcal N }}}_{A\to E}$ of ${{ \mathcal N }}_{A\to B}$ is defined as ${\hat{{ \mathcal N }}}_{A\to E}={\mathrm{Tr}}_{B}\{U{\omega }_{A}{U}^{\dagger }\}$. A quantum channel ${{ \mathcal N }}_{A\to B}$ is degradable [DS05] if there exists a quantum channel ${{ \mathcal D }}_{B\to E}$ such that ${{ \mathcal D }}_{B\to E}({{ \mathcal N }}_{A\to B}({\omega }_{A}))={\hat{{ \mathcal N }}}_{A\to E}({\omega }_{A})$, for all ${\omega }_{A}\in { \mathcal T }({{ \mathcal H }}_{A})$.

The quantum entropy of a state $\rho \in { \mathcal D }(H)$ is defined as $H(\rho )\equiv -\mathrm{Tr}\{\rho {\mathrm{log}}_{2}\rho \}$. 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

$$\begin{eqnarray}&&{h}_{2}(x)\equiv -x{\mathrm{log}}_{2}x-(1-x){\mathrm{log}}_{2}(1-x).\end{eqnarray} \tag{ 3.1 }$$

Throughout the paper we use a function g(x), which is the entropy of a bosonic thermal state with mean photon number x ≥ 0:

$$\begin{eqnarray}&&g(x)\equiv (x+1){\mathrm{log}}_{2}(x+1)-x{\mathrm{log}}_{2}x.\end{eqnarray} \tag{ 3.2 }$$

By continuity, we have that ${h}_{2}(0)={\mathrm{lim}}_{x\to 0}{h}_{2}(x)=0$ and $g(0)={\mathrm{lim}}_{x\to 0}g(x)=0$. The quantum relative entropy $D(\rho \parallel \sigma )$ of $\rho ,\sigma \in { \mathcal D }({ \mathcal H })$ is defined as [Fal70, Lin73]

$$\begin{eqnarray}&&D(\rho \parallel \sigma )\equiv \displaystyle \sum _{i}\langle i| \rho {\mathrm{log}}_{2}\rho -\rho {\mathrm{log}}_{2}\sigma +\sigma -\rho | i\rangle ,\end{eqnarray} \tag{ 3.3 }$$

where $\{| i\rangle \}{}_{i=1}^{\infty }$ is an orthonormal basis of eigenvectors of the state ρ, if $\mathrm{supp}(\rho )\subseteq \mathrm{supp}(\sigma )$ and $D(\rho \parallel \sigma )=\infty$ otherwise. The quantum relative entropy $D(\rho \parallel \sigma )$ is non-negative for $\rho ,\sigma \in { \mathcal D }({ \mathcal H })$ and is monotone with respect to a quantum channel [Lin75] ${ \mathcal N }\,:{ \mathcal T }({{ \mathcal H }}_{A})\to { \mathcal T }({{ \mathcal H }}_{B})$:

$$\begin{eqnarray}&&D(\rho \parallel \sigma )\geqslant D({ \mathcal N }(\rho )\parallel { \mathcal N }(\sigma )).\end{eqnarray} \tag{ 3.4 }$$

The quantum mutual information $I{(A;B)}_{\rho }$ of a bipartite state ${\rho }_{{AB}}\in { \mathcal D }({{ \mathcal H }}_{A}\otimes {{ \mathcal H }}_{B})$ is defined as [Lin73]

$$\begin{eqnarray}&&I{(A;B)}_{\rho }\equiv D({\rho }_{{AB}}\parallel {\rho }_{A}\otimes {\rho }_{B}).\end{eqnarray} \tag{ 3.5 }$$

The coherent information $I{(A\rangle B)}_{\rho }$ of ${\rho }_{{AB}}$ is defined as [HS10, Kuz11, SN96]

$$\begin{eqnarray}&&I{(A\rangle B)}_{\rho }\equiv I{(A;B)}_{\rho }-H{(A)}_{\rho },\end{eqnarray} \tag{ 3.6 }$$

when $H{(A)}_{\rho }\lt \infty$. This expression reduces to

$$\begin{eqnarray}&&I{(A\rangle B)}_{\rho }=H{(B)}_{\rho }-H{({AB})}_{\rho },\end{eqnarray} \tag{ 3.7 }$$

if $H{(B)}_{\rho }\lt \infty$.

The fidelity of two quantum states $\rho ,\sigma \in { \mathcal D }({ \mathcal H })$ is defined as [Uhl76] $F(\rho ,\sigma )\equiv {\parallel \sqrt{\rho }\sqrt{\sigma }\parallel }_{1}^{2}$. The trace distance between two density operators $\rho ,\sigma \in { \mathcal D }({ \mathcal H })$ is equal to ${\parallel \rho -\sigma \parallel }_{1}$. 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 ${ \mathcal S }$ is defined as ${\parallel { \mathcal S }\parallel }_{\diamond }\equiv {\sup }_{{\rho }_{{RA}}\in { \mathcal D }({{ \mathcal H }}_{R}\otimes {{ \mathcal H }}_{A})}{\parallel ({\mathrm{id}}_{R}\otimes {{ \mathcal S }}_{A\to B})({\rho }_{{RA}})\parallel }_{1}$, where ${\mathrm{id}}_{R}$ is the identity map acting on a Hilbert space ${{ \mathcal H }}_{R}$ corresponding to an arbitrarily large reference system [Kit97]. It suffices to optimize with respect to input states ρ that are pure. The diamond norm distance ${\parallel { \mathcal N }-{ \mathcal M }\parallel }_{\diamond }$ is a measure of the distinguishability of two quantum channels ${ \mathcal N }$ and ${ \mathcal M }$.

The concept of approximate degradability was introduced in [SSWR17]. The following two definitions of approximate degradability will be useful in our paper.

Definition 1 A channel ${{ \mathcal N }}_{A\to B}$ is ε-degradable if there exists a channel ${{ \mathcal D }}_{B\to E}$ such that $\tfrac{1}{2}{\parallel \hat{{ \mathcal N }}-{ \mathcal D }\circ { \mathcal N }\parallel }_{\diamond }\leqslant \varepsilon$, where $\hat{{ \mathcal N }}$ denotes a complementary channel of ${ \mathcal N }$.

Definition 2 A channel ${{ \mathcal N }}_{A\to B}$ is $\varepsilon$-close-degradable if there exists a degradable channel ${{ \mathcal M }}_{A\to B}$ such that $\tfrac{1}{2}{\parallel { \mathcal N }-{ \mathcal M }\parallel }_{\diamond }\leqslant \varepsilon$.

Remark 3. Let ${{ \mathcal N }}_{A\to B}$ be a quantum channel that is $\varepsilon$-close-degradable. Then ${{ \mathcal N }}_{A\to B}$ is $\varepsilon +2\sqrt{\varepsilon }$-degradable by [SSWR17], proposition A.5. A converse implication is not known to hold.

Next, we recall the definition of an energy observable and a Gibbs observable [Hol12, Win16]. We also review the uniform continuity of conditional quantum entropy with energy constraints [Win16]. When defining a Gibbs observable, we follow [Hol12, Win16].

Definition 4 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 $\{| {e}_{k}\rangle \}{}_{k}$ be an orthonormal basis for a Hilbert space ${ \mathcal H }$, and let ${\{{g}_{k}\}}_{k}$ be a sequence of non-negative real numbers. Then

$$\begin{eqnarray}&&G=\displaystyle \sum _{k=1}^{\infty }{g}_{k}| {e}_{k}\rangle \langle {e}_{k}| \end{eqnarray} \tag{ 3.8 }$$

is a self-adjoint operator that we call an energy observable.

Definition 5 The nth extension ${\overline{G}}_{n}$ of an energy observable G is defined as

$$\begin{eqnarray}&&{\overline{G}}_{n}=\displaystyle \frac{1}{n}(G\otimes I\otimes \cdots \otimes I+\cdots +I\otimes \cdots \otimes I\otimes G),\end{eqnarray} \tag{ 3.9 }$$

where n is the number of factors in each tensor product above.

Definition 6 An energy observable G is a Gibbs observable if for all $\beta \gt 0$, we have $\mathrm{Tr}\{\exp (-\beta G)\}\lt \infty$, so that the partition function $Z(\beta )\equiv$ $\mathrm{Tr}\{\exp (-\beta G)\}$ has a finite value and hence $\exp (-\beta G)/\mathrm{Tr}\{\exp (-\beta G)\}$ is a well defined thermal state.

For a Gibbs observable G, let us consider a quantum state ρ such that $\mathrm{Tr}\{G\rho \}\leqslant W$. There exists a unique state that maximizes the entropy $H(\rho )$, and this unique maximizer has the Gibbs form $\gamma (W)=\exp (-\beta (W)G)/Z(\beta (W))$, where $\beta (W)$ is the solution of the equation:

$$\begin{eqnarray}&&\mathrm{Tr}\{\exp (-\beta G)(G-W)\}=0.\end{eqnarray} \tag{ 3.10 }$$

In particular, for the Gibbs observable $G={\rm{\hslash }}\omega \hat{n}$, where $\hat{n}={\hat{a}}^{\dagger }\hat{a}$ is the photon number operator, a thermal state (mean photon number $\bar{n}$) that saturates the energy-constrained inequality $\mathrm{Tr}\{G\rho \}\leqslant W$, gives the maximum value of the entropy:

$$\begin{eqnarray}&&H(\gamma (W))=g(\bar{n})=(\bar{n}+1){\mathrm{log}}_{2}(\bar{n}+1)-\bar{n}{\mathrm{log}}_{2}\bar{n}.\end{eqnarray} \tag{ 3.11 }$$

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]:

Lemma 1 For a Gibbs observable $G\in { \mathcal P }({{ \mathcal H }}_{A})$, and states ${\omega }_{{AB}},{\tau }_{{AB}}\in { \mathcal D }({{ \mathcal H }}_{A}\otimes {{ \mathcal H }}_{B})$, such that $\tfrac{1}{2}{\parallel {\omega }_{{AB}}-{\tau }_{{AB}}\parallel }_{1}\leqslant \varepsilon \lt \varepsilon ^{\prime} \leqslant 1,\mathrm{Tr}\{(G\otimes {I}_{B}){\omega }_{{AB}}\},\mathrm{Tr}\{(G\otimes {I}_{B}){\tau }_{{AB}}\}\leqslant W$, where $W\in [0,\infty )$ and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$, the following inequality holds

$$\begin{eqnarray}&&| H{(A| B)}_{\omega }-H{(A| B)}_{\tau }| \leqslant (2\varepsilon ^{\prime} +4\delta )H(\gamma (W/\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta ).\end{eqnarray} \tag{ 3.12 }$$

Throughout the paper, we consider only those quantum channels that satisfy the following finite output entropy condition:

Condition 7 Let G be a Gibbs observable and $W\in [0,\infty )$. A quantum channel ${ \mathcal N }$ satisfies the finite output entropy condition with respect to G and W if

$$\begin{eqnarray}&&\mathop{\sup }\limits_{\rho :\,\mathrm{Tr}\{G\rho \}\leqslant W}H({ \mathcal N }(\rho ))\lt \infty .\end{eqnarray} \tag{ 3.13 }$$

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 7 when the Gibbs observable for m modes is taken to be

$$\begin{eqnarray}&&{\hat{E}}_{m}\equiv \displaystyle \sum _{j=1}^{m}{\omega }_{j}{\hat{a}}_{j}^{\dagger }{\hat{a}}_{j},\end{eqnarray} \tag{ 3.14 }$$

where ${\omega }_{j}\gt 0$ is the frequency of the jth mode and ${\hat{a}}_{j}$ is the photon annihilation operator for the jth mode, so that ${\hat{a}}_{j}^{\dagger }{\hat{a}}_{j}$ is the photon number operator for the jth mode.

Let

$$\begin{eqnarray}&&\hat{x}\equiv [{\hat{q}}_{1},\,\ldots ,\,{\hat{q}}_{m},{\hat{p}}_{1},\,\ldots ,\,{\hat{p}}_{m}]\equiv [{\hat{x}}_{1},\,\ldots ,\,{\hat{x}}_{2m}]\end{eqnarray} \tag{ 3.15 }$$

denote a vector of position- and momentum-quadrature operators, satisfying the canonical commutation relations:

$$\begin{eqnarray}[{\hat{x}}_{j},{\hat{x}}_{k}]={\rm{i}}{{\rm{\Omega }}}_{j,k},\quad {\rm{where}}\quad {\rm{\Omega }}\equiv \left[\begin{array}{cc}0 & 1\\ -1 & 0\end{array}\right]\otimes {I}_{m},\end{eqnarray} \tag{ 3.16 }$$

and I_m denotes the m × m identity matrix. We take the annihilation operator for the jth mode as ${\hat{a}}_{j}=({\hat{q}}_{j}+{\rm{i}}{\hat{p}}_{j})/\sqrt{2}$. For $\xi \in {{\mathbb{R}}}^{2m}$, we define the unitary displacement operator $D(\xi )\equiv \exp ({\rm{i}}{\xi }^{T}{\rm{\Omega }}\hat{x})$. Displacement operators satisfy the following relation:

$$\begin{eqnarray}&&D{(\xi )}^{\dagger }D({\xi }^{{\prime} })=D({\xi }^{{\prime} })D{(\xi )}^{\dagger }\exp ({\rm{i}}{\xi }^{T}{\rm{\Omega }}{\xi }^{{\prime} }).\end{eqnarray} \tag{ 3.17 }$$

Every state $\rho \in { \mathcal D }({ \mathcal H })$ has a corresponding Wigner characteristic function, defined as

$$\begin{eqnarray}&&{\chi }_{\rho }(\xi )\equiv \mathrm{Tr}\{D(\xi )\rho \},\end{eqnarray} \tag{ 3.18 }$$

and from which we can obtain the state ρ as

$$\begin{eqnarray}&&\rho =\displaystyle \frac{1}{{(2\pi )}^{m}}\int {{\rm{d}}}^{2m}\xi \ {\chi }_{\rho }(\xi )\ {D}^{\dagger }(\xi ).\end{eqnarray} \tag{ 3.19 }$$

A quantum state ρ is Gaussian if its Wigner characteristic function has a Gaussian form as

$$\begin{eqnarray}&&{\chi }_{\rho }(\xi )=\exp \left(-\displaystyle \frac{1}{4}{[{\rm{\Omega }}\xi ]}^{T}{V}^{\rho }{\rm{\Omega }}\xi +{[{\rm{\Omega }}{\mu }^{\rho }]}^{T}\xi \right),\end{eqnarray} \tag{ 3.20 }$$

where ${\mu }^{\rho }$ is the $2{m}\times 1$ mean vector of ρ, whose entries are defined by ${\mu }_{j}^{\rho }\equiv {\langle {\hat{x}}_{j}\rangle }_{\rho }$ and ${V}^{\rho }$ is the $2{m}\times 2{m}$ covariance matrix of ρ, whose entries are defined as

$$\begin{eqnarray}&&{V}_{j,k}^{\rho }\equiv \langle \{{\hat{x}}_{j}-{\mu }_{j}^{\rho },{\hat{x}}_{k}-{\mu }_{k}^{\rho }\}\rangle .\end{eqnarray} \tag{ 3.21 }$$

The following condition holds for a valid covariance matrix: $V+{\rm{i}}{\rm{\Omega }}\geqslant 0$, which is a manifestation of the uncertainty principle [SMD94].

A thermal Gaussian state ${\theta }_{\beta }$ of m modes with respect to ${\hat{E}}_{m}$ from (3.14) and having inverse temperature $\beta \gt 0$ thus has the following form:

$$\begin{eqnarray}&&{\theta }_{\beta }={{\rm{e}}}^{-\beta {\hat{E}}_{m}}/\mathrm{Tr}\{{{\rm{e}}}^{-\beta {\hat{E}}_{m}}\},\end{eqnarray} \tag{ 3.22 }$$

and has a mean vector equal to zero and a diagonal $2{m}\times 2{m}$ covariance matrix. One can calculate that the photon number in this state is equal to

$$\begin{eqnarray}&&\displaystyle \sum _{j}\displaystyle \frac{1}{{{\rm{e}}}^{\beta {\omega }_{j}}-1}.\end{eqnarray} \tag{ 3.23 }$$

A single-mode thermal state with mean photon number $\bar{n}=1/({{\rm{e}}}^{\beta \omega }-1)$ has the following representation in the photon number basis:

$$\begin{eqnarray}&&\theta (\bar{n})\equiv \displaystyle \frac{1}{1+\bar{n}}\displaystyle \sum _{n=0}^{\infty }{\left(\displaystyle \frac{\bar{n}}{\bar{n}+1}\right)}^{n}| n\rangle \langle n| .\end{eqnarray} \tag{ 3.24 }$$

It is also well known that thermal states can be written as a Gaussian mixture of displacement operators acting on the vacuum state:

$$\begin{eqnarray}&&{\theta }_{\beta }=\int {{\rm{d}}}^{2m}\xi \ p(\xi )\ D(\xi ){[| 0\rangle \langle 0| ]}^{\otimes m}{D}^{\dagger }(\xi ),\end{eqnarray} \tag{ 3.25 }$$

where $p(\xi )$ 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:

$$\begin{eqnarray}&&{\theta }_{\beta }=\int {{\rm{d}}}^{2m}\xi \ q(\xi )\ D(\xi ){\theta }_{{\beta }^{{\prime} }}{D}^{\dagger }(\xi ),\end{eqnarray} \tag{ 3.26 }$$

where ${\beta }^{{\prime} }\geqslant \beta$ and $q(\xi )$ is a particular circularly symmetric Gaussian distribution.

In our paper, we employ the two-mode squeezed vacuum state with parameter $\bar{n}$, which is equivalent to a purification of the thermal state in (3.24) and is defined as

$$\begin{eqnarray}&&| {\psi }_{\mathrm{TMS}}(\bar{n})\rangle \equiv \displaystyle \frac{1}{\sqrt{\bar{n}+1}}\displaystyle \sum _{n=0}^{\infty }\sqrt{{\left(\displaystyle \frac{\bar{n}}{\bar{n}+1}\right)}^{n}}| n{\rangle }_{R}| n{\rangle }_{A}.\end{eqnarray} \tag{ 3.27 }$$

A $2{m}\times 2{m}$ matrix S is symplectic if it preserves the symplectic form: $S{\rm{\Omega }}{S}^{T}={\rm{\Omega }}$. According to Williamson's theorem [Wil36], there is a diagonalization of the covariance matrix ${V}^{\rho }$ of the form

$$\begin{eqnarray}&&{V}^{\rho }={S}^{\rho }({D}^{\rho }\oplus {D}^{\rho }){({S}^{\rho })}^{T},\end{eqnarray} \tag{ 3.28 }$$

where ${S}^{\rho }$ is a symplectic matrix and ${D}^{\rho }\equiv \mathrm{diag}({\nu }_{1},\,\ldots ,\,{\nu }_{m})$ is a diagonal matrix of symplectic eigenvalues such that ${\nu }_{i}\geqslant 1$ for all $i\in \{1,\,\ldots ,\,m\}$. Computing this decomposition is equivalent to diagonalizing the matrix ${{\rm{i}}{V}}^{\rho }{\rm{\Omega }}$ [WTLB17, appendix A].

The entropy $H(\rho )$ of a quantum Gaussian state ρ is a direct function of the symplectic eigenvalues of its covariance matrix ${V}^{\rho }$ [Ser17]:

$$\begin{eqnarray}&&H(\rho )=\displaystyle \sum _{j=1}^{m}g(({\nu }_{j}-1)/2),\end{eqnarray} \tag{ 3.29 }$$

where $g(\cdot )$ is defined in (3.2).

The Hilbert–Schmidt adjoint of a Gaussian quantum channel ${{ \mathcal N }}_{X,Y}$ from l modes to m modes has the following effect on a displacement operator $D(\xi )$ [Ser17]:

$$\begin{eqnarray}&&D(\xi )\mapsto D({X}^{T}\xi )\exp \left(-\displaystyle \frac{1}{2}{\xi }^{T}Y\xi +{\rm{i}}{\xi }^{T}{\rm{\Omega }}d\right),\end{eqnarray} \tag{ 3.30 }$$

where X is a real $2{m}\times 2l$ matrix, Y is a real $2{m}\times 2{m}$ positive semi-definite matrix, and $d\in {{\mathbb{R}}}^{2m}$, such that they satisfy

$$\begin{eqnarray}&&Y+{\rm{i}}{\rm{\Omega }}-{\rm{i}}{X}{\rm{\Omega }}{X}^{T}\geqslant 0.\end{eqnarray} \tag{ 3.31 }$$

The effect of the channel on the mean vector ${\mu }^{\rho }$ and the covariance matrix ${V}^{\rho }$ is thus as follows:

$$\begin{eqnarray}&&{\mu }^{\rho }\,\longmapsto \,X{\mu }^{\rho }+d,\end{eqnarray} \tag{ 3.32 }$$

$$\begin{eqnarray}&&{V}^{\rho }\,\longmapsto \,{{XV}}^{\rho }{X}^{T}+Y.\end{eqnarray} \tag{ 3.33 }$$

A phase-insensitive, single-mode bosonic Gaussian channel adds an equal amount of noise to each quadrature of the electromagnetic field, such that

$$\begin{eqnarray}&&X=\mathrm{diag}(\sqrt{\tau },\sqrt{\tau }),\end{eqnarray} \tag{ 3.34 }$$

$$\begin{eqnarray}&&Y=\mathrm{diag}(\nu ,\nu ),\,\end{eqnarray} \tag{ 3.35 }$$

$$\begin{eqnarray}&&d=0,\,\end{eqnarray} \tag{ 3.36 }$$

where $\tau \in [0,1]$ corresponds to attenuation, $\tau \geqslant 1$ amplification, and ν is the variance of an additive-noise. Moreover, the following inequalities should hold

$$\begin{eqnarray}&&\nu \geqslant 0,\,\end{eqnarray} \tag{ 3.37 }$$

$$\begin{eqnarray}&&{\nu }^{2}\geqslant {(1-\tau )}^{2},\end{eqnarray} \tag{ 3.38 }$$

in order for the map to be a legitimate completely positive and trace preserving map. The channel is entanglement breaking [HSR03] if the following inequality holds [Hol08]

$$\begin{eqnarray}&&\nu \geqslant \tau +1.\end{eqnarray} \tag{ 3.39 }$$

All Gaussian channels are covariant with respect to displacement operators. That is, the following relation holds

$$\begin{eqnarray}&&{{ \mathcal N }}_{X,Y}(D(\xi )\rho {D}^{\dagger }(\xi ))=D(X\xi ){{ \mathcal N }}_{X,Y}(\rho ){D}^{\dagger }(X\xi ),\end{eqnarray} \tag{ 3.40 }$$

and note that $D(X\xi )$ 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 ${{ \mathcal N }}_{X,Y}$ 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

$$\begin{eqnarray}S=\left[\begin{array}{cc}X & Z\\ {X}_{E} & {Z}_{E}\end{array}\right],\end{eqnarray} \tag{ 3.41 }$$

which corresponds to the Gaussian unitary transformation on a larger space. The complementary channel ${\hat{{ \mathcal N }}}_{{X}_{E},{Y}_{E}}$ from input to the environment then effects the following transformation on mean vectors and covariance matrices:

$$\begin{eqnarray}&&{\mu }^{\rho }\,\longmapsto \,{X}_{E}{\mu }^{\rho },\end{eqnarray} \tag{ 3.42 }$$

$$\begin{eqnarray}&&\,{V}^{\rho }\,\longmapsto \,{X}_{E}{V}^{\rho }{X}_{E}^{T}+{Y}_{E},\end{eqnarray} \tag{ 3.43 }$$

where ${Y}_{E}\equiv {Z}_{E}{Z}_{E}^{T}$.

A quantum thermal channel is a Gaussian channel that can be characterized by a beamsplitter of transmissivity $\eta \in (0,1)$, coupling the signal input state with a thermal state with mean photon number ${N}_{B}\geqslant 0$, and followed by a partial trace over the environment. In the Heisenberg picture, the beamsplitter transformation is given by the following Bogoliubov transformation:

$$\begin{eqnarray}&&\hat{b}=\sqrt{\eta }\hat{a}-\sqrt{1-\eta }\hat{e},\end{eqnarray} \tag{ 3.44 }$$

$$\begin{eqnarray}&&\hat{e}^{\prime} =\sqrt{1-\eta }\hat{a}+\sqrt{\eta }\hat{e},\,\end{eqnarray} \tag{ 3.45 }$$

where $\hat{a},\hat{b},\hat{e}$, and $\hat{e}^{\prime}$ 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 ${{ \mathcal L }}_{\eta ,{N}_{B}}$. 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 $\eta {N}_{S}+(1-\eta ){N}_{B}$.

A quantum amplifier channel is a Gaussian channel that can be characterized by a two-mode squeezer with parameter $G\gt 1$, coupling the signal input state with a thermal state with mean photon number ${N}_{B}\geqslant 0$, and followed by a partial trace over the environment. In the Heisenberg picture, the two-mode squeezer implementing a quantum amplifier channel has the following Bogoliubov transformation:

$$\begin{eqnarray}&&\hat{b}=\sqrt{G}\hat{a}+\sqrt{G-1}{\hat{e}}^{\dagger },\end{eqnarray} \tag{ 3.46 }$$

$$\begin{eqnarray}&&\hat{e}^{\prime} =\sqrt{G-1}{\hat{a}}^{\dagger }+\sqrt{G}\hat{e},\,\end{eqnarray} \tag{ 3.47 }$$

where $\hat{a},\hat{b},\hat{e}$, and $\hat{e}^{\prime}$ correspond to the same parties as discussed above. Throughout the paper, we represent the noisy amplifier channel by ${{ \mathcal A }}_{G,{N}_{B}}$, and the quantum-limited amplifier channel (with ${N}_{B}=0$) by ${{ \mathcal A }}_{G,0}$.

An additive-noise channel is specified by the following completely positive and trace preserving map:

$$\begin{eqnarray}&&{{ \mathcal N }}_{\bar{n}}(\rho )\equiv \int {{\rm{d}}}^{2}\alpha \ {P}_{\bar{n}}(\alpha )D(\alpha )\rho {D}^{\dagger }(\alpha ),\end{eqnarray} \tag{ 3.48 }$$

where ${P}_{\bar{n}}=\exp (-| \alpha {| }^{2}/\bar{n})/(\pi \bar{n})$ and $D(\alpha )$ is a displacement operator for the input mode. The variance $\bar{n}\gt 0$ completely characterizes the channel ${{ \mathcal N }}_{\bar{n}}$, and it roughly represents the number of noise photons added to the input mode by the channel.

The following theorem on continuity of output entropy for infinite-dimensional systems with finite average energy constraints is a direct consequence of [LS09, theorem 11] and lemma 1.

Theorem 8. Let ${{ \mathcal N }}_{A\to B}$ and ${{ \mathcal M }}_{A\to B}$ be quantum channels, $G\in { \mathcal P }({{ \mathcal H }}_{B})$ be a Gibbs observable, such that

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}{{ \mathcal N }}^{\otimes n}({\rho }_{{A}^{n}})\},\mathrm{Tr}\{{\overline{G}}_{n}{{ \mathcal M }}^{\otimes n}({\rho }_{{A}^{n}})\}\leqslant W,\end{eqnarray} \tag{ 3.49 }$$

where $W\in [0,\infty )$ and ${\rho }_{{{RA}}^{n}}\in { \mathcal D }({{ \mathcal H }}_{R}\otimes {{ \mathcal H }}_{A}^{\otimes n})$. If $\tfrac{1}{2}{\parallel { \mathcal N }-{ \mathcal M }\parallel }_{\diamond }\leqslant \varepsilon \lt \varepsilon ^{\prime} \leqslant 1$ and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$, then the following inequality holds

$$\begin{eqnarray}| H(({\mathrm{id}}_{R}\otimes {{ \mathcal N }}_{A\to B}^{\otimes n})({\rho }_{{{RA}}^{n}}))-H(({\mathrm{id}}_{R}\otimes {{ \mathcal M }}_{A\to B}^{\otimes n})({\rho }_{{{RA}}^{n}}))| & \\ &&\,\leqslant n[(2\varepsilon ^{\prime} +4\delta )H(\gamma (W/\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )].\end{eqnarray} \tag{ 3.50 }$$

Proof. Let

$$\begin{eqnarray}&&{\rho }^{j}=({\mathrm{id}}_{R}\otimes {{ \mathcal M }}_{A\to B}^{\otimes j}\otimes {{ \mathcal N }}_{A\to B}^{\otimes (n-j)})({\rho }_{{{RA}}^{n}}),\end{eqnarray} \tag{ 3.51 }$$

and consider the following chain of inequalities:

$$\begin{eqnarray}&&\begin{array}{l}| H{({{RB}}^{n})}_{{\rho }^{0}}-H{({{RB}}^{n})}_{{\rho }^{n}}| \,\\ \,=\left|\displaystyle \sum _{j=1}^{n}H{({{RB}}^{n})}_{{\rho }^{j-1}}-H{({{RB}}^{n})}_{{\rho }^{j}}\right|\end{array}\,\end{eqnarray} \tag{ 3.52 }$$

$$\begin{eqnarray}&&\leqslant \displaystyle \sum _{j=1}^{n}| H{({{RB}}^{n})}_{{\rho }^{j-1}}-H{({{RB}}^{n})}_{{\rho }^{j}}| \,\end{eqnarray} \tag{ 3.53 }$$

$$\begin{eqnarray}&&=\displaystyle \sum _{j=1}^{n}| H{({B}_{j}| {{RB}}_{1}\cdots {B}_{j-1}{B}_{j+1}\cdots {B}_{n})}_{{\rho }^{j-1}}-H{({B}_{j}| {{RB}}_{1}\cdots {B}_{j-1}{B}_{j+1}\cdots {B}_{n})}_{{\rho }^{j}}| \end{eqnarray} \tag{ 3.54 }$$

$$\begin{eqnarray}&&\leqslant n[(2\varepsilon ^{\prime} +4\delta )\left(\displaystyle \sum _{j=1}^{n}\displaystyle \frac{1}{n}H(\gamma ({W}_{j}/\delta ))\right)+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )]\,\end{eqnarray} \tag{ 3.55 }$$

$$\begin{eqnarray}&&\leqslant n[(2\varepsilon ^{\prime} +4\delta )H\left(\displaystyle \frac{1}{n}\displaystyle \sum _{j=1}^{n}\gamma ({W}_{j}/\delta )\right)+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )]\,\end{eqnarray} \tag{ 3.56 }$$

$$\begin{eqnarray}&&\leqslant n[(2\varepsilon ^{\prime} +4\delta )H(\gamma (W/\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )].\,\end{eqnarray} \tag{ 3.57 }$$

The first inequality follows from the triangle inequality. The second equality follows from the fact that the states ${\rho }^{j}$ and ${\rho }^{j-1}$ 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 ${ \mathcal N }$ and ${ \mathcal M }$, i.e., $\mathrm{Tr}\{G{ \mathcal N }({\rho }_{{A}_{j}})\},\mathrm{Tr}\{G{ \mathcal M }({\rho }_{{A}_{j}})\}\leqslant {W}_{j}$ and $\tfrac{1}{n}{\sum }_{j}{W}_{j}\leqslant W$. Then the second inequality follows because $\tfrac{1}{2}\parallel {\rho }^{j}-{\rho }^{j-1}{\parallel }_{1}\leqslant \varepsilon$ 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

$$\begin{eqnarray}&&\mathrm{Tr}\left\{\displaystyle \frac{1}{n}\displaystyle \sum _{j=1}^{n}G\,\gamma ({W}_{j}/\delta )\right\}=\displaystyle \frac{1}{n}\displaystyle \sum _{j=1}^{n}\mathrm{Tr}\{G\,\gamma ({W}_{j}/\delta )\}\leqslant W/\delta ,\end{eqnarray} \tag{ 3.58 }$$

and $\gamma (W/\delta )$ is the Gibbs state that maximizes the entropy corresponding to the energy $W/\delta$. ■

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 ${ \mathcal N }$ can be defined as $F({ \mathcal N })={\mathrm{lim}}_{n\to \infty }\tfrac{1}{n}{\sup }_{{P}^{(n)}}{f}_{n}({{ \mathcal N }}^{\otimes n},{P}^{(n)})$, 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].

Lemma 2 If $F({ \mathcal N })={\mathrm{lim}}_{n\to \infty }\tfrac{1}{n}{\sup }_{{P}^{(n)}}{f}_{n}({{ \mathcal N }}^{\otimes n},{P}^{(n)})$ for a channel ${ \mathcal N }$ and $\forall$ $n,{P}^{(n)},| {f}_{n}({{ \mathcal N }}^{\otimes n},{P}^{(n)})-{f}_{n}({{ \mathcal M }}^{\otimes n},{P}^{(n)})| \leqslant {nc}$, then $| F({ \mathcal N })-F({ \mathcal M })| \leqslant c$.

The energy-constrained quantum and private capacities of quantum channels have been defined in [WQ16, section 3]. 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.

An $(n,M,G,W,\varepsilon )$ code for energy-constrained quantum communication consists of an encoding channel ${{ \mathcal E }}^{n}\,:{ \mathcal T }({{ \mathcal H }}_{S})\to { \mathcal T }({{ \mathcal H }}_{A}^{\otimes n})$ and a decoding channel ${{ \mathcal D }}^{n}\,:{ \mathcal T }({{ \mathcal H }}_{B}^{\otimes n})\to { \mathcal T }({{ \mathcal H }}_{S})$, where $M=\dim ({{ \mathcal H }}_{S})$. The energy constraint is such that the following bound holds for all states resulting from the output of the encoding channel ${{ \mathcal E }}^{n}$:

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}{{ \mathcal E }}^{n}({\rho }_{S})\}\leqslant W,\end{eqnarray} \tag{ 3.59 }$$

where ${\rho }_{S}\in { \mathcal D }({{ \mathcal H }}_{S})$. Note that

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}{{ \mathcal E }}^{n}({\rho }_{S})\}=\mathrm{Tr}\{G{\overline{\rho }}_{n}\},\end{eqnarray} \tag{ 3.60 }$$

where

$$\begin{eqnarray}&&{\overline{\rho }}_{n}\equiv \displaystyle \frac{1}{n}\displaystyle \sum _{i=1}^{n}{\mathrm{Tr}}_{{A}^{n}\setminus {A}_{i}}\{{{ \mathcal E }}^{n}({\rho }_{S})\}\end{eqnarray} \tag{ 3.61 }$$

due to the i.i.d. nature of the observable ${\overline{G}}_{n}$. Furthermore, the quantum communication code satisfies the following reliability condition such that for all pure states ${\phi }_{{RS}}\in { \mathcal D }({{ \mathcal H }}_{R}\otimes {{ \mathcal H }}_{S})$,

$$\begin{eqnarray}&&F({\phi }_{{RS}},({\mathrm{id}}_{R}\otimes [{{ \mathcal D }}^{n}\,\circ \,{{ \mathcal N }}^{\otimes n}\,\circ \,{{ \mathcal E }}^{n}])({\phi }_{{RS}}))\geqslant 1-\varepsilon ,\end{eqnarray} \tag{ 3.62 }$$

where ${{ \mathcal H }}_{R}$ is isomorphic to ${{ \mathcal H }}_{S}$. A rate R is achievable for quantum communication over ${ \mathcal N }$ subject to the energy constraint W if for all $\varepsilon \in (0,1)$, $\delta \gt 0$, and sufficiently large n, there exists an $(n,{2}^{n[R-\delta ]},G,W,\varepsilon )$ energy-constrained quantum communication code. The energy-constrained quantum capacity $Q({ \mathcal N },G,W)$ of ${ \mathcal N }$ is equal to the supremum of all achievable rates.

If the channel ${ \mathcal N }$ satisfies condition 7 and G is a Gibbs observable, then the quantum capacity $Q({ \mathcal N },G,W)$ is equal to the regularized energy-constrained coherent information of the channel ${ \mathcal N }$ [WQ16]

$$\begin{eqnarray}&&Q({ \mathcal N },G,W)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}{I}_{c}({{ \mathcal N }}^{\otimes n},{\overline{G}}_{n},W),\end{eqnarray} \tag{ 3.63 }$$

where the energy-constrained coherent information of the channel is defined as [WQ16]

$$\begin{eqnarray}&&{I}_{c}({ \mathcal N },G,W)\equiv \mathop{\sup }\limits_{\rho :\mathrm{Tr}\{\rho G\}\leqslant W}H({ \mathcal N }(\rho ))-H(\hat{{ \mathcal N }}(\rho )),\end{eqnarray} \tag{ 3.64 }$$

and $\hat{{ \mathcal N }}$ denotes a complementary channel of ${ \mathcal N }$. 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].

An $(n,M,G,W,\varepsilon )$ code for private communication consists of a set $\{{\rho }_{{A}^{n}}^{m}\}{}_{m=1}^{M}$ of quantum states, each in ${ \mathcal D }({{ \mathcal H }}_{A}^{\otimes n})$, and a POVM $\{{{\rm{\Lambda }}}_{{B}^{n}}^{m}\}{}_{m=1}^{M}$ such that

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}{\rho }_{{A}^{n}}^{m}\}\leqslant W,\,\end{eqnarray} \tag{ 3.65 }$$

$$\begin{eqnarray}&&\mathrm{Tr}\{{{\rm{\Lambda }}}_{{B}^{n}}^{m}{{ \mathcal N }}^{\otimes n}({\rho }_{{A}^{n}}^{m})\}\geqslant 1-\varepsilon ,\,\end{eqnarray} \tag{ 3.66 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\parallel {\hat{{ \mathcal N }}}^{\otimes n}({\rho }_{{A}^{n}}^{m})-{\omega }_{{E}^{n}}\parallel }_{1}\leqslant \varepsilon ,\,\end{eqnarray} \tag{ 3.67 }$$

for all $m\in \{1,\,\ldots ,\,M\}$, with ${\omega }_{{E}^{n}}$ some fixed state in ${ \mathcal D }({{ \mathcal H }}_{E}^{\otimes n})$. In the above, $\hat{{ \mathcal N }}$ is a channel complementary to ${ \mathcal N }$. A rate R is achievable for private communication over ${ \mathcal N }$ subject to energy constraint W if for all $\varepsilon \in (0,1),\delta \gt 0$, and sufficiently large n, there exists an $(n,{2}^{n[R-\delta ]},G,W,\varepsilon )$ private communication code. The energy-constrained private capacity $P({ \mathcal N },G,W)$ of ${ \mathcal N }$ 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 ${ \mathcal N }$ satisfies condition 7 and G is a Gibbs observable, then the energy-constrained private capacity $P({ \mathcal N },G,W)$ is given by the regularized energy-constrained private information of the channel:

$$\begin{eqnarray}&&P({ \mathcal N },G,W)=\mathop{\mathrm{lim}}\limits_{n\to \infty }\displaystyle \frac{1}{n}{P}^{(1)}({{ \mathcal N }}^{\otimes n},{\overline{G}}_{n},W),\end{eqnarray} \tag{ 3.68 }$$

where the energy-constrained private information is defined as

$$\begin{eqnarray}&&{P}^{(1)}({ \mathcal N },G,W)\equiv \mathop{\sup }\limits_{{\bar{\rho }}_{{{ \mathcal E }}_{A}}:\mathrm{Tr}\{G{\bar{\rho }}_{{{ \mathcal E }}_{A}}\}\leqslant W}\int {\rm{d}}{x}\,{p}_{X}(x)[D({ \mathcal N }({\rho }_{A}^{x})\parallel { \mathcal N }({\bar{\rho }}_{{{ \mathcal E }}_{A}}))-D(\hat{{ \mathcal N }}({\rho }_{A}^{x})\parallel \hat{{ \mathcal N }}({\bar{\rho }}_{{{ \mathcal E }}_{A}}))],\end{eqnarray} \tag{ 3.69 }$$

and ${\bar{\rho }}_{{{ \mathcal E }}_{A}}\equiv \int {\rm{d}}{x}\,{p}_{X}(x){\rho }_{A}^{x}$ is an average state of the ensemble

$$\begin{eqnarray}&&{{ \mathcal E }}_{A}\equiv \{{p}_{X}(x),{\rho }_{A}^{x}\},\end{eqnarray} \tag{ 3.70 }$$

and $\hat{{ \mathcal N }}$ denotes a complementary channel of ${ \mathcal N }$. 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].

Remark 9. The unconstrained quantum and private capacities of a quantum channel ${ \mathcal N }$ are defined in the same way as above but without the energy constraints demanded in (3.59) and (3.65). As a consequence of these definitions and the fact that the set of states with finite but arbitrarily large energy is dense in the set of all states, for channels satisfying the finite output entropy condition for every energy $W\geqslant 0$, the unconstrained quantum and private capacities are respectively given by

$$\begin{eqnarray}&&\mathop{\sup }\limits_{W\geqslant 0}Q({ \mathcal N },G,W),\qquad \mathop{\sup }\limits_{W\geqslant 0}P({ \mathcal N },G,W).\end{eqnarray} \tag{ 3.71 }$$

An upper bound on the quantum capacity of an ε-degradable channel was established as [SSWR17, theorem 3.1(ii)] for the finite-dimensional case. Here, we prove a related bound for the infinite-dimensional case with finite average energy constraints on the input and output states of the channels.

Theorem 11. Let ${{ \mathcal N }}_{A\to B}$ be an ε-degradable channel with a degrading channel ${{ \mathcal D }}_{B\to E^{\prime} }$, and let $G\in { \mathcal P }({{ \mathcal H }}_{A})$ and $G^{\prime} \in { \mathcal P }({{ \mathcal H }}_{E^{\prime} })$ be Gibbs observables, such that for all input states ${\rho }_{{A}^{n}}\in { \mathcal D }({H}_{A}^{\otimes n})$ satisfying input average energy constraints $\mathrm{Tr}\{{\overline{G}}_{n}{\rho }_{{A}^{n}}\}\leqslant W$, the following output average energy constraints are satisfied:

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }{\hat{{ \mathcal N }}}^{\otimes n}({\rho }_{{A}^{n}})\},\quad \mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }({{ \mathcal D }}^{\otimes n}\,\circ \,{{ \mathcal N }}^{\otimes n})({\rho }_{{A}^{n}})\}\leqslant W^{\prime} ,\end{eqnarray} \tag{ 4.9 }$$

where ${\hat{{ \mathcal N }}}_{A\to E}$ is a complementary channel of ${ \mathcal N }$ and $E^{\prime} \simeq E$. Then the energy-constrained quantum capacity $Q({ \mathcal N },G,W)$ is bounded from above as

$$\begin{eqnarray}&&Q({ \mathcal N },G,W)\leqslant {U}_{{ \mathcal D }}({ \mathcal N },G,W)+(2\varepsilon ^{\prime} +4\delta )H(\gamma (W^{\prime} /\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta ),\end{eqnarray} \tag{ 4.10 }$$

with $\varepsilon ^{\prime} \in (\varepsilon ,1],W,W^{\prime} \in [0,\infty )$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. Let

$$\begin{eqnarray*} & & \,{\sigma }_{{B}^{n}}={{ \mathcal N }}^{\otimes n}({\rho }_{{A}^{n}}),\\ & & {\rho }_{E{{\prime} }^{j}{E}^{(n-j)}}^{j}=({{ \mathcal D }}^{\otimes j}\,\circ \,{{ \mathcal N }}^{\otimes j})\otimes {\hat{{ \mathcal N }}}^{\otimes (n-j)}({\rho }_{{A}^{n}}),\end{eqnarray*}$$

and consider the following chain of inequalities:

$$\begin{eqnarray}&&H{({B}^{n})}_{\sigma }-H{({E}^{n})}_{{\rho }^{0}}\\ &&\quad =\,H{({B}^{n})}_{\sigma }-H{({E}^{{\prime} n})}_{{\rho }^{n}}+H{({E}^{{\prime} n})}_{{\rho }^{n}}-H{({E}^{n})}_{{\rho }^{0}}\,\,\,\end{eqnarray} \tag{ 4.11 }$$

$$\begin{eqnarray}&&\,\leqslant \,{U}_{{{ \mathcal D }}^{\otimes n}}({{ \mathcal N }}^{\otimes n},{\overline{G}}_{n},W)+H{({E}^{{\prime} n})}_{{\rho }^{n}}-H{({E}^{n})}_{{\rho }^{0}}\,\,\end{eqnarray} \tag{ 4.12 }$$

$$\begin{eqnarray} & = & n\,{U}_{{ \mathcal D }}({ \mathcal N },G,W)\\ & & +\displaystyle \sum _{j=1}^{n}[H{({E}_{j}^{{\prime} }| {E}_{1}^{{\prime} }...{E}_{j-1}^{{\prime} }{E}_{j+1}...{E}_{n})}_{{\rho }^{j}}-H{({E}_{j}| {E}_{1}^{{\prime} }...{E}_{j-1}^{{\prime} }{E}_{j+1}...{E}_{n})}_{{\rho }^{j-1}}]\,\end{eqnarray} \tag{ 4.13 }$$

$$\begin{eqnarray}&&\,\leqslant \,n[{U}_{{ \mathcal D }}({ \mathcal N },G,W)+(2\varepsilon ^{\prime} +4\delta )\left(\displaystyle \sum _{j=1}^{n}\displaystyle \frac{1}{n}H(\gamma ({W}_{j}^{{\prime} }/\delta ))\right)+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )]\end{eqnarray} \tag{ 4.14 }$$

$$\begin{eqnarray}&&\leqslant \,n[{U}_{{ \mathcal D }}({ \mathcal N },G,W)+(2\varepsilon ^{\prime} +4\delta )H\left(\displaystyle \frac{1}{n}\displaystyle \sum _{j=1}^{n}\gamma ({W}_{j}^{{\prime} }/\delta )\right)+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )]\,\end{eqnarray} \tag{ 4.15 }$$

$$\begin{eqnarray}&&\leqslant \,n[{U}_{{ \mathcal D }}({ \mathcal N },G,W)+(2\varepsilon ^{\prime} +4\delta )H(\gamma (W^{\prime} /\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )].\,\end{eqnarray} \tag{ 4.16 }$$

The first inequality follows from the definition in (4.1). The second equality follows from lemma 3 and the telescoping technique. Let ${W}_{j}^{{\prime} }$ denote the energy constraint on the jth output state of both the channels ${ \mathcal D }\,\circ \,{ \mathcal N }$ and $\hat{{ \mathcal N }}$, i.e., $\mathrm{Tr}\{G^{\prime} ({ \mathcal D }\,\circ \,{ \mathcal N })({\rho }_{{A}_{j}})\},\mathrm{Tr}\{G^{\prime} \hat{{ \mathcal N }}({\rho }_{{A}_{j}})\}\leqslant {W}_{j}^{{\prime} }$ where $\tfrac{1}{n}{\sum }_{j}{W}_{j}^{\prime} \leqslant W^{\prime}$. Then the second inequality holds because $\tfrac{1}{2}\parallel {\rho }^{j}-{\rho }^{j-1}{\parallel }_{1}\leqslant \varepsilon$ 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 $\mathrm{Tr}\{\tfrac{1}{n}{\sum }_{j=1}^{n}G\gamma ({W}_{j}^{{\prime} }/\delta )\}=\tfrac{1}{n}{\sum }_{j=1}^{n}\mathrm{Tr}\{G\gamma ({W}_{j}^{{\prime} }/\delta )\}\leqslant W^{\prime} /\delta$, and $\gamma (W^{\prime} /\delta )$ is the Gibbs state that maximizes the entropy corresponding to the energy $W^{\prime} /\delta$. Since the chain of inequalities is true for all ${\rho }_{{A}^{n}}$ satisfying the input average energy constraint, from (3.64) and the above, we get that

$$\begin{eqnarray}&&\displaystyle \frac{1}{n}{I}_{c}({{ \mathcal N }}^{\otimes n},{\overline{G}}_{n},W)\leqslant {U}_{{ \mathcal D }}({ \mathcal N },G,W)+(2\varepsilon ^{\prime} +4\delta )H(\gamma (W^{\prime} /\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta ).\end{eqnarray} \tag{ 4.17 }$$

Since the last inequality holds for all n, we obtain the desired result by taking the limit $n\to \infty$ and applying (3.63). ■

An upper bound on the quantum capacity of an ε-close-degradable channel was established as [SSWR17, 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 12. Let ${{ \mathcal N }}_{A\to B}$ be an $\varepsilon$-close-degradable channel, i.e., $\tfrac{1}{2}{\parallel { \mathcal N }-{ \mathcal M }\parallel }_{\diamond }\leqslant \varepsilon \lt \varepsilon ^{\prime} \leqslant 1$, where ${{ \mathcal M }}_{A\to B}$ is a degradable channel. Let $G\in { \mathcal P }({{ \mathcal H }}_{A}),G^{\prime} \in { \mathcal P }({{ \mathcal H }}_{B})$ be Gibbs observables, such that for all input states ${\rho }_{{{RA}}^{n}}\in { \mathcal D }({{ \mathcal H }}_{R}\otimes {{ \mathcal H }}_{A}^{\otimes n})$ satisfying the input average energy constraint $\mathrm{Tr}\{{\overline{G}}_{n}{\rho }_{{A}^{n}}\}\leqslant W$, the following output average energy constraints are satisfied:

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }{{ \mathcal N }}^{\otimes n}({\rho }_{{A}^{n}})\},\mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }{{ \mathcal M }}^{\otimes n}({\rho }_{{A}^{n}})\}\leqslant W^{\prime} ,\end{eqnarray} \tag{ 4.18 }$$

where $W,W^{\prime} \in [0,\infty )$. Then the energy-constrained quantum capacity $Q({ \mathcal N },G,W)$ is bounded from above as

$$\begin{eqnarray}&&Q({ \mathcal N },G,W)\leqslant {I}_{c}({ \mathcal M },G,W)+(4\varepsilon ^{\prime} +8\delta )H(\gamma (W^{\prime} /\delta ))+2g(\varepsilon ^{\prime} )+4{h}_{2}(\delta ),\end{eqnarray} \tag{ 4.19 }$$

with $\varepsilon ^{\prime} \in (\varepsilon ,1]$ and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. Let ${\omega }_{{{RB}}^{n}}=({\mathrm{id}}_{R}\otimes {{ \mathcal N }}^{\otimes n})({\rho }_{{{RA}}^{n}})$ and ${\tau }_{{{RB}}^{n}}=({\mathrm{id}}_{R}\otimes {{ \mathcal M }}^{\otimes n})({\rho }_{{{RA}}^{n}})$, and consider the following chain of inequalities:

$$\begin{eqnarray}&&H{({B}^{n})}_{\omega }-H{({{RB}}^{n})}_{\omega }-H{({B}^{n})}_{\tau }+H{({{RB}}^{n})}_{\tau }\\ &&\quad =\,H{({B}^{n})}_{\omega }-H{({B}^{n})}_{\tau }+H{({{RB}}^{n})}_{\tau }-H{({{RB}}^{n})}_{\omega }\end{eqnarray} \tag{ 4.20 }$$

$$\begin{eqnarray}&&\,\leqslant \,2n[(2\varepsilon ^{\prime} +4\delta )H(\gamma (W/\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )].\end{eqnarray} \tag{ 4.21 }$$

The first inequality follows from applying theorem 8 twice. Then from lemma 2,

$$\begin{eqnarray}&&Q({ \mathcal N },G,W)\leqslant Q({ \mathcal M },G,W)+(4\varepsilon ^{\prime} +8\delta )H(\gamma (W^{\prime} /\delta ))+2g(\varepsilon ^{\prime} )+4{h}_{2}(\delta ).\end{eqnarray} \tag{ 4.22 }$$

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]. ■

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 [SSWR17]. 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 13. Let ${{ \mathcal N }}_{A\to B}$ be a finite-dimensional $\varepsilon$-degradable channel with a degrading channel ${{ \mathcal D }}_{B\to E^{\prime} }$, and let $\hat{{ \mathcal N }}:{ \mathcal T }(A)\to { \mathcal T }(E)$ be a complementary channel of ${ \mathcal N }$, such that $E^{\prime} \simeq E$. If

$$\begin{eqnarray}&&{U}_{{ \mathcal D }}({ \mathcal N })=\mathop{\max }\limits_{\rho \in { \mathcal D }({{ \mathcal H }}_{A})}[H({ \mathcal N }(\rho ))-H(({ \mathcal D }\,\circ \,{ \mathcal N })(\rho ))],\end{eqnarray} \tag{ 4.23 }$$

then the private capacity $P({ \mathcal N })$ of ${ \mathcal N }$ is bounded from above as

$$\begin{eqnarray}&&P({ \mathcal N })\leqslant {U}_{{ \mathcal D }}({ \mathcal N })+6\varepsilon {\mathrm{log}}_{2}\dim ({{ \mathcal H }}_{E})+3g(\varepsilon ).\end{eqnarray} \tag{ 4.24 }$$

Proof. Consider Stinespring dilations ${ \mathcal U }\,:{ \mathcal T }(A)\to { \mathcal T }(B)\otimes { \mathcal T }(E)$ and ${ \mathcal V }\,:{ \mathcal T }(B)\to { \mathcal T }(E^{\prime} )\otimes { \mathcal T }(F)$ of the channel ${ \mathcal N }$ and the degrading channel ${ \mathcal D }$, respectively. Let ${\rho }_{{{XA}}^{n}}$ be a classical-quantum state in correspondence with an ensemble $\{{p}_{X}(x),{\rho }_{{A}^{n}}^{x}\}$:

$$\begin{eqnarray}&&{\rho }_{{{XA}}^{n}}=\displaystyle \sum _{x}{p}_{X}(x)| x\rangle \langle x{| }_{X}\otimes {\rho }_{{A}^{n}}^{x},\end{eqnarray} \tag{ 4.25 }$$

and let

$$\begin{eqnarray}&&{\omega }_{{{XE}}^{n}{E}^{{\prime} n}{F}^{n}}=\displaystyle \sum _{x}{p}_{X}(x)| x\rangle \langle x{| }_{X}\otimes ({\mathrm{id}}_{E}^{\otimes n}\otimes {{ \mathcal V }}^{\otimes n})\,\circ \,{{ \mathcal U }}^{\otimes n}({\rho }_{{A}^{n}}^{x}).\end{eqnarray} \tag{ 4.26 }$$

Consider the following extension of ${\omega }_{{{XE}}^{n}{E}^{{\prime} n}{F}^{n}}$:

$$\begin{eqnarray}&&{\sigma }_{{{XYE}}^{n}{E}^{{\prime} n}{F}^{n}}=\displaystyle \sum _{x,y}{p}_{X}(x){p}_{Y| X}(y| x)| x\rangle \langle x{| }_{X}\otimes | y\rangle \langle y{| }_{Y}\otimes ({\mathrm{id}}_{E}^{\otimes n}\otimes {{ \mathcal V }}^{\otimes n})\,\circ \,{{ \mathcal U }}^{\otimes n}({\psi }_{{A}^{n}}^{x,y}),\end{eqnarray} \tag{ 4.27 }$$

where ${\psi }_{{A}^{n}}^{x,y}$ is a pure state, and let ${\sigma }_{{E}^{n}{E}^{{\prime} n}{F}^{n}}^{x,y}=({\mathrm{id}}_{E}^{\otimes n}\otimes {{ \mathcal V }}^{\otimes n})\circ {{ \mathcal U }}^{\otimes n}({\psi }_{{A}^{n}}^{x,y})$. Consider the following chain of inequalities:

$$\begin{eqnarray}&&I{(X;{B}^{n})}_{\omega }-I{(X;{E}^{n})}_{\omega }=\,I{(X;{F}^{n}| {E}^{{\prime} n})}_{\omega }+I{(X;{E}^{{\prime} n})}_{\omega }-I{(X;{E}^{n})}_{\omega }\,\,\end{eqnarray} \tag{ 4.28 }$$

$$\begin{eqnarray}&&\,=\,I{(X;{F}^{n}| {E}^{{\prime} n})}_{\omega }+H{({E}^{{\prime} n})}_{\omega }-H{({E}^{n})}_{\omega }+H{({E}^{n}| X)}_{\omega }-H{({E}^{{\prime} n}| X)}_{\omega }\end{eqnarray} \tag{ 4.29 }$$

$$\begin{eqnarray}&&\,\leqslant \,I{(X;{F}^{n}| {E}^{{\prime} n})}_{\omega }+2n[2\varepsilon {\mathrm{log}}_{2}\dim ({{ \mathcal H }}_{E})+g(\varepsilon )]\,\end{eqnarray} \tag{ 4.30 }$$

$$\begin{eqnarray}&&\,\leqslant \,I{({XY};{F}^{n}| {E}^{{\prime} n})}_{\sigma }+n[4\varepsilon {\mathrm{log}}_{2}\dim ({{ \mathcal H }}_{E})+2g(\varepsilon )]\end{eqnarray} \tag{ 4.31 }$$

$$\begin{eqnarray}&&\,=\,H{({F}^{n}| {E}^{{\prime} n})}_{\sigma }-H{({F}^{n}{E}^{{\prime} n}| {XY})}_{\sigma }+H{({E}^{{\prime} n}| {XY})}_{\sigma }\\ &&\qquad +n[4\varepsilon {\mathrm{log}}_{2}\dim ({{ \mathcal H }}_{E})+2g(\varepsilon )]\end{eqnarray} \tag{ 4.32 }$$

$$\begin{eqnarray}&&=\,H{({F}^{n}| {E}^{{\prime} n})}_{\sigma }-H{({E}^{n}| {XY})}_{\sigma }+H{({E}^{{\prime} n}| {XY})}_{\sigma }\\ &&\quad +n[4\varepsilon {\mathrm{log}}_{2}\dim ({{ \mathcal H }}_{E})+2g(\varepsilon )]\end{eqnarray} \tag{ 4.33 }$$

$$\begin{eqnarray}&&\,\leqslant n[{U}_{{ \mathcal D }}({ \mathcal N })+6\varepsilon {\mathrm{log}}_{2}\dim ({{ \mathcal H }}_{E})+3g(\varepsilon )].\,\end{eqnarray} \tag{ 4.34 }$$

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 ${\sigma }_{{E}^{n}{E}^{{\prime} n}{F}^{n}}^{x,y}$ is a pure state, so that $H{({F}^{n}{E}^{{\prime} n})}_{{\sigma }^{x,y}}=H{({E}^{n})}_{{\sigma }^{x,y}}$. The last inequality follows from the definition in (4.23), and additivity of ${U}_{{ \mathcal D }}({ \mathcal N })$ [SSWR17]. Also, we applied the telescoping technique for each ${\sigma }^{x,y}$ 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 $\{{p}_{X}(x),{\rho }_{{A}^{n}}^{x}\}$, the final result follows from the definition of private information of the channel, dividing by n, taking the limit $n\to \infty$, 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 14. Let ${{ \mathcal N }}_{A\to B}$ be an $\varepsilon$-degradable channel with a degrading channel ${{ \mathcal D }}_{B\to E^{\prime} }$, and let $G\in { \mathcal P }({{ \mathcal H }}_{A}),G^{\prime} \in { \mathcal P }({{ \mathcal H }}_{E^{\prime} })$ be Gibbs observables, such that for all input states ${\rho }_{{A}^{n}}\in { \mathcal D }({H}_{A}^{\otimes n})$ satisfying input average energy constraints $\mathrm{Tr}\{{\overline{G}}_{n}{\rho }_{{A}^{n}}\}\leqslant W$, the following output average energy constraints are satisfied:

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }{\hat{{ \mathcal N }}}^{\otimes n}({\rho }_{{A}^{n}})\},\mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }({{ \mathcal D }}^{\otimes n}\,\circ \,{{ \mathcal N }}^{\otimes n})({\rho }_{{A}^{n}})\}\leqslant W^{\prime} ,\end{eqnarray} \tag{ 4.35 }$$

where ${\hat{{ \mathcal N }}}_{A\to E}$ is a complementary channel of ${ \mathcal N }$, and $E^{\prime} \simeq E$. Then the energy-constrained private capacity is bounded from above as

$$\begin{eqnarray}&&P({ \mathcal N },G,W)\leqslant {U}_{{ \mathcal D }}({ \mathcal N },G,W)+(6\varepsilon ^{\prime} +12\delta )H(\gamma (W^{\prime} /\delta ))+3g(\varepsilon ^{\prime} )+6{h}_{2}(\delta ),\end{eqnarray} \tag{ 4.36 }$$

with $\varepsilon ^{\prime} \in (\varepsilon ,1],W,W^{\prime} \in [0,\infty )$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. Since the proof is similar to the above one and previous ones, we just summarize it briefly below. Consider Stinespring dilations ${ \mathcal U }\,:{ \mathcal T }(A)\to { \mathcal T }(B)\otimes { \mathcal T }(E)$ and ${ \mathcal V }\,:{ \mathcal T }(B)\to { \mathcal T }(E^{\prime} )\otimes { \mathcal T }(F)$ of the channel ${ \mathcal N }$ and the degrading channel ${ \mathcal D }$, respectively. Then the action of ${{ \mathcal U }}^{\otimes n}$ followed by ${{ \mathcal V }}^{\otimes n}$ on the ensemble $\{{p}_{X}(x),{\rho }_{{A}^{n}}^{x}\}$ leads to the following ensemble:

$$\begin{eqnarray}&&\{{p}_{X}(x),{\omega }_{{E}^{n}{E}^{{\prime} n}{F}^{n}}^{x}\equiv ({\mathrm{id}}_{E}^{\otimes n}\otimes {{ \mathcal V }}^{\otimes n})\,\circ \,{{ \mathcal U }}^{\otimes n}({\rho }_{{A}^{n}}^{x})\}.\end{eqnarray} \tag{ 4.37 }$$

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:

$$\begin{eqnarray}&&I{(X;{B}^{n})}_{\omega }-I{(X;{E}^{n})}_{\omega }\leqslant n[{U}_{{ \mathcal D }}({ \mathcal N },G,W)+(6\varepsilon ^{\prime} +12\delta )H(\gamma (W^{\prime} /\delta ))+3g(\varepsilon ^{\prime} )+6{h}_{2}(\delta )].\end{eqnarray} \tag{ 4.38 }$$

The desired result follows from dividing by n, taking the limit $n\to \infty$, the definition of the energy-constrained private information of the channel, and using the fact that the regularized energy-constrained private information is an upper bound on the energy-constrained private capacity of a quantum channel [WQ16]. ■

An upper bound on the private capacity of an ε-close-degradable channel was established as [SSWR17, 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 15. Let ${{ \mathcal N }}_{A\to B}$ be an $\varepsilon$-close-degradable channel, i.e., $\tfrac{1}{2}{\parallel { \mathcal N }-{ \mathcal M }\parallel }_{\diamond }\leqslant \varepsilon \lt \varepsilon ^{\prime} \leqslant 1$, where ${{ \mathcal M }}_{A\to B}$ is a degradable channel. Let $G\in { \mathcal P }({{ \mathcal H }}_{A}),G^{\prime} \in { \mathcal P }({{ \mathcal H }}_{B})$ be Gibbs observables, such that for all input states ${\rho }_{{A}^{n}}\in { \mathcal D }({{ \mathcal H }}_{A}^{\otimes n})$ satisfying input average energy constraints $\mathrm{Tr}\{{\overline{G}}_{n}{\rho }_{{A}^{n}}\}\leqslant W$, the following output average energy constraints are satisfied:

$$\begin{eqnarray}&&\mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }{{ \mathcal N }}^{\otimes n}({\rho }_{{A}^{n}})\},\mathrm{Tr}\{{\overline{G}}_{n}^{{\prime} }{{ \mathcal M }}^{\otimes n}({\rho }_{{A}^{n}})\}\leqslant W^{\prime} ,\end{eqnarray} \tag{ 4.39 }$$

where $W,W^{\prime} \in [0,\infty )$. Then

$$\begin{eqnarray}&&P({ \mathcal N },G,W)\leqslant {I}_{c}({ \mathcal M },G,W)+(8\varepsilon ^{\prime} +16\delta )H(\gamma (W^{\prime} /\delta ))+4g(\varepsilon ^{\prime} )+8{h}_{2}(\delta ),\end{eqnarray} \tag{ 4.40 }$$

with $\varepsilon ^{\prime} \in (\varepsilon ,1]$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. We follow the proof of [LS09, corollary 15] closely, but incorporate energy constraints. Consider Stinespring dilations ${ \mathcal U }\,:{ \mathcal T }(A)\to { \mathcal T }(B)\otimes { \mathcal T }(E)$ and ${ \mathcal V }\,:{ \mathcal T }(A)\to { \mathcal T }(B)\otimes { \mathcal T }(E)$ of the channels ${ \mathcal N }$ and ${ \mathcal M }$, respectively. Consider an input ensemble $\{{p}_{X}(x),{\rho }_{{A}^{n}}^{x}\}$, which leads to the output ensembles

$$\begin{eqnarray}&&\{{p}_{X}(x),{\omega }^{x}\equiv {{ \mathcal U }}^{\otimes n}({\rho }_{{A}^{n}}^{x})\},\,\end{eqnarray} \tag{ 4.41 }$$

$$\begin{eqnarray}&&\{{p}_{X}(x),{\tau }^{x}\equiv {{ \mathcal V }}^{\otimes n}({\rho }_{{A}^{n}}^{x})\}.\end{eqnarray} \tag{ 4.42 }$$

Supposing at first that the index x is discrete, from four times applying theorem 8 and employing the same expansions as in the proof of [LS09, corollary 15] , we get

$$\begin{eqnarray}&&I{(X;{B}^{n})}_{\omega }-I{(X;{E}^{n})}_{\omega }-[I{(X;{B}^{n})}_{\tau }-I{(X;{E}^{n})}_{\tau }]\leqslant 4n[(2\varepsilon ^{\prime} +4\delta )H(\gamma (W/\delta ))+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta )].\end{eqnarray} \tag{ 4.43 }$$

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

$$\begin{eqnarray}&&P({ \mathcal N },G,W)\leqslant P({ \mathcal M },G,W)+(8\varepsilon ^{\prime} +16\delta )H(\gamma (W^{\prime} /\delta ))+4g(\varepsilon ^{\prime} )+8{h}_{2}(\delta )\end{eqnarray} \tag{ 4.44 }$$

$$\begin{eqnarray}&&\,=\,{I}_{c}({ \mathcal M },G,W)+(8\varepsilon ^{\prime} +16\delta )H(\gamma (W^{\prime} /\delta ))+4g(\varepsilon ^{\prime} )+8{h}_{2}(\delta ).\,\end{eqnarray} \tag{ 4.45 }$$

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]. ■

In this section, we provide an upper bound using the theorem that any thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ can be decomposed as the concatenation of a pure-loss channel ${{ \mathcal L }}_{\eta ^{\prime} ,0}$ with transmissivity $\eta ^{\prime}$ followed by a quantum-limited amplifier channel ${{ \mathcal A }}_{G,0}$ with gain G [CGH06, GPNBL⁺12], i.e.,

$$\begin{eqnarray}&&{{ \mathcal L }}_{\eta ,{N}_{B}}={{ \mathcal A }}_{G,0}\,\circ \,{{ \mathcal L }}_{\eta ^{\prime} ,0},\end{eqnarray} \tag{ 5.1 }$$

where $G=(1-\eta ){N}_{B}+1$, and $\eta ^{\prime} =\eta /G$. In theorem 26, we prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound.

Theorem 16. An upper bound on the quantum capacity of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}$, and input mean photon number constraint ${N}_{S}$ is given by

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant \max \{0,{Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\},\end{eqnarray} \tag{ 5.2 }$$

$$\begin{eqnarray}&&{Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\equiv g(\eta ^{\prime} {N}_{S})-g[(1-\eta ^{\prime} ){N}_{S}],\end{eqnarray} \tag{ 5.3 }$$

with $\eta ^{\prime} =\eta /((1-\eta ){N}_{B}+1)$.

Proof. An upper bound on the energy-constrained quantum capacity can be established by using (5.1) and a data-processing argument. We find that

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})=Q({{ \mathcal A }}_{G}\,\circ \,{{ \mathcal L }}_{\eta ^{\prime} ,0},{N}_{S})\,\,\end{eqnarray} \tag{ 5.4 }$$

$$\begin{eqnarray}&&\leqslant \,Q({{ \mathcal L }}_{\eta ^{\prime} ,0},{N}_{S})\,\end{eqnarray} \tag{ 5.5 }$$

$$\begin{eqnarray}&&=\,\max \{0,g(\eta ^{\prime} {N}_{S})-g[(1-\eta ^{\prime} ){N}_{S}]\}.\,\end{eqnarray} \tag{ 5.6 }$$

The first inequality follows from definitions and data-processing—the energy-constrained capacity of ${{ \mathcal A }}_{G}\,\circ \,{{ \mathcal L }}_{\eta ^{\prime} ,0}$ cannot exceed that of ${{ \mathcal L }}_{\eta ^{\prime} ,0}$. The second equality follows from the formula for the energy-constrained quantum capacity of a pure-loss bosonic channel with transmissivity $\eta ^{\prime}$ and input mean photon number N_S [WHG12, WQ16]. ■

Remark 17. Applying remark 9, we find the following data-processing bound ${Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}})$ on the unconstrained quantum capacity of bosonic thermal channels:

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}})\leqslant {Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}})=\mathop{\sup }\limits_{{N}_{S}:{N}_{S}\in [0,\infty ]}{Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\,\end{eqnarray} \tag{ 5.7 }$$

$$\begin{eqnarray}&&\,\,=\mathop{\mathrm{lim}}\limits_{{N}_{S}\to \infty }{Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\,\end{eqnarray} \tag{ 5.8 }$$

$$\begin{eqnarray}&&\,\,=\,{\mathrm{log}}_{2}(\eta /(1-\eta ))-{\mathrm{log}}_{2}({N}_{B}+1),\end{eqnarray} \tag{ 5.9 }$$

where the second equality follows from the monotonicity of $g(\eta {N}_{S})-g[(1-\eta ){N}_{S}]$ with respect to N_S for $\eta \geqslant 1/2$ [GSE08].

The bound

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant -{\mathrm{log}}_{2}([1-\eta ]{\eta }^{{N}_{B}})-g({N}_{B})\end{eqnarray} \tag{ 5.10 }$$

was found in [PLOB17, WTB17]. Moreover, the following bound was established quite recently in [RMG18, equation (40)]:

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant \max \left\{0,{\mathrm{log}}_{2}\displaystyle \frac{\eta -N(1-\eta )}{(1+N)(1-\eta )}\right\}.\end{eqnarray} \tag{ 5.11 }$$

As discussed in [RMG18], a comparison of (5.9) with the bounds from (5.10) and (5.11) leads to the conclusion that the bound given in (5.11) is always tighter than (5.9). However, (5.9) and the bound in (5.10) are incomparable as one is better than the other for certain parameter regimes. Also, (5.10) is tighter than (5.11) for certain parameter regimes.

We note that the upper bound in (5.11) was independently established in [NAJ18].

Remark 18. The data-processing bound ${Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})$ on the energy-constrained quantum capacity $Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})$ places a strong restriction on the channel parameters η and N_B. Since the quantum capacity of a pure-loss channel with transmissivity $\eta ^{\prime}$ is non-zero only for $\eta ^{\prime} \gt 1/2$, the energy-constrained quantum capacity $Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})$ is non-zero only for

$$\begin{eqnarray}&&1\geqslant \eta \gt \displaystyle \frac{{N}_{B}+1}{{N}_{B}+2}.\end{eqnarray} \tag{ 5.12 }$$

However, [CGH06, section 4] provides a stronger restriction on η and N_B than (5.12) does.

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 11, 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 ${ \mathcal D }$ given in (5.20), such that the concatenation of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ followed by ${ \mathcal D }$ is close in diamond distance to the complementary channel ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$ of the thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$.

We start by motivating the reason for choosing the particular degrading channel in (5.20), which is depicted in figure 1, and then we find an upper bound on the diamond distance between ${ \mathcal D }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$ and ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$. 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 (5.24). 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 19). 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 11. We finally provide an upper bound on the energy-constrained quantum capacity of a thermal channel by using all these tools and invoking theorem 11.

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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 ${ \mathcal B }$ and ${ \mathcal B }^{\prime}$ represent beamsplitter transformations with transmissivity η and $(1-\eta )/\eta$, respectively. In the Heisenberg picture, the beamsplitter transformation ${{ \mathcal B }}_{{C}_{1}{D}_{1}\to {C}_{2}{D}_{2}}$ is given by

$$\begin{eqnarray}&&{\hat{c}}_{2}=\sqrt{\eta }{\hat{c}}_{1}-\sqrt{1-\eta }{\hat{d}}_{1},\end{eqnarray} \tag{ 5.13 }$$

$$\begin{eqnarray}&&\,{\hat{d}}_{2}=\sqrt{1-\eta }{\hat{c}}_{1}+\sqrt{\eta }{\hat{d}}_{1}.\,\end{eqnarray} \tag{ 5.14 }$$

Similarly, the beamsplitter transformation ${{ \mathcal B }}_{{C}_{1}{D}_{1}\to {C}_{2}{D}_{2}}^{{\prime} }$ is given by

$$\begin{eqnarray}&&{\hat{c}}_{2}=\sqrt{(1-\eta )/\eta }{\hat{c}}_{1}+\sqrt{(2\eta -1)/\eta }{\hat{d}}_{1},\,\end{eqnarray} \tag{ 5.15 }$$

$$\begin{eqnarray}&&{\hat{d}}_{2}=-\sqrt{(2\eta -1)/\eta }{\hat{c}}_{1}+\sqrt{(1-\eta )/\eta }{\hat{d}}_{1},\end{eqnarray} \tag{ 5.16 }$$

where ${\hat{c}}_{1},{\hat{c}}_{2},{\hat{d}}_{1}$, and ${\hat{d}}_{2}$ are annihilation operators representing various modes involved in the beamsplitter transformations. Here, $\eta \in [1/2,1]$. It is important to stress that there is a difference in phase between ${ \mathcal B }$ and ${ \mathcal B }^{\prime}$ beamsplitter transformations, which is crucial in our development.

Consider the following action of the thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ on an input state ${\phi }_{{RA}}$:

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes {{ \mathcal L }}_{\eta ,{N}_{B}})({\phi }_{{RA}})={\mathrm{Tr}}_{{E}_{1}{E}_{2}}\{{{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}})\},\end{eqnarray} \tag{ 5.17 }$$

where R is a reference system and ${\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}}$ is a two-mode squeezed vacuum state with parameter N_B, as defined in (3.27).

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 ${\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}}$ has the following form:

$$\begin{eqnarray}V=\left[\begin{array}{cc}2{N}_{B}+1 & 2\sqrt{{N}_{B}(1+{N}_{B})}\\ 2\sqrt{{N}_{B}(1+{N}_{B})} & 2{N}_{B}+1\end{array}\right].\end{eqnarray} \tag{ 5.18 }$$

The action of a complementary channel ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$ on an input state ${\phi }_{{RA}}$ is given by

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes {\hat{{ \mathcal L }}}_{\eta ,{N}_{B}})({\phi }_{{RA}})={\mathrm{Tr}}_{B}\{{{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}})\}.\end{eqnarray} \tag{ 5.19 }$$

It can be understood from figure 1 that the system R is correlated with the input system A for the channel, and the system $E^{\prime}$ is the environment's input. The beamsplitter transformation ${ \mathcal B }$ then leads to systems B and E₂. Hence, the output of the thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ is system B, and the outputs of the complementary channel ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$ are systems E₁ and E₂.

Our aim is to introduce a degrading channel ${ \mathcal D }$, such that the combined state of R and the output of ${ \mathcal D }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$ emulate the combined state of $R,{E}_{1}$, and E₂, to an extent. This will then allow us to bound the diamond distance between ${ \mathcal D }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$ and ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$ from above. For the case when there 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 $(1-\eta )/\eta$ [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 $(1-\eta )/\eta$ and the beamsplitter transformation ${ \mathcal B }^{\prime}$ from (5.15)–(5.16). As described in figure 1, the output B of the thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ becomes an input to the beamsplitter ${ \mathcal B }^{\prime}$. We consider one mode (F in figure 1) of the two-mode squeezed vacuum state ${\psi }_{\mathrm{TMS}}{({N}_{B})}_{{{FE}}_{1}^{{\prime} }}$ as an environmental input for ${ \mathcal B }^{\prime}$, so that the subsystem ${E}_{1}^{{\prime} }$ mimics E₁. Hence, our choice of degrading channel seems reasonable, as the combined state of system R and output systems ${E}_{1}^{{\prime} },{E}_{2}^{{\prime} }$ of ${ \mathcal D }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$ emulates the combined state of $R,{E}_{1}$, and E₂, 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 ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\,:{ \mathcal T }(B)\to { \mathcal T }({E}_{1}^{{\prime} })\otimes { \mathcal T }({E}_{2}^{{\prime} })$. More formally, ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}$ has the following action on the output state ${{ \mathcal L }}_{\eta ,{N}_{B}}({\phi }_{{RA}})$:

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes [{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}])({\phi }_{{RA}})={\mathrm{Tr}}_{G}\{{{ \mathcal B }}_{{BF}\to {E}_{2}^{{\prime} }G}^{{\prime} }({{ \mathcal L }}_{\eta ,{N}_{B}}({\phi }_{{RA}})\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{{{FE}}_{1}^{{\prime} }}))\}.\end{eqnarray} \tag{ 5.20 }$$

Next, we provide a strategy to bound the diamond distance between ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$ and ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$. Consider the following submatrix corresponding to the position-quadrature operators of the covariance matrix of an input state ${\phi }_{{RA}}$:

$$\begin{eqnarray}\gamma =\left[\begin{array}{cc}a & c\\ c & b\end{array}\right]\end{eqnarray} \tag{ 5.21 }$$

where $a,b,c\in {\mathbb{R}}$ are such that the above is the position-quadrature part of a legitimate covariance matrix. Let ${\xi }_{{{RE}}_{2}^{{\prime} }{E}_{2}{E}_{1}{{GE}}_{1}^{{\prime} }}$ denote the state after the beamsplitter transformations act on an input state ${\phi }_{{RA}}$:

$$\begin{eqnarray}&&{\xi }_{{{RE}}_{2}^{{\prime} }{E}_{2}{E}_{1}{{GE}}_{1}^{{\prime} }}={{ \mathcal B }}_{{BF}\to {E}_{2}^{{\prime} }G}^{{\prime} }[{{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}[{\phi }_{{RA}}\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}}]\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{{{FE}}_{1}^{{\prime} }})].\end{eqnarray} \tag{ 5.22 }$$

Then the submatrix corresponding to the position-quadrature operators of the covariance matrix of the output state in (5.20) is given by [Mat]:

$$\begin{eqnarray}\gamma ^{\prime} =\left[\begin{array}{ccc}a & c\sqrt{1-\eta } & 0\\ c\sqrt{1-\eta } & b+\eta (1-b+2{N}_{B}) & 2\sqrt{{N}_{B}(1+{N}_{B})(2-1/\eta )}\\ 0 & 2\sqrt{{N}_{B}(1+{N}_{B})(2-1/\eta )} & 2{N}_{B}+1\end{array}\right].\end{eqnarray} \tag{ 5.23 }$$

Now, we introduce a particular channel that simulates the action of ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$ on an input state ${\phi }_{{RA}}$. We denote this channel by Ξ, and it has the following action on an input state ${\phi }_{{RA}}$:

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes {\rm{\Xi }})({\phi }_{{RA}})={\mathrm{Tr}}_{B}\{{{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes \omega {({N}_{B})}_{E^{\prime} {E}_{1}})\},\end{eqnarray} \tag{ 5.24 }$$

where $\omega {({N}_{B})}_{E^{\prime} {E}_{1}}$ 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:

$$\begin{eqnarray}V^{\prime} =\left[\begin{array}{cc}2{N}_{B}+1 & 2\sqrt{[{N}_{B}(1+{N}_{B})(2\eta -1)]/{\eta }^{2}}\\ 2\sqrt{[{N}_{B}(1+{N}_{B})(2\eta -1)]/{\eta }^{2}} & 2{N}_{B}+1\end{array}\right].\end{eqnarray} \tag{ 5.25 }$$

The matrix $V^{\prime}$ in (5.25) is a well defined submatrix of the covariance matrix for the noisy version of a two-mode squeezed vacuum state, because $(2\eta -1)/{\eta }^{2}\in [0,1]$ for $\eta \in [1/2,1]$. The submatrix of the covariance matrix corresponding to the state in (5.24) is the same as the submatrix in (5.23) [Mat]. In other words, the covariance matrix for the systems $R,{E}_{1}^{{\prime} }$, and ${E}_{2}^{{\prime} }$ in figure 1 is exactly the same as the covariance matrix for the systems $R,{E}_{1}$, and E₂ in figure 2. This equality of covariance matrices is sufficient to conclude that the following equivalence holds for any quantum input state ${\phi }_{{RA}}$ (see [Ser17, chapter 5] for a proof):

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes [{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}])({\phi }_{{RA}})=({\mathrm{id}}_{R}\otimes {\rm{\Xi }})({\phi }_{{RA}}).\end{eqnarray} \tag{ 5.26 }$$

Thus, the channels ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$ and Ξ are indeed the same.

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From (5.19), (5.24), and (5.26), the action of both ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$ and Ξ can be understood as tensoring the state of the environment with the input state of the channel, performing the beamsplitter transformation ${ \mathcal B }$, 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 (5.19) and the concatenation of the thermal channel followed by the degrading channel in (5.20).

Theorem 19. Fix $\eta \in [1/2,1]$. Let ${{ \mathcal L }}_{\eta ,{N}_{B}}$ be a thermal channel with transmissivity $\eta$, and let ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}$ be a degrading channel as defined in (5.20). Then

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\parallel {\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}-{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\circ {{ \mathcal L }}_{\eta ,{N}_{B}}\parallel }_{\diamond }\leqslant \sqrt{1-{\eta }^{2}/\kappa (\eta ,{N}_{B})},\end{eqnarray} \tag{ 5.27 }$$

with

$$\begin{eqnarray}&&\kappa (\eta ,{N}_{B})={\eta }^{2}+{N}_{B}({N}_{B}+1)[1+3{\eta }^{2}-2\eta (1+\sqrt{2\eta -1})].\end{eqnarray} \tag{ 5.28 }$$

Proof. Consider the following chain of inequalities:

$$\begin{eqnarray}&&{\parallel ({\mathrm{id}}_{R}\otimes {\hat{{ \mathcal L }}}_{\eta ,{N}_{B}})({\phi }_{{RA}})-({\mathrm{id}}_{R}\otimes [{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\circ {{ \mathcal L }}_{\eta ,{N}_{B}}])({\phi }_{{RA}})\parallel }_{1}\\ &&=\,{\parallel ({\mathrm{id}}_{R}\otimes {\hat{{ \mathcal L }}}_{\eta ,{N}_{B}})({\phi }_{{RA}})-({\mathrm{id}}_{R}\otimes {\rm{\Xi }})({\phi }_{{RA}})\parallel }_{1}\,\,\end{eqnarray} \tag{ 5.29 }$$

$$\begin{eqnarray}&&=\,{\parallel {\mathrm{Tr}}_{B}\{{{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}})-{{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes \omega {({N}_{B})}_{E^{\prime} {E}_{1}})\}\parallel }_{1}\end{eqnarray} \tag{ 5.30 }$$

$$\begin{eqnarray}&&\leqslant \,{\parallel {{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}})-{{ \mathcal B }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes \omega {({N}_{B})}_{E^{\prime} {E}_{1}})\parallel }_{1}\,\end{eqnarray} \tag{ 5.31 }$$

$$\begin{eqnarray}&&=\,{\parallel {\phi }_{{RA}}\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}}-{\phi }_{{RA}}\otimes \omega {({N}_{B})}_{E^{\prime} {E}_{1}}\parallel }_{1}\,\end{eqnarray} \tag{ 5.32 }$$

$$\begin{eqnarray}&&=\,{\parallel {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}}-\omega {({N}_{B})}_{E^{\prime} {E}_{1}}\parallel }_{1}\,\end{eqnarray} \tag{ 5.33 }$$

$$\begin{eqnarray}&&\leqslant \,2\sqrt{1-F({\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}},\omega {({N}_{B})}_{E^{\prime} {E}_{1}})}.\,\end{eqnarray} \tag{ 5.34 }$$

The first equality follows from (5.26). The second equality follows from (5.19) and (5.24). 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 ${\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}}$ and $\omega {({N}_{B})}_{E^{\prime} {E}_{1}}$ by using their respective covariance matrices in (5.18) and (5.25), in the Uhlmann fidelity formula for two-mode Gaussian states [MM12]. We find [Mat]

$$\begin{eqnarray}&&F({\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}},\omega {({N}_{B})}_{E^{\prime} {E}_{1}})=\displaystyle \frac{{\eta }^{2}}{{\eta }^{2}+{N}_{B}({N}_{B}+1)[1+3{\eta }^{2}-2\eta (1+\sqrt{2\eta -1})]}.\end{eqnarray} \tag{ 5.35 }$$

Since these inequalities hold for any input state ${\phi }_{{RA}}$, the final result follows from the definition of the diamond norm.■

Theorem 20. An upper bound on the quantum capacity of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}$, and input mean photon number constraint ${N}_{S}$ is given by

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant {Q}_{{U}_{2}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\equiv g(\eta {N}_{S}+(1-\eta ){N}_{B})-g({\zeta }_{+})-g({\zeta }_{-})\\ &&\,\,+\,(2\varepsilon ^{\prime} +4\delta )g([(1-\eta ){N}_{S}+(1+\eta ){N}_{B}]/\delta )+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta ),\end{eqnarray} \tag{ 5.36 }$$

with

$$\begin{eqnarray}&&\varepsilon =\sqrt{1-{\eta }^{2}/({\eta }^{2}+{N}_{B}({N}_{B}+1)[1+3{\eta }^{2}-2\eta (1+\sqrt{2\eta -1})])},\,\end{eqnarray} \tag{ 5.37 }$$

$$\begin{eqnarray}&&{\zeta }_{\pm }=\displaystyle \frac{1}{2}(-1+\sqrt{[{(1+2{N}_{B})}^{2}-2\varrho +{(1+2\vartheta )}^{2}\pm 4(\vartheta -{N}_{B})\sqrt{{[1+{N}_{B}+\vartheta ]}^{2}-\varrho }]/2}),\end{eqnarray} \tag{ 5.38 }$$

$$\begin{eqnarray}&&\varrho =4{N}_{B}({N}_{B}+1)(2\eta -1)/\eta ,\,\,\end{eqnarray} \tag{ 5.39 }$$

$$\begin{eqnarray}&&\vartheta =\eta {N}_{B}+(1-\eta ){N}_{S},\,\,\,\end{eqnarray} \tag{ 5.40 }$$

$\varepsilon ^{\prime} \in (\varepsilon ,1]$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. From theorem 19, 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 by the degrading channel, i.e.,

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\parallel {\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}-{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\circ {{ \mathcal L }}_{\eta ,{N}_{B}}\parallel }_{\diamond }\\ &&\,\leqslant \sqrt{1-{\eta }^{2}/({\eta }^{2}+{N}_{B}({N}_{B}+1)[1+3{\eta }^{2}-2\eta (1+\sqrt{2\eta -1})])}\lt \varepsilon ^{\prime} \leqslant 1.\end{eqnarray} \tag{ 5.41 }$$

Due to the input mean photon number constraint N_S, and environment photon number N_B for both ${{ \mathcal L }}_{\eta ,{N}_{B}}$ and ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}$, there is a total photon number constraint $(1-\eta ){N}_{S}+(1+\eta ){N}_{B}$ for the average output of n channel uses of both ${\hat{{ \mathcal L }}}_{\eta ,{N}_{B}}$ and ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}}$. Using these results in theorem 11, we find the following upper bound on the energy-constrained quantum capacity of a thermal channel:

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant {U}_{{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})+(2\varepsilon ^{\prime} +4\delta )g([(1-\eta ){N}_{S}+(1+\eta ){N}_{B}]/\delta )+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta ).\end{eqnarray} \tag{ 5.42 }$$

Using proposition 21, we find that the thermal state with mean photon number N_S optimizes the conditional entropy of degradation ${U}_{{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})$. For the given thermal channel in (5.17) and the degrading channel in (5.20), we find the following analytical expression [Mat]:

$$\begin{eqnarray}&&{U}_{{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})=g(\eta {N}_{S}+(1-\eta ){N}_{B})-g({\zeta }_{+})-g({\zeta }_{-}),\end{eqnarray} \tag{ 5.43 }$$

with ${\zeta }_{\pm }$ defined as in the theorem statement. ■

Proposition 21. Let ${{ \mathcal L }}_{\eta ,{N}_{B}}$ be a thermal channel with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}$, and input mean photon number constraint ${N}_{S}$. Let ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}$ be the degrading channel from (5.20). Then the thermal state with mean photon number ${N}_{S}$ optimizes the conditional entropy of degradation ${U}_{{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})$, defined from (4.2).

Proof. Consider the Stinespring dilation in (5.22) of the degrading channel ${{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}$ from (5.20), and denote it by ${ \mathcal W }$. Then according to (4.2),

$$\begin{eqnarray}&&{U}_{{{ \mathcal D }}_{(1-\eta )/\eta ,{N}_{B}}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})=\mathop{\sup }\limits_{\rho \,:\,\mathrm{Tr}\{\hat{n}\rho \}\leqslant {N}_{S}}H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})(\rho )}.\end{eqnarray} \tag{ 5.44 }$$

Our aim is to find an input state ρ with a certain photon number ${N}_{t}\leqslant {N}_{S}$, such that it maximizes the conditional entropy in (5.44). From the extremality of Gaussian states applied to the conditional entropy [EW07], it suffices to perform the optimization in (5.44) 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 (5.44). 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

$$\begin{eqnarray}&&{\sigma }_{{E}_{2}^{{\prime} }{{GE}}_{1}^{{\prime} }}(\phi )=({{\rm{e}}}^{{\rm{i}}\phi \hat{n}}\otimes {{\rm{e}}}^{{\rm{i}}\phi \hat{n}}\otimes {{\rm{e}}}^{-{\rm{i}}\phi \hat{n}})({ \mathcal W }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}})(\rho )({{\rm{e}}}^{-{\rm{i}}\phi \hat{n}}\otimes {{\rm{e}}}^{-{\rm{i}}\phi \hat{n}}\otimes {{\rm{e}}}^{{\rm{i}}\phi \hat{n}}),\end{eqnarray} \tag{ 5.45 }$$

and let

$$\begin{eqnarray}&&{\xi }_{{E}_{2}^{{\prime} }{{GE}}_{1}^{{\prime} }}=\displaystyle \frac{1}{2\pi }{\int }_{0}^{2\pi }{\rm{d}}\phi \,({ \mathcal W }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}})({{\rm{e}}}^{{\rm{i}}\phi \hat{n}}\rho {{\rm{e}}}^{-{\rm{i}}\phi \hat{n}}).\end{eqnarray} \tag{ 5.46 }$$

Note that the phase covariance property mentioned above is the statement that the following equality holds for all $\phi \in [0,2\pi )$ [Mat]:

$$\begin{eqnarray}&&{\sigma }_{{E}_{2}^{{\prime} }{{GE}}_{1}^{{\prime} }}(\phi )=({ \mathcal W }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}})({{\rm{e}}}^{{\rm{i}}\phi \hat{n}}\rho {{\rm{e}}}^{-{\rm{i}}\phi \hat{n}}).\end{eqnarray} \tag{ 5.47 }$$

Consider the following chain of inequalities for a Gaussian input state ρ:

$$\begin{eqnarray}&&H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})(\rho )}=\displaystyle \frac{1}{2\pi }{\int }_{0}^{2\pi }{\rm{d}}\phi \,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{\sigma (\phi )}\,\end{eqnarray} \tag{ 5.48 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{1}{2\pi }{\int }_{0}^{2\pi }{\rm{d}}\phi \,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})({{\rm{e}}}^{{\rm{i}}\phi \hat{n}}\rho {{\rm{e}}}^{-{\rm{i}}\phi \hat{n}})}\end{eqnarray} \tag{ 5.49 }$$

$$\begin{eqnarray}&&\leqslant \,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{\xi }\,\end{eqnarray} \tag{ 5.50 }$$

$$\begin{eqnarray}&&=\,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})(\theta ({N}_{t}))}.\,\end{eqnarray} \tag{ 5.51 }$$

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:

$$\begin{eqnarray}&&\theta ({N}_{t})=\displaystyle \frac{1}{2\pi }{\int }_{0}^{2\pi }{\rm{d}}\phi \,{{\rm{e}}}^{{\rm{i}}\phi \hat{n}}\rho {{\rm{e}}}^{-{\rm{i}}\phi \hat{n}}.\end{eqnarray} \tag{ 5.52 }$$

In (5.52), 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 $\theta ({N}_{t})$ 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

$$\begin{eqnarray}&&{\tau }_{{E}_{2}^{{\prime} }{{GE}}_{1}^{{\prime} }}(\alpha )\\ &&=[D(\sqrt{1-\eta }\alpha )\otimes D(\sqrt{2\eta -1}\alpha )\otimes I][({ \mathcal W }\,\circ \,{{ \mathcal L }}_{\eta ,{N}_{B}})(\theta ({N}_{t}))][{D}^{\dagger }(\sqrt{1-\eta }\alpha )\otimes {D}^{\dagger }(\sqrt{2\eta -1}\alpha )\otimes I].\end{eqnarray} \tag{ 5.53 }$$

Consider the following chain of inequalities:

$$\begin{eqnarray}&&H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})(\theta ({N}_{t}))}=\int {{\rm{d}}}^{2}\alpha \,{q}_{({N}_{S}-{N}_{t})}(\alpha )\,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})(\theta ({N}_{t}))}\,\end{eqnarray} \tag{ 5.54 }$$

$$\begin{eqnarray}&&=\int {{\rm{d}}}^{2}\alpha \,{q}_{({N}_{S}-{N}_{t})}(\alpha )\,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{\tau (\alpha )}\,\end{eqnarray} \tag{ 5.55 }$$

$$\begin{eqnarray}&&=\int {{\rm{d}}}^{2}\alpha \,{q}_{({N}_{S}-{N}_{t})}(\alpha )\,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})(D(\alpha )\theta ({N}_{t}){D}^{\dagger }(\alpha ))}\,\end{eqnarray} \tag{ 5.56 }$$

$$\begin{eqnarray}&&\leqslant \,H{(G| {E}_{1}^{{\prime} }{E}_{2}^{{\prime} })}_{({ \mathcal W }\circ {{ \mathcal L }}_{\eta ,{N}_{B}})\theta ({N}_{S})},\,\,\,\end{eqnarray} \tag{ 5.57 }$$

where ${q}_{N}(\alpha )=\exp \{-| \alpha {| }^{2}/N\}/\pi N$ is a complex-centered Gaussian distribution with variance $N\geqslant 0$. 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 (3.40). 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 (3.26). 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 22. The arguments used in the proof of proposition 21 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 26.

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 bound on the energy-constrained quantum capacity of a thermal channel directly follows from theorem 12.

Theorem 23. If a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ and a pure-loss bosonic channel ${{ \mathcal L }}_{\eta ,0}$ have the same transmissivity parameter $\eta \in [0,1]$, then

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\parallel {{ \mathcal L }}_{\eta ,{N}_{B}}-{{ \mathcal L }}_{\eta ,0}\parallel }_{\diamond }\leqslant \displaystyle \frac{{N}_{B}}{{N}_{B}+1}.\end{eqnarray} \tag{ 5.58 }$$

Proof. Let ${ \mathcal B }$ represent the beamsplitter transformation, and let ${\theta }_{E}({N}_{B})$ and ${\theta }_{E}^{{\prime} }(0)$ denote the states of the environment for the thermal channel and pure-loss channel, respectively. For any input state ${\psi }_{{RA}}$ to both thermal and pure-loss channels, the following inequalities hold:

$$\begin{eqnarray}&&\,{\parallel ({\mathrm{id}}_{R}\otimes {{ \mathcal L }}_{\eta ,{N}_{B}})({\psi }_{{RA}})-({\mathrm{id}}_{R}\otimes {{ \mathcal L }}_{\eta ,0})({\psi }_{{RA}})\parallel }_{1}\\ &&\quad =\,{\parallel {\mathrm{Tr}}_{E^{\prime} }\{{{ \mathcal B }}_{{AE}\to {BE}^{\prime} }({\psi }_{{RA}}\otimes {\theta }_{E}({N}_{B}))-{{ \mathcal B }}_{{AE}\to {BE}^{\prime} }({\psi }_{{RA}}\otimes {\theta }_{E}^{{\prime} }(0))\}\parallel }_{1}\end{eqnarray} \tag{ 5.59 }$$

$$\begin{eqnarray}&&\leqslant \,{\parallel {{ \mathcal B }}_{{AE}\to {BE}^{\prime} }({\psi }_{{RA}}\otimes {\theta }_{E}({N}_{B}))-{{ \mathcal B }}_{{AE}\to {BE}^{\prime} }({\psi }_{{RA}}\otimes {\theta }_{E}^{{\prime} }(0))\parallel }_{1}\,\end{eqnarray} \tag{ 5.60 }$$

$$\begin{eqnarray}&&=\,{\parallel {\psi }_{{RA}}\otimes {\theta }_{E}({N}_{B})-{\psi }_{{RA}}\otimes {\theta }_{E}^{{\prime} }(0)\parallel }_{1}\,\end{eqnarray} \tag{ 5.61 }$$

$$\begin{eqnarray}&&=\,{\parallel {\theta }_{E}({N}_{B})-{\theta }_{E}^{{\prime} }(0)\parallel }_{1}\,\end{eqnarray} \tag{ 5.62 }$$

$$\begin{eqnarray}&&=\,{\parallel \displaystyle \sum _{n=0}^{\infty }\displaystyle \frac{{({N}_{B})}^{n}}{{({N}_{B}+1)}^{n+1}}| n\rangle \langle n| -| 0\rangle \langle 0| \parallel }_{1}\,\end{eqnarray} \tag{ 5.63 }$$

$$\begin{eqnarray}&&=\,\displaystyle \frac{2{N}_{B}}{{N}_{B}+1}.\,\end{eqnarray} \tag{ 5.64 }$$

The first equality follows from the definition of the channel in terms of its environment and a unitary interaction (beamsplitter). The first inequality follows from monotonicity of the trace distance. The second equality follows from invariance of the trace distance under a unitary operator (beamsplitter). The last equality follows from basic algebra. Since these inequalities hold for any state ${\psi }_{{RA}}$, the final result follows from the definition of the diamond norm. ■

Remark 24. In [TW16], it has been shown that the optimal strategy to distinguish two quantum thermal channels ${{ \mathcal L }}_{\eta ,{N}_{B}^{1}}$ and ${{ \mathcal L }}_{\eta ,{N}_{B}^{2}}$, each having the same transmissivity parameter η, and thermal noises ${N}_{B}^{1}$ and ${N}_{B}^{2}$, respectively, is to use a highly squeezed, two-mode squeezed vacuum state ${\psi }_{\mathrm{TMS}}{({N}_{S})}_{{RA}}$ as input to the channels. According to [TW16, equation (35)],

$$\begin{eqnarray}&&\mathop{\mathrm{lim}}\limits_{{N}_{S}\to \infty }F({\sigma }_{{N}_{B}^{1}},{\sigma }_{{N}_{B}^{2}})=F(\theta ({N}_{B}^{1}),\theta ({N}_{B}^{2})),\end{eqnarray} \tag{ 5.65 }$$

where ${\sigma }_{{N}_{B}^{i}}\equiv {({\mathrm{id}}_{R}\otimes {{ \mathcal L }}_{\eta ,{N}_{B}^{i}})({\psi }_{\mathrm{TMS}}({N}_{S})}_{{RA}})$, and $\theta ({N}_{B}^{i})$ is a thermal state with mean photon number ${N}_{B}^{i}$. Hence, a lower bound on the diamond distance in theorem 23 is given by

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\parallel {{ \mathcal L }}_{\eta ,{N}_{B}}-{{ \mathcal L }}_{\eta ,0}\parallel }_{\diamond }\geqslant 1-\sqrt{F(\theta ({N}_{B}),\theta (0))}=1-1/\sqrt{{N}_{B}+1},\end{eqnarray} \tag{ 5.66 }$$

where the inequality follows from the Powers–Stormer inequality [PS70]. We also suspect that the upper bound in theorem 23 is achievable, but we are not aware of a method for computing the trace distance of general quantum Gaussian states, which is what it seems would be needed to verify this suspicion.

Theorem 25. An upper bound on the quantum capacity of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}$, and input mean photon number constraint ${N}_{S}$ is given by

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant {Q}_{{U}_{3}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\equiv g(\eta {N}_{S})-g[(1-\eta ){N}_{S}]\\ &&\,\,+(4\varepsilon ^{\prime} +8\delta )g[(\eta {N}_{S}+(1-\eta ){N}_{B})/\delta ]+2g(\varepsilon ^{\prime} )+4{h}_{2}(\delta ),\end{eqnarray} \tag{ 5.67 }$$

with $\varepsilon ={N}_{B}/({N}_{B}+1),\varepsilon ^{\prime} \in (\varepsilon ,1]$ and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. From theorem 23, we have that $\tfrac{1}{2}{\parallel {{ \mathcal L }}_{\eta ,{N}_{B}}-{{ \mathcal L }}_{\eta ,0}\parallel }_{\diamond }\leqslant \tfrac{{N}_{B}}{{N}_{B}+1}\lt \varepsilon ^{\prime} \leqslant 1$. Due to the input mean photon number constraint N_S for n channel uses, the output mean photon number cannot exceed $\eta {N}_{S}+(1-\eta ){N}_{B}$ for the thermal channel and $\eta {N}_{S}$ for the pure-loss channel. Hence, there is a photon number constraint $\eta {N}_{S}+(1-\eta ){N}_{B}$ for the output of both the thermal and pure-loss channels. Since the pure-loss channel is a degradable channel for $\eta \in [1/2,1]$ [GSE08, WPGG07], the final result follows directly from theorem 12. ■

Theorem 27. An upper bound on the private capacity of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}\geqslant 0$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&P({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant \max \{0,{P}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\}\end{eqnarray} \tag{ 7.1 }$$

$$\begin{eqnarray}&&\,{P}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\equiv g(\eta ^{\prime} {N}_{S})-g[(1-\eta ^{\prime} ){N}_{S}],\end{eqnarray} \tag{ 7.2 }$$

with $\eta ^{\prime} =\eta /((1-\eta ){N}_{B}+1)$.

Proof. A proof follows from arguments similar to those in the proof of theorem 16. Since a pure-loss channel is a degradable channel [GSE08, WPGG07], its energy-constrained private capacity is the same as its energy-constrained quantum capacity [WQ16]. ■

Remark 28. Applying remarks 9 and 17, we find the following data-processing bound ${P}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}})$ on the unconstrained private capacity $P({{ \mathcal L }}_{\eta ,{N}_{B}})$ of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$:

$$\begin{eqnarray}&&P({{ \mathcal L }}_{\eta ,{N}_{B}})\leqslant {P}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}})={\mathrm{log}}_{2}(\eta /(1-\eta ))-{\mathrm{log}}_{2}({N}_{B}+1).\end{eqnarray} \tag{ 7.3 }$$

Theorem 29. An upper bound on the private capacity of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}\geqslant 0$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&P({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant {P}_{{U}_{2}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\equiv g(\eta {N}_{S}+(1-\eta ){N}_{B})-g({\zeta }_{+})-g({\zeta }_{-})\\ &&\,\,\,+(6\varepsilon ^{\prime} +12\delta )g([(1-\eta ){N}_{S}+(1+\eta ){N}_{B}]/\delta )+3g(\varepsilon ^{\prime} )+6{h}_{2}(\delta ),\end{eqnarray} \tag{ 7.4 }$$

with

$$\begin{eqnarray}&&\varepsilon =\sqrt{1-{\eta }^{2}/({\eta }^{2}+{N}_{B}({N}_{B}+1)[1+3{\eta }^{2}-2\eta (1+\sqrt{2\eta -1})])},\,\,\end{eqnarray} \tag{ 7.5 }$$

$$\begin{eqnarray}&&{\zeta }_{\pm }=\displaystyle \frac{1}{2}(-1+\sqrt{[{(1+2{N}_{B})}^{2}-2\varrho +{(1+2\vartheta )}^{2}\pm 4(\vartheta -{N}_{B})\sqrt{{[1+{N}_{B}+\vartheta ]}^{2}-\varrho }]/2}),\end{eqnarray} \tag{ 7.6 }$$

$$\begin{eqnarray}&&\varrho =4{N}_{B}({N}_{B}+1)(2-1/\eta ),\,\,\,\,\,\,\end{eqnarray} \tag{ 7.7 }$$

$$\begin{eqnarray}&&\vartheta =\eta {N}_{B}+(1-\eta ){N}_{S},\,\,\,\,\,\,\end{eqnarray} \tag{ 7.8 }$$

$\varepsilon ^{\prime} \in (\varepsilon ,1]$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. A proof follows from arguments similar to those in the proof of theorem 20. The final result is obtained using theorem 14. ■

Theorem 30. An upper bound on the private capacity of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}\geqslant 0$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&P({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant {P}_{{U}_{3}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\equiv g(\eta {N}_{S})-g[(1-\eta ){N}_{S}]\\ &&\,\,\,\,+(8\varepsilon ^{\prime} +16\delta )g[(\eta {N}_{S}+(1-\eta ){N}_{B})/\delta ]+4g(\varepsilon ^{\prime} )+8{h}_{2}(\delta ),\end{eqnarray} \tag{ 7.9 }$$

with $\varepsilon ={N}_{B}/({N}_{B}+1),\varepsilon ^{\prime} \in (\varepsilon ,1]$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. A proof follows from arguments similar to those in the proof of theorem 25. The final result is obtained using theorem 15. ■

In this section, we provide an upper bound using theorem 31 below, which states that any phase-insensitive single-mode bosonic Gaussian channel can be decomposed as a pure-amplifier channel followed by a pure-loss channel, if the original channel is not entanglement breaking. This theorem was independently proven in [NAJ18, RMG18] (see also [SWAT17] in this context).

Before we state the theorem, let us recall that the action of a phase-insensitive channel ${ \mathcal N }$ on the covariance matrix Γ of a single-mode, bosonic quantum state is given by

$$\begin{eqnarray}&&{\rm{\Gamma }}\,\longmapsto \,\tau \,{\rm{\Gamma }}+\nu {I}_{2},\end{eqnarray} \tag{ 9.1 }$$

where ν is the variance of an additive-noise, I₂ is the 2 × 2 identity matrix, and τ and ν satisfy the conditions in (3.37)–(3.38). Moreover, as mentioned previously, a phase-insensitive channel ${ \mathcal N }$ is entanglement breaking [Hol08, HSR03] if

$$\begin{eqnarray}&&\tau +1\leqslant \nu .\end{eqnarray} \tag{ 9.2 }$$

Theorem 31. Any single-mode, phase-insensitive bosonic Gaussian channel ${ \mathcal N }$ that is not entanglement breaking (i.e., satisfies $\tau +1\gt \nu$) can be decomposed as the concatenation of a quantum-limited amplifier channel ${{ \mathcal A }}_{G,0}$ with gain $G\gt 1$ followed by a pure-loss channel ${{ \mathcal L }}_{\eta ,0}$ with transmissivity $\eta \in (0,1]$, i.e.,

$$\begin{eqnarray}&&{ \mathcal N }={{ \mathcal L }}_{\eta ,0}\,\circ \,{{ \mathcal A }}_{G,0},\end{eqnarray} \tag{ 9.3 }$$

where $\eta =(\tau +1-\nu )/2$ and $G=\tau /\eta$.

Proof. The action of a quantum-limited amplifier channel ${{ \mathcal A }}_{G,0}$ with gain G followed by a pure-loss channel ${{ \mathcal L }}_{\eta ,0}$ with transmissivity η, on the convariance matrix Γ is given by

$$\begin{eqnarray}&&\eta (G\,{\rm{\Gamma }}+[G-1]{I}_{2})+[1-\eta ]{I}_{2}.\end{eqnarray} \tag{ 9.4 }$$

By comparing (9.1) and (9.4), we find that it is necessary for the following equalities to hold

$$\begin{eqnarray}&&\eta G=\tau ,\end{eqnarray} \tag{ 9.5 }$$

$$\begin{eqnarray}&&\eta (G-1)+1-\eta =\nu .\,\end{eqnarray} \tag{ 9.6 }$$

Solving these equations for η and G in terms of τ and ν then gives $\eta =(\tau +1-\nu )/2$ and $G=\tau /\eta$. By the assumption that ${ \mathcal N }$ is not entanglement breaking, which is that $\tau +1\gt \nu$, we find that

$$\begin{eqnarray}&&\eta =(\tau +1-\nu )/2\gt 0.\end{eqnarray} \tag{ 9.7 }$$

Now applying the conditions in (3.37) and (3.38) for the channel ${ \mathcal N }$ to be a CPTP map, we find that

$$\begin{eqnarray}\eta =(\tau +1-\nu )/2\leqslant (\tau +1-| 1-\tau | )/2=\left\{\begin{array}{ll}\tau & {\rm{}}\,{\rm{for}}\,{\rm{}}\tau \in [0,1)\\ 1 & {\rm{}}\,{\rm{for}}\,{\rm{}}\tau \geqslant 1\end{array}\right..\end{eqnarray} \tag{ 9.8 }$$

By the fact that $G=\tau /\eta$, the above implies that $G\gt 1$, so that the decomposition in (9.3) is valid under the stated conditions. ■

We now apply theorem 31 and a data-processing argument to a noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ with gain $G\gt 1$, environment photon number ${N}_{B}\geqslant 0$, for which $\tau =G$ and $\nu =(G-1)(2{N}_{B}+1)$. This channel is entanglement breaking when $(G-1){N}_{B}\geqslant 1$ [Hol08].

Theorem 32. An upper bound on the energy-constrained quantum and private capacities of a noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ with gain $G\gt 1$ and environment photon number ${N}_{B}\geqslant 0$, such that $(G-1){N}_{B}\lt 1$, and input photon number constraint ${N}_{S}\geqslant 0$, is given by

$$\begin{eqnarray}&&Q({{ \mathcal A }}_{G,{N}_{B}},{N}_{S}),P({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\leqslant \max \{0,{Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\},\end{eqnarray} \tag{ 9.9 }$$

where

$$\begin{eqnarray}&&{Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\equiv g(G^{\prime} {N}_{S}+G^{\prime} -1)-g[(G^{\prime} -1)({N}_{S}+1)],\end{eqnarray} \tag{ 9.10 }$$

$$\begin{eqnarray}&&G^{\prime} =G/(1+{N}_{B}(1-G)).\,\end{eqnarray} \tag{ 9.11 }$$

Proof. An upper bound on the energy-constrained quantum and private capacities can be established by using (9.3) and a data-processing argument. We find that

$$\begin{eqnarray}&&Q({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})=Q({{ \mathcal L }}_{\eta ,0}\,\circ \,{{ \mathcal A }}_{G^{\prime} ,0},{N}_{S})\,\,\,\,\,\end{eqnarray} \tag{ 9.12 }$$

$$\begin{eqnarray}&&\,\leqslant Q({{ \mathcal A }}_{G^{\prime} ,0},{N}_{S})\,\,\,\,\,\end{eqnarray} \tag{ 9.13 }$$

$$\begin{eqnarray}&&=\max \{0,g(G^{\prime} \,{N}_{S}+G^{\prime} -1)-g[(G^{\prime} -1)({N}_{S}+1)]\}.\end{eqnarray} \tag{ 9.14 }$$

The first inequality follows from the definition and data-processing—the energy-constrained capacity of ${{ \mathcal L }}_{\eta ,0}\,\circ \,{{ \mathcal A }}_{G^{\prime} ,0}$ cannot exceed that of ${{ \mathcal A }}_{G^{\prime} ,0}$. The second equality follows from the formula for the energy-constrained quantum capacity of a quantum-limited amplifier channel with gain $G^{\prime}$ and input mean photon number N_S [QW17]. Since a quantum-limited amplifier channel is a degradable channel [CG06, WPGG07], its energy-constrained private capacity is the same as its energy-constrained quantum capacity. ■

Remark 33. Applying remark 9, we find the following data-processing bound ${Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}})$ on the unconstrained quantum and private capacities of amplifier channels for which $(G-1){N}_{B}\lt 1$:

$$\begin{eqnarray}&&Q({{ \mathcal A }}_{G,{N}_{B}}),P({{ \mathcal A }}_{G,{N}_{B}})\leqslant {Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}})=\mathop{\sup }\limits_{{N}_{S}:{N}_{S}\in [0,\infty ]}{Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\end{eqnarray} \tag{ 9.15 }$$

$$\begin{eqnarray}&&\,\,\,=\mathop{\mathrm{lim}}\limits_{{N}_{S}\to \infty }{Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\end{eqnarray} \tag{ 9.16 }$$

$$\begin{eqnarray}&&\,\,\,\,=\,{\mathrm{log}}_{2}(G/(G-1))-{\mathrm{log}}_{2}({N}_{B}+1).\end{eqnarray} \tag{ 9.17 }$$

The second equality follows from the monotonicity of ${Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})$ with respect to N_S, which in turn follows from the fact that the first derivative of ${Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})$ with respect to N_S goes to zero as ${N}_{S}\to \infty$, and the second derivative is always negative.

The bound

$$\begin{eqnarray}&&Q({{ \mathcal A }}_{G,{N}_{B}}),P({{ \mathcal A }}_{G,{N}_{B}})\leqslant {\mathrm{log}}_{}\left(\displaystyle \frac{{G}^{{N}_{B}+1}}{G-1}\right)-g({N}_{B})\end{eqnarray} \tag{ 9.18 }$$

was given in [PLOB17, WTB17]. From a comparison of (9.17) with (9.18), we find that the bound given in (9.18) is always tighter than (9.17). Both the bounds in (9.17) and (9.18) converge to the true unconstrained quantum and private capacity in the limit as ${N}_{B}\to 0$, but (9.18) is tighter for ${N}_{B}\gt 0$.

Remark 34. The data-processing bound ${Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})$ on the energy-constrained quantum capacity of amplifier channels places a strong restriction on the channel parameters G and N_B. Since the quantum capacity of a quantum-limited amplifier channel with gain $G^{\prime}$ is non-zero only for $G^{\prime} \ne \infty$, the energy-constrained quantum capacity of an amplifier channel will be non-zero only for

$$\begin{eqnarray}&&1\leqslant G\lt (1+{N}_{B})/{N}_{B},\end{eqnarray} \tag{ 9.19 }$$

which is same as the condition given in [CGH06] and is equivalent to the condition $(G-1){N}_{B}\lt 1$, that the channel is not entanglement breaking.

We now study the closeness of the data-processing bound ${Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})$ when compared to a known lower bound. In particular, we use the following lower bound on the energy-constrained quantum and private capacities of an amplifier channel [HW01, WQ16] and denote it by ${Q}_{L}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})$:

$$\begin{eqnarray}&&Q({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\geqslant {Q}_{L}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\equiv g({{GN}}_{S}+(G-1)({N}_{B}+1))\\ &&\,-g([D+(G-1)({N}_{S}+{N}_{B}+1)-1]/2)-g([D-(G-1)({N}_{S}+{N}_{B}+1)-1]/2),\end{eqnarray} \tag{ 9.20 }$$

where

$$\begin{eqnarray}&&{D}^{2}\equiv {[(1+G){N}_{S}+(G-1)({N}_{B}+1)+1]}^{2}-4{{GN}}_{S}({N}_{S}+1).\end{eqnarray} \tag{ 9.21 }$$

Theorem 35. Let ${{ \mathcal A }}_{G,{N}_{B}}$ be an amplifier channel with gain $G\gt 1$ and environment photon number ${N}_{B}\geqslant 0$, such that $(G-1){N}_{B}\lt 1$, and input photon number constraint ${N}_{S}\geqslant 0$. Then the following relation holds between the data-processing bound ${Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})$ in (9.9) and the lower bound ${Q}_{L}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})$ in (9.20) on the energy-constrained quantum and private capacities of an amplifier channel:

$$\begin{eqnarray}&&{Q}_{L}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\leqslant {Q}_{{U}_{1}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\leqslant {Q}_{L}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})+1/\mathrm{ln}2.\end{eqnarray} \tag{ 9.22 }$$

Proof. A proof follows from arguments similar to those in the proof of theorem 26. ■

In this section, we provide an upper bound on the energy-constrained quantum and private capacities of a quantum amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ using the idea of ε-degradability. We first construct an approximate degrading channel ${ \mathcal D }$ by following arguments similar to those in section 5.2. Furthermore, we introduce a particular channel that simulates the serial concatenation of the amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ and the approximate degrading channel ${ \mathcal D }$. We finally provide an upper bound on the energy-constrained quantum and private capacities of an amplifier channel by using all these tools and invoking theorem 11.

Similar to section 5.2, we first establish an upper bound on the diamond distance between the complementary channel of the amplifier channel and the concatenation of the amplifier channel followed by a particular approximate degrading channel. Let ${ \mathcal T }$ and ${ \mathcal T }^{\prime}$ represent transformations of two-mode squeezers with parameter G and $(2G-1)/G$, respectively. In the Heisenberg picture, the unitary transformation corresponding to ${ \mathcal T }$ and ${ \mathcal T }^{\prime}$ follow from (3.46).

Consider the following action of the noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ on an input state ${\phi }_{{RA}}$:

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes {{ \mathcal A }}_{G,{N}_{B}})({\phi }_{{RA}})={\mathrm{Tr}}_{{E}_{1}{E}_{2}}\{{{ \mathcal T }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}})\},\end{eqnarray} \tag{ 9.23 }$$

where R is a reference system and ${\psi }_{\mathrm{TMS}}{({N}_{B})}_{E^{\prime} {E}_{1}}$ is a two-mode squeezed vacuum state with parameter N_B, as defined in (3.27). It is evident from (9.23) that the output of the noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ is system B, and the outputs of the complementary channel ${\hat{{ \mathcal A }}}_{G,{N}_{B}}$ are systems E₁ and E₂.

Consider a two-mode squeezer ${ \mathcal T }^{\prime}$ with parameter $(2G-1)/G$, such that the output of the amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ becomes an environmental input for ${ \mathcal T }^{\prime}$. We consider one mode of the two-mode squeezed vacuum state ${\psi }_{\mathrm{TMS}}{({N}_{B})}_{{{FE}}_{1}^{{\prime} }}$ as an input for ${ \mathcal T }^{\prime}$, so that the subsystem ${E}_{1}^{{\prime} }$ mimics E₁. We denote our choice of degrading channel by ${{ \mathcal D }}_{(2G-1)/G,{N}_{B}}\,:{ \mathcal T }(B)\to { \mathcal T }({E}_{1}^{{\prime} })\otimes { \mathcal T }({E}_{2}^{{\prime} })$. More formally, ${{ \mathcal D }}_{(2G-1)/G,{N}_{B}}$ has the following action on the output state ${{ \mathcal A }}_{G,{N}_{B}}({\phi }_{{RA}})$:

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes [{{ \mathcal D }}_{(2G-1)/G,{N}_{B}}\,\circ \,{{ \mathcal A }}_{G,{N}_{B}}])({\phi }_{{RA}})={\mathrm{Tr}}_{G}\{{{ \mathcal T }}_{{BF}\to {E}_{2}^{{\prime} }G}^{{\prime} }({{ \mathcal A }}_{G,{N}_{B}}({\phi }_{{RA}})\otimes {\psi }_{\mathrm{TMS}}{({N}_{B})}_{{{FE}}_{1}^{{\prime} }}))\}.\end{eqnarray} \tag{ 9.24 }$$

Now, similar to section 5.2, we introduce a particular channel that simulates the action of ${{ \mathcal D }}_{(2G-1)/G}\,\circ \,{{ \mathcal A }}_{G,{N}_{B}}$ on an input state ${\phi }_{{RA}}$. We denote this channel by Λ, and it has the following action on an input state ${\phi }_{{RA}}$:

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes {\rm{\Lambda }})({\phi }_{{RA}})={\mathrm{Tr}}_{B}\{{{ \mathcal T }}_{{AE}^{\prime} \to {{BE}}_{2}}({\phi }_{{RA}}\otimes \omega {({N}_{B})}_{E^{\prime} {E}_{1}})\},\end{eqnarray} \tag{ 9.25 }$$

where $\omega {({N}_{B})}_{E^{\prime} {E}_{1}}$ represents a noisy version of a two-mode squeezed vacuum state with parameter N_B, and is same as (5.25), except η is replaced by G. Similar to (5.26), the following equivalence holds for any quantum input state ${\phi }_{{RA}}$:

$$\begin{eqnarray}&&({\mathrm{id}}_{R}\otimes [{{ \mathcal D }}_{(2G-1)/G,{N}_{B}}\,\circ \,{{ \mathcal A }}_{G,{N}_{B}}])({\phi }_{{RA}})=({\mathrm{id}}_{R}\otimes {\rm{\Lambda }})({\phi }_{{RA}}).\end{eqnarray} \tag{ 9.26 }$$

Thus, the channels ${{ \mathcal D }}_{(2G-1)/G,{N}_{B}}\,\circ \,{{ \mathcal A }}_{G,{N}_{B}}$ and Λ are indeed the same.

Similar to theorem 19, we now establish an upper bound on the diamond distance between the complementary channel of a noisy amplifier channel and the concatenation of the amplifier channel followed by the degrading channel in (9.24).

Theorem 36. Fix $G\gt 1$. Let ${{ \mathcal A }}_{G,{N}_{B}}$ be an amplifier channel with gain $G$, and let ${{ \mathcal D }}_{(2G-1)/G,{N}_{B}}$ be a degrading channel as defined in (9.24). Then

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}\parallel {\hat{{ \mathcal A }}}_{G,{N}_{B}}-{{ \mathcal D }}_{(2G-1)/G,{N}_{B}}\,\circ \,{{ \mathcal A }}_{G,{N}_{B}}{\parallel }_{\diamond }\leqslant \sqrt{1-{G}^{2}/\kappa (G,{N}_{B})},\end{eqnarray} \tag{ 9.27 }$$

with

$$\begin{eqnarray}&&\kappa (G,{N}_{B})={G}^{2}+{N}_{B}({N}_{B}+1)[1+3{G}^{2}-2G(1+\sqrt{2G-1})].\end{eqnarray} \tag{ 9.28 }$$

Proof. A proof follows from arguments similar to those in the proof of theorem 19. ■

Theorem 37. An upper bound on the energy-constrained quantum capacity of a noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ with gain $G\gt 1$, environment photon number N_B, such that $(G-1){N}_{B}\lt 1$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&Q({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\leqslant {Q}_{{U}_{2}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\equiv g({{GN}}_{S}+(G-1){N}_{B})-g({\zeta }_{+})-g({\zeta }_{-})\\ &&\,+(2\varepsilon ^{\prime} +4\delta )g([(G-1){N}_{S}+(1+G){N}_{B}]/\delta )+g(\varepsilon ^{\prime} )+2{h}_{2}(\delta ),\,\end{eqnarray} \tag{ 9.29 }$$

with

$$\begin{eqnarray}&&\varepsilon =\sqrt{1-{G}^{2}/({G}^{2}+{N}_{B}({N}_{B}+1)[1+3{G}^{2}-2G(1+\sqrt{2G-1})])},\,\,\end{eqnarray} \tag{ 9.30 }$$

$$\begin{eqnarray}&&{\zeta }_{\pm }=\displaystyle \frac{1}{2}(-1+\sqrt{[{(1+2{N}_{B})}^{2}-2\varrho +{(2\vartheta -1)}^{2}\pm 4(\vartheta -{N}_{B}-1)\sqrt{{[{N}_{B}+\vartheta ]}^{2}-\varrho }]/2}),\,\end{eqnarray} \tag{ 9.31 }$$

$$\begin{eqnarray}&&\varrho =4{N}_{B}({N}_{B}+1)(2G-1)/G,\,\,\,\,\,\,\end{eqnarray} \tag{ 9.32 }$$

$$\begin{eqnarray}&&\vartheta =G(1+{N}_{B})+(G-1){N}_{S},\,\,\,\,\,\,\end{eqnarray} \tag{ 9.33 }$$

$\varepsilon ^{\prime} \in (\varepsilon ,1]$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. A proof follows from arguments similar to those in the proof of theorem 20. ■

Theorem 38. An upper bound on the energy-constrained private capacity of a noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ with with gain $G\gt 1$, environment photon number ${N}_{B}$, such that $(G-1){N}_{B}\lt 1$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&\begin{array}{l}P({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\leqslant {P}_{{U}_{2}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\equiv g({{GN}}_{S}+(G-1){N}_{B})-g({\zeta }_{+})-g({\zeta }_{-})\\ \,+(6\varepsilon ^{\prime} +12\delta )g([(G-1){N}_{S}+(1+G){N}_{B}]/\delta )+3g(\varepsilon ^{\prime} )+6{h}_{2}(\delta ),\end{array}\end{eqnarray} \tag{ 9.34 }$$

with

$$\begin{eqnarray}&&\varepsilon =\sqrt{1-{G}^{2}/({G}^{2}+{N}_{B}({N}_{B}+1)[1+3{G}^{2}-2G(1+\sqrt{2G-1})])},\,\end{eqnarray} \tag{ 9.35 }$$

$$\begin{eqnarray}&&{\zeta }_{\pm }=\displaystyle \frac{1}{2}(-1+\sqrt{[{(1+2{N}_{B})}^{2}-2\varrho +{(2\vartheta -1)}^{2}\pm 4(\vartheta -{N}_{B}-1)\sqrt{{[{N}_{B}+\vartheta ]}^{2}-\varrho }]/2}),\,\end{eqnarray} \tag{ 9.36 }$$

$$\begin{eqnarray}&&\varrho =4{N}_{B}({N}_{B}+1)(2G-1)/G,\,\,\,\,\,\,\end{eqnarray} \tag{ 9.37 }$$

$$\begin{eqnarray}&&\vartheta =G(1+{N}_{B})+(G-1){N}_{S},\,\,\,\,\,\,\end{eqnarray} \tag{ 9.38 }$$

$\varepsilon ^{\prime} \in (\varepsilon ,1]$, and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. A proof follows from arguments similar to those in the proof of theorem 20. The final result is obtained using theorem 14. ■

In this section, we first establish an upper bound on the diamond distance between a noisy amplifier channel and a quantum-limited amplifier channel. Since a quantum-limited amplifier channel is a degradable channel, an upper bound on the energy-constrained quantum capacity of a noisy amplifier channel directly follows from theorem 12.

Theorem 39. If a noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ and a quantum-limited amplifier channel ${{ \mathcal A }}_{G,0}$ have the same gain $G\gt 1$, then

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\parallel {{ \mathcal A }}_{G,{N}_{B}}-{{ \mathcal A }}_{G,0}\parallel }_{\diamond }\leqslant \displaystyle \frac{{N}_{B}}{{N}_{B}+1}.\end{eqnarray} \tag{ 9.39 }$$

Proof. A proof follows from arguments similar to those in the proof of theorem 23. ■

Theorem 40. An upper bound on the energy-constrained quantum capacity of a noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ with gain $G\gt 1$, environment photon number ${N}_{B}$, such that $(G-1){N}_{B}\lt 1$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&Q({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\leqslant {Q}_{{U}_{3}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\equiv g({{GN}}_{S}+G-1)-g[(G-1)({N}_{S}+1)]\\ &&\,\,+(4\varepsilon ^{\prime} +8\delta )g[({{GN}}_{S}+(G-1){N}_{B})/\delta ]+2g(\varepsilon ^{\prime} )+4{h}_{2}(\delta ),\end{eqnarray} \tag{ 9.40 }$$

with $\varepsilon ={N}_{B}/({N}_{B}+1),\varepsilon ^{\prime} \in (\varepsilon ,1]$ and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. A proof follows from arguments similar to those in the proof of theorem 25. ■

Theorem 41. An upper bound on the energy-constrained private capacity of a noisy amplifier channel ${{ \mathcal A }}_{G,{N}_{B}}$ with gain $G\gt 1$, environment photon number N_B, such that $(G-1){N}_{B}\lt 1$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&P({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\leqslant {P}_{{U}_{3}}({{ \mathcal A }}_{G,{N}_{B}},{N}_{S})\equiv g({{GN}}_{S}+G-1)-g[(G-1)({N}_{S}+1)]\\ &&\,\,+(8\varepsilon ^{\prime} +16\delta )g[({{GN}}_{S}+(G-1){N}_{B})/\delta ]+4g(\varepsilon ^{\prime} )+8{h}_{2}(\delta ),\end{eqnarray} \tag{ 9.41 }$$

with $\varepsilon ={N}_{B}/({N}_{B}+1),\varepsilon ^{\prime} \in (\varepsilon ,1]$ and $\delta =(\varepsilon ^{\prime} -\varepsilon )/(1+\varepsilon ^{\prime} )$.

Proof. A proof follows from arguments similar to those in the proof of theorem 25. The final result is obtrained using theorem 15. ■

Recently, the following upper bound on the unconstrained quantum capacity of a thermal channel for which $\eta \gt (1-\eta ){N}_{B}$ was introduced in [RMG18, equation (40)]:

$$\begin{eqnarray}&&{Q}_{{U}_{1}}({{ \mathcal L }}_{\eta ,{N}_{B}})=\max \left\{0,{\mathrm{log}}_{2}\left(\displaystyle \frac{\eta -(1-\eta ){N}_{B}}{(1-\eta )({N}_{B}+1)}\right)\right\}.\end{eqnarray} \tag{ 11.1 }$$

This bound was obtained by using the decomposition ${{ \mathcal L }}_{\eta ,{N}_{B}}={{ \mathcal L }}_{\eta ^{\prime} ,0}\,\circ \,{{ \mathcal A }}_{G,0}$ from theorem 31 (found independently in [RMG18]) and the bottleneck inequality $Q({{ \mathcal L }}_{\eta ^{\prime} ,0}\,\circ \,{{ \mathcal A }}_{G,0})\leqslant \min \{Q({{ \mathcal L }}_{\eta ^{\prime} ,0}),Q({{ \mathcal A }}_{G,0})\}$ for the unconstrained quantum capacity. Note that (11.1) is slightly tighter than (5.9) for all parameter regimes. These findings were independently discovered in [NAJ18].

We now introduce a new upper bound on the energy-constrained quantum and private capacities of thermal channels (independently discovered in [NAJ18] as well). In the energy-constrained scenario, one cannot directly apply the bottleneck inequality in order to obtain a bound for the finite-energy case, due to an important physical consideration discussed below. However, we introduce a method to tackle this issue and establish an upper bound in the following theorem:

Theorem 46. An upper bound on the energy-constrained quantum and private capacities of a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ with transmissivity $\eta \in [1/2,1]$, environment photon number ${N}_{B}\geqslant 0$, such that $\eta \gt (1-\eta ){N}_{B}$, and input mean photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S}),P({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\leqslant \max \{0,{Q}_{{U}_{4}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\},\end{eqnarray} \tag{ 11.2 }$$

where

$$\begin{eqnarray}&&{Q}_{{U}_{4}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})\equiv g(\eta {N}_{S}+(1-\eta ){N}_{B})-g[(1/\eta ^{\prime} -1)(\eta {N}_{S}+(1-\eta ){N}_{B})],\end{eqnarray} \tag{ 11.3 }$$

and $\eta ^{\prime} =\eta -(1-\eta ){N}_{B}$.

Proof. Using theorem 31, a thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$ satisfying $\eta \gt (1-\eta ){N}_{B}$ can be decomposed as the concatenation of a quantum-limited amplifier channel ${{ \mathcal A }}_{G,0}$ followed by a pure-loss channel ${{ \mathcal L }}_{\eta ^{\prime} ,0}$, such that

$$\begin{eqnarray}&&G=\eta /\eta ^{\prime} ,\end{eqnarray} \tag{ 11.4 }$$

$$\begin{eqnarray}&&\eta ^{\prime} =\eta -(1-\eta ){N}_{B}.\end{eqnarray} \tag{ 11.5 }$$

Consider the following chain of inequalities:

$$\begin{eqnarray}&&Q({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})=Q({{ \mathcal L }}_{\eta ^{\prime} ,0}\,\circ \,{{ \mathcal A }}_{G,0},{N}_{S})\,\,\,\,\end{eqnarray} \tag{ 11.6 }$$

$$\begin{eqnarray}&&\leqslant \,Q({{ \mathcal L }}_{\eta ^{\prime} ,0},{{GN}}_{S}+G-1)\,\,\end{eqnarray} \tag{ 11.7 }$$

$$\begin{eqnarray}&&\,=\,g(\eta ^{\prime} [{{GN}}_{S}+G-1])-g[(1-\eta ^{\prime} )({{GN}}_{S}+G-1)]\end{eqnarray} \tag{ 11.8 }$$

$$\begin{eqnarray}&&\,=\,g(\eta {N}_{S}+(1-\eta ){N}_{B})-g[(1/\eta ^{\prime} -1)(\eta {N}_{S}+(1-\eta ){N}_{B})].\end{eqnarray} \tag{ 11.9 }$$

The first inequality is a consequence of the following argument: consider an arbitrary encoding and decoding scheme for energy-constrained quantum communication over the thermal channel ${{ \mathcal L }}_{\eta ,{N}_{B}}$, which satisfies the mean input photon number constraint ${N}_{S}\geqslant 0$. Due to the decomposition of ${{ \mathcal L }}_{\eta ,{N}_{B}}$ as ${{ \mathcal L }}_{\eta ^{\prime} ,0}\,\circ \,{{ \mathcal A }}_{G,0}$, this encoding, followed by many uses of the pure-amplifier channel ${{ \mathcal A }}_{G,0}$ can be considered as an encoding for the channel ${{ \mathcal L }}_{\eta ^{\prime} ,0}$, which also satisfies the mean photon number constraint ${{GN}}_{S}+G-1$, due to the fact that the pure-amplifier channel ${{ \mathcal A }}_{G,0}$ introduces a gain. Since the energy-constrained quantum capacity of the channel ${{ \mathcal L }}_{\eta ^{\prime} ,0}$ involves an optimization over all such encodings that satisfies the mean photon number constraint ${{GN}}_{S}+G-1$, we arrive at the desired inequality. The second equality follows from the formula for the energy-constrained quantum capacity of a pure-loss bosonic channel with transmissivity $\eta ^{\prime}$ and input mean photon number ${{GN}}_{S}+G-1$ [WHG12, WQ16]. ■

Now, we conduct a numerical evaluation in order to compare the bound in (11.3) with our other bounds on the energy-constrained quantum and private capacities of a thermal channel. Since there is a free parameter $\varepsilon ^{\prime}$ in both the ε-degradable bound in (5.36) and the ε-close-degradable bound in (5.67), we optimize these bounds with respect to $\varepsilon ^{\prime}$ [Mat]. In figure 6(a), we find that both the data-processing bound ${Q}_{{U}_{1}}$ and ${Q}_{{U}_{4}}$ are close to the lower bound for medium transimissivity and high thermal noise. Moreover, ${Q}_{{U}_{4}}$ is slightly tighter than ${Q}_{{U}_{1}}$ for some parameter regimes. In figure 6(b), we find that the ε-degradable bound is tighter than all other bounds for high transmissivity and high thermal noise.

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Remark 47. The upper bound ${Q}_{{U}_{4}}({{ \mathcal L }}_{\eta ,{N}_{B}},{N}_{S})$ on the energy-constrained quantum and private capacities of thermal channels places a strong restriction on the channel parameters η and N_B. Since the quantum and private capacities of a pure-loss channel with $\eta ^{\prime}$ are non-zero only for $\eta ^{\prime} \gt 1/2$, the energy-constrained quantum and private capacities of a thermal channel will be non-zero only for

$$\begin{eqnarray}&&1\geqslant \eta \gt \displaystyle \frac{1+2{N}_{B}}{2(1+{N}_{B})},\end{eqnarray} \tag{ 11.10 }$$

which is same as the condition given in [CGH06, section 4].

In this section, we establish another upper bound on the energy-constrained quantum and private capacities of an additive-noise channel, by using theorem 46.

Theorem 48. An upper bound on the energy-constrained quantum and private capacities of an additive-noise channel ${N}_{\bar{n}}$ with noise parameter $\bar{n}\in (0,1)$, and input photon number constraint ${N}_{S}\geqslant 0$ is given by

$$\begin{eqnarray}&&Q({{ \mathcal N }}_{\bar{n}},{N}_{S}),P({{ \mathcal N }}_{\bar{n}},{N}_{S})\leqslant \max \{0,{Q}_{{U}_{4}}({{ \mathcal N }}_{\bar{n}},{N}_{S})\},\end{eqnarray} \tag{ 11.11 }$$

where

$$\begin{eqnarray}&&{Q}_{{U}_{4}}({{ \mathcal N }}_{\bar{n}},{N}_{S})\equiv g({N}_{S}+\bar{n})-g[\bar{n}({N}_{S}+\bar{n})/(1-\bar{n})].\end{eqnarray} \tag{ 11.12 }$$

Proof. A proof follows from arguments similar to those in the proof of theorem 42. The final result is obtained using theorem 46. ■

Remark 49. From a comparison of (11.12) and (10.2), we find that ${Q}_{{U}_{1}}({{ \mathcal N }}_{\bar{n}},{N}_{S})$ is tighter than ${Q}_{{U}_{4}}({{ \mathcal N }}_{\bar{n}},{N}_{S})$ only for low-noise and low input mean photon number. The bound ${Q}_{{U}_{4}}({{ \mathcal N }}_{\bar{n}},{N}_{S})$ is tighter than ${Q}_{{U}_{1}}({{ \mathcal N }}_{\bar{n}},{N}_{S})$ for all other parameter regimes.

Remark 50. Applying remarks 9 and 17, and theorem 48, we find the following data-processing bound ${Q}_{{U}_{4}}({{ \mathcal N }}_{\bar{n}})$ on the unconstrained quantum and private capacities of additive-noise channels:

$$\begin{eqnarray}&&{Q}_{{U}_{4}}({{ \mathcal N }}_{\bar{n}})={\mathrm{log}}_{2}[(1-\bar{n})/\bar{n}].\end{eqnarray} \tag{ 11.13 }$$

Remark 51. From a comparison of (11.13) with the bound in (10.4), we find that (11.13) is tighter than (10.4) for high noise.