Narrow bounds for the quantum capacity of thermal attenuators

Thermal attenuator channels model the decoherence of quantum systems interacting with a thermal bath, e.g., a two-level system subject to thermal noise and an electromagnetic signal traveling through a fiber or in free-space. Hence determining the quantum capacity of these channels is an outstanding open problem for quantum computation and communication. Here we derive several upper bounds on the quantum capacity of qubit and bosonic thermal attenuators. We introduce an extended version of such channels which is degradable and hence has a single-letter quantum capacity, bounding that of the original thermal attenuators. Another bound for bosonic attenuators is given by the bottleneck inequality applied to a particular channel decomposition. With respect to previously known bounds we report better results in a broad range of attenuation and noise: we can now approximate the quantum capacity up to a negligible uncertainty for most practical applications, e.g., for low thermal noise.

the limit of small environmental noise. This result applies to any weakly degradable channel but, in order to be more explicit, here we compute the associated upper bound for both the qubit and continuous-variable thermal attenuators. Moreover, for the Gaussian attenuator, we provide an additional bound based on the bottleneck inequality applied to a twisted version of the channel decomposition used in 15,31,44 . Eventually we compare all our bounds with those already known in the literature and in particular with the recent results of Pirandola et al. 42 and Sharma et al. 43 We show a significant improvement with respect to the state-of-the-art for a large region of attenuation and noise parameters, shrinking the unknown value of the quantum capacity within an error bar which is so narrow to be irrelevant for many practical applications.

Results
Thermal attenuator channels. Let us consider a communication channel that connects two parties. The sender, Alice, wants to transmit the quantum stateρ A 2 S H A ð Þ, represented by a positive density operator of unit trace on the Hilbert space of the system, H A . At the other end of the channel Bob receives a transformed quantum state on the Hilbert space H B . Any physical transformation applied to the state during the transmission can be represented by a quantum channel Φ, i.e., a linear, completely-positive and trace-preserving map, which evolves the initial state asρ B ¼ Φρ A À Á . The class of channels studied in this article is that of thermal attenuators, Φ η,N , originating from the following physical representation (see Fig. 1). An energy-preserving interaction U ðηÞ AE!BF , parametrized by a transmissivity parameter η ∈ [0, 1], couples the input systemρ A with an environment described by a ðηÞ AE!BF . The parameter 1−η determines the fraction of energy dispersed from the system into the environment, while the total energy is preserved. The output of the channel Φ η,N is recovered by discarding the output environment F, while that of its weak complementary Φ η;N by discarding the output system B. The two channels are called weakly complementary to each other because E is in a mixed state and hence this unitary representation is not a proper Stinespring dilation. By purifying the input environment via an ancilla E′, entangled with E, we obtain the extended channel Φ e η;N with output BE' and its strong complementaryΦ e η;N with output F. Note that the latter coincides with Φ η;N thermal stateτ E / exp ÀĤ E À Á , whereĤ E is the bath hamiltonian, of dimension equal to that of the system, and the state has mean energy N ¼ TrĤ EτE Â Ã ! 0 in dimensionless units. The total output state can be written aŝ while the action of the channel is obtained by tracing out the output environmental system F: This general framework is particularly relevant in the two paradigmatic cases in which the system is represented by a twolevel system or by a single bosonic mode. Both situations are quite important for practical applications since they model the effect of damping and thermal noise on common information carriers, typically used in experimental implementations of quantum information and quantum communication protocols. In the following, we introduce in detail these two kinds of physical systems.
In order to describe a two-level system, we fix as a basis 0 j i and 1 j i corresponding to the ground and excited states respectively. In this basis we can represent a generally mixed quantum state with a density matrix:ρ where p ∈ [0, 1] is the mean population of the excited state and γ 2 C is a complex coherence term, with γ j j 2 pð1 À pÞ. The thermal attenuator channel Γ η;N with η ∈ [0, 1] and N ∈ [0, 1/2], also known in the literature as generalized amplitude damping channel 7,19 , acts on the matrix elements in the following way: and admits a representation according to the general structure given in Eqs. (1) and (2). In this case, the environment is given by a thermal two-level system: where N ∈ [0, 1/2] represents the dimensionless mean energy and the bath hamiltonian isĤ E / 1 j i 1 h j. The unitary interaction is instead given by an energy-preserving rotation on the subspace 01 j i; 10 j i f gof the joint Hilbert space of the system: physically inducing the hopping of excitations between the system and the environment. It is easy to check that, tracing out the environmental two-level system, we obtain the single qubit thermal attenuator channel defined in (4) and (5). A single mode of electromagnetic radiation instead is formally equivalent to an infinite-dimensional quantum harmonic oscillator. It can be described in terms of bosonic annihilation and creation operatorsâ andâ y , obeying the bosonic commutation relationâ;â y Â Ã ¼ 1 or, equivalently, in terms of the quadratureŝ q ¼â þâ y Given n modes and introducing the vector of quadrature operatorsr ¼q 1 ;p 1 ; ;q n ;p n ð Þ , one can define the characteristic function 37,45 of a bosonic stateρ: where ξ 2 R 2n and Ω ¼ 0 1 À1 0 ! Èn is the symplectic form. The most common bosonic states are Gaussian states, i.e. those whose characteristic function is Gaussian: uniquely determined by the first and second moments of the statê ρ G , given respectively by the vector of mean values m and by the symmetric covariance matrix V of the quadrature operators. The single-mode thermal attenuator channel E η;N has been extensively studied in the context of Gaussian quantum information 3,27,37 and its action on the first and second moments is given by: where η ∈ [0, 1] and N ≥ 0. As for the qubit case, also this infinitedimensional channel can be represented using the general picture of Eqs. (1) and (2). Indeed, introducing the single-mode Gaussian thermal stateτ characterized by mτ ¼ 0 and Vτ ¼ ð2N þ 1Þ1, the thermal attenuator channel can be generated by a passive unitary interactionÛ that acts on the bosonic operators of the system,â, and of the environment,â E , as a beam splitter: It is easy to check that, tracing out the environmental mode, one recovers the definition of the single-mode thermal attenuator given in Eqs. (10) and (11).
Known bounds for the quantum capacity of thermal attenuators. The quantum capacity Q(Φ) of a channel Φ is defined as the maximum rate at which quantum information can be transferred by using the channel N times with vanishing error in the limit N → ∞. It is well known [23][24][25][26] that this quantity can be expressed as: where is the coherent information 27,46 and SðρÞ ¼ Àtrρ log 2ρ È É is the Von Neumann entropy; note that all logarithms henceforth are understood as base-2. Finally,Φ is the complementary channel 28 , obtained from the Stinespring dilation of Φ by tracing out the system instead of the environment, as detailed in the next subsection. Because of the peculiar superadditivity phenomenon [32][33][34][35][36] , the so called single-letter capacity is in general smaller than the actual capacity of the channel.
This fact directly gives a lower bound for the quantum capacity of the qubit thermal attenuator: where the maximization is only with respect to p ∈ [0, 1], since one can check numerically that for this channel optimal states are diagonal; this quantity can be easily numerically computed for all values of η and N. Similarly, a lower bound can be obtained also for the bosonic counterpart of the thermal attenuator by restricting the optimization over the class of Gaussian input states 41 : where ρ G varies over the set Gaussian states and gðNÞ = ðN þ 1Þlog 2 ðN þ 1Þ À N log 2 N corresponds to the entropy of the thermal state of the environment. Whether the right-hand-sides of (17) and (18) are equal or not to the true quantum capacity of the associated channels is still an important open problem in quantum information. It can be shown 19,28,40 , that for a zero-temperature environment this is the case, i.e., for N = 0 all inequalities in (17) and (18) are saturated, giving the quantum capacity of both the qubit and the Gaussian attenuators. For N > 0 instead, the capacity is still unknown, apart from some upper bounds. For the qubit case, we are not aware of any upper bounds proposed in the literature, while for the Gaussian thermal attenuator the best bounds at the moment are those recently introduced in refs. 42,43 , which can be combined to get the following expression: Let us note that Q PLOB is actually a bound on the quantum capacity assisted by two-way classical communication 42 and thus trivially bounds also the simpler unassisted capacity discussed in this article (strong-converse bounds for the two-way capacity were derived, e.g., in refs. 47,48 ). The other bound instead, Q SWAT , is itself a bound on the unassisted capacity and it has been shown 43 to beat other possible bounds based on ϵ-degradability 49 .
In the next sections we are going to derive new upper bounds which are significantly closer to the lower limits (17) and (18), especially in the low temperature regime.
The extended channel. In this subsection we first review the notions of degradability and weak degradability and then we introduce an extended version of thermal attenuator maps, whose quantum capacity is easier to compute and can be used as a useful upper bound.
Let us come back to the description of a generic attenuator map Φ η,N , valid both for qubit and bosonic systems, that we have previously defined in Eqs. (1) and (2). If, instead of tracing out the environment as done in Eq. (2), we trace out the system, we get what has been defined in refs. 29,31 as the weakly complementary channel: physically representing the flow of information from the system into the environment. Notice that this is different form the standard notion of complementary channel: and τ j i EE′ is a purification of the environment, i.e., The weak and standard complementary channels become equivalent (up to a trivial isometry) only in the particular case in which the environment is initially pure (zerotemperature limit). Finally, we also remark that the two different types of complementarity induce different definitions of degradability. A generic channel Φ is degradable 28,30 , if there exists another quantum channel Δ such thatΦ ¼ Δ Φ. Similarly, a generic channel Φ is weakly degradable 29,31 , if there exists another quantum channel Δ such that Φ ¼ Δ Φ. For degradable channels, the capacity is additive and much easier to determine. Unfortunately, typical models of quantum attenuators, as the qubit and the bosonic examples considered in this work, are degradable only for N = 0 but become only weakly degradable for N > 0 and this is the main reason behind the hardness in computing their quantum capacity.
In order to circumvent this problem, we define the extended version of a thermal attenuator channel as whereρ BFE′ is the global state defined in (22). In other words, Φ e η;N represents a situation in which Bob has access not only to the output system B but also to the purifying part E′ of the environment, see Fig. 1. A remarkable fact is that, locally, the auxiliary system E′ remains always in the initial thermal state because it is unaffected by the dynamics; however, E′ can be correlated with B and this fact can be exploited by Bob to retrieve more quantum information. In general, since trowing away E′ can only reduce the quantum capacity, one always has The advantage of dealing with the extended channel Φ e η;N is that it is degradable whenever the original channel Φ η,N is weakly degradable. This fact follows straightforwardly from the observation that the complementary channel of Φ e η;N is the weakly The degradability of Φ e η;N significantly simplifies the evaluation of Q Φ e η;N and provides a very useful upper bound for the quantum capacity of thermal attenuators: where in the last step we used the additivity property valid for all degradable channels 28,30 .
Another useful property of the extended channel, which follows from Eq. (24) and the definition of coherent information, Eq. (15), is the following: relating the coherent information of the extended and of the weakly complementary channels. Below we compute more explicitly the previous bound (25) for the specific cases of discrete-and continuous-variable thermal attenuators.
In the case of two-level systems, the purification of the thermal state given in Eq. (6) is The channel Γ η,N can be weakly degraded to Γ η;N by the composite map Δ = Ψ μ Γ η′, N , where η′ = (1 − η)/η. Here Ψ μ is a phase-damping channel 7 of parameter μ = 1 − 2N, which acts on the generic qubit state of Eq. (3) by damping the coherence matrix element as γ7 !μγ while leaving the population p constant. Hence, the extended channel Γ e η;N defined as in (23) is degradable and from (25) we get where the optimization over the single parameter p can be efficiently performed numerically for all values of η and N, giving the result presented in Fig. 2. In this case, the fact that we can reduce the optimization over diagonal input states follows from the symmetry of the coherent information under the matrixelement flipping γ → −γ and from the concavity of the coherent information for degradable channels 50 . We note that, by construction, the gap between the lower (17) and the upper (28) bounds closes in the limit N → 0, where we recover the zero-temperature capacity of the amplitude damping channel consistently with 19,41 .
In the case of bosonic systems instead, the purification of the thermal environmental modeτ with first moments mτ ¼ 0 and covariance matrix Vτ ¼ ð2N þ 1Þ1 is a Gaussian two-mode squeezed state τ j i 37 , characterized by m jτi ¼ 0 and where σ 3 = diag(1, −1) is the third Pauli matrix. The thermal attenuator E η;N is weakly degradable 29,31 , since its weakly complementary channel can be expressed as η;N is degradable and we can apply the general upper bound (25). Moreover, as shown in 40,51 , the quantum capacity of a degradable Gaussian channel is maximized by Gaussian states with fixed second moments, so that we can write reducing the problem to a tractable Gaussian optimization. Since the coherent information of the extended channel is concave and symmetric with respect to phase-space rotations and translations, for a fixed energy n it is maximized by the thermal stateτ n with covariance matrix Vτ n ¼ ð1 þ 2nÞ1 2 , see ref. 52 Therefore, without energy constraint we have The last term can be explicitly computed by using the standard formalism of Gaussian states; more simply, we can relate this quantity to the results of ref. 41 Indeed, from the property given in Eq. (26), we have Jτ n ; E e η;N ¼ ÀJτ n ; E η;N ¼ ÀJτ n ; E 1Àη;N ; ð32Þ but the last term is simply the negative of the coherent information of a thermal attenuator of transmissivity η′ = 1 − η and a thermal input state, a quantity which has been already computed in ref. 41 In the limit of n → ∞, we get our desired upper bound: where gðNÞ = ðN þ 1Þlog 2 ðN þ 1Þ À N log 2 N, shown in Fig. 3.
Comparing the upper bound (33) with the lower bound (18) we observe that we can determine the quantum capacity of the thermal attenuator up to an uncertainty of 2g(N), which vanishes in the limit of small thermal noise. For the special case N = 0, the gap closes and we recover the capacity of the pure lossy channel consistently with the previous results of ref. 40 Twisted decomposition of Gaussian attenuators. In this section, through a completely different method, we derive a bound for the quantum capacity which is tighter than Q Φ e η;N but applies only to the bosonic version of thermal attenuators.
Let us first introduce a second kind of thermal Gaussian channel: the single-mode amplifier A κ;N 37 , which combines the input state and the usual thermal stateτ of energy N through a two-mode squeezing interaction with gain κ > 1. Tracing out the environment, the first and second moments of the quantum state transform in the following way: In the particular case in which the environment is at zero temperature, i.e., for N = 0, the channel A κ;0 is called quantumlimited amplifier.
It can be shown that all phase-insensitive Gaussian channels can be decomposed as a quantum-limited attenuator followed by a quantum-limited amplifier 31,44 , with important implications for their classical capacity [15][16][17] . In this work we introduce a twisted version of this decomposition in which the order of the attenuator and of the amplifier is inverted, which is quite useful for bounding the quantum capacity of thermal attenuators. Lemma 1.
Every thermal attenuator E η;N that is not entanglement-breaking can be decomposed as a quantum-limited amplifier followed by a quantum-limited attenuator: with attenuation and gain coefficients given by The proof can be obtained by direct substitution and using the fact that non-entanglement-breaking attenuators are characterized by the condition N < η/(1 − η) 27,53 and so both coefficients in (37) are positive and well defined. As shown in the Methods Section, the previous decomposition can be generalized to all phase-insensitive Gaussian channels including thermal amplifiers and additive Gaussian noise channels. It is important to remark that, differently from the decomposition introduced in ref. 44 and employed in ref. 43 , our twisted version does not apply to entanglement-breaking channels. For the purposes of this work, this is not a restriction since all entanglement-breaking channels trivially have zero quantum capacity and we can exclude them from our analysis. Now, given a thermal attenuator with N < η/(1 − η), we make use of the twisted decomposition (36) obtaining where we used the "bottleneck" inequality Q(Φ 1 ο Φ 2 ) ≤ min{Q (Φ 1 ), Q(Φ 2 )} and the exact expression for the capacity of the quantum-limited attenuator 40 . Substituting the value of η′ of Eq. (37) into (38), we get our desired upper bound One can easily check that this last bound is always better than the one derived in Eq. (33) and the bound Q SWAT of Eq. (19). Moreover, for sufficiently small η or for sufficiently small N, it outperforms also the bound Q PLOB of Eq. (19), see Fig. 3. By combining our result, Q twist , with Q PLOB and with the lower bound Q 1 (Φ η,N ), the quantum capacity is now constrained within a very small uncertainty window. Figure 4 shows the tiny gap existing between our new upper bound (40) based on the twisted decomposition and the lower bound (18).

Discussion
In this article we computed some upper bounds on the quantum capacity of thermal attenuator channels, making use of an extended channel whose degradability properties are preserved when the environment has non-zero mean energy. This method gave interesting bounds in both the qubit and bosonic case, which are tight in the low temperature limit. Our method is quite general since it can be applied to any weakly degradable channel that admits a physical dilation with a mixed environmental state, not necessarily thermal. For example, one can apply this method straightforwardly to the bosonic thermal amplifier, though in this case the previously known upper bound 42 is very tight and cannot be improved in this way. The second method we employed is less general, since it relies on a specific decomposition of thermal attenuators, but provides a better upper bound. Moreover, the twisted decomposition of Gaussian channels that we introduced  43 (light-blue dashed line). Note that the best upper bound at small noise values is provided by our Q twist . As the noise increases, Q PLOB starts beating the former for weak attenuation, while Q SWAT remains always strictly larger than our bound in this work is an interesting result in itself which could find application in other contexts.
Our methods are of general interest for computing also other information capacities. Indeed, the channels that we introduce, i.e., the extended channel and those constituting the twisted decomposition, are by construction less noisy than the TA, in the sense that the latter can be obtained by any of the former via concatenation with another channel. This key property allows in principle to upper bound any information capacity of a TA with that of any of the channels we introduced. Of course, this may turn out to be very difficult in practice, depending on the kind of capacity that we are interested in. For example, a bound on the private capacity seems straightforward to derive, whereas it would require more efforts to bound the two-way and the strongconverse quantum capacities, the former because of the lack of a closed expression and the latter because of the difficult regularization involved in its formula. For these reasons, we believe that the channels we introduced are worth investigating also in the context of bounding other information capacities and may provide further interesting results.
Combining the results of this work with other previously known bounds, we can now estimate the value of the quantum capacity of thermal attenuator channels up to corrections, which are irrelevant for most practical purposes.

Methods
Computation of the upper bound for Qubit TA. The capacity of the extended qubit attenuator Γ e η;N can be computed by maximizing the coherent information of the channel, which is additive, as discussed in the main text: and we have defined for simplicity of notation depending on the parameters p, γ of the input qubit and on the channel parameters η, N. Recall that the extended channel Γ e η;N maps states of the system A to states of the joint system BE' that includes the purifying part of the environment. Conversely, its complementaryΓ e η;N maps states of A to states of the interacting part of the environment F. Therefore to compute their entropies we start by writing the joint state of the system AEE', in the basis 0 j i; 1 j i f g 3 : Next, we compute its evolvedρ BFE′ under the action of U ðηÞ AE!BF I E′ and the marginals with respect to the bipartition BE′-F, which correspond to the output states of the two channels: The entropies of these states can be computed numerically and it can be checked that they are invariant under phase-flip, i.e., γ → −γ. Hence we obtain an expression of the coherent information that is an even function of γ and write, following 5 : where the inequality follows from the concavity of the mutual information as a function of the input state, which holds since the channel is degradable 50 . Hence we have restricted the optimization to diagonal states in the chosen basis, i.e., on the single parameter p for fixed η, N: for η > 1/2 and 0 otherwise. The latter expression can be easily solved numerically, as shown by the plots in the main text.
Twisted decomposition of phase-insensitive Gaussian channels. Here we generalize the twisted decomposition introduced in the main text for bosonic thermal attenuators to the more general class of phase-insensitive Gaussian channels G τ;y , defined by the following action on the first and second moments of single-mode Gaussian states 37 : where τ ≥ 0 is a generalized transmissivity and y ! 1 À τ j jis a noise parameter 27 . This family includes the thermal attenuator for 0 ≤ τ < 1, the thermal amplifier for τ > 1, and the additive Gaussian noise channel for τ = 1. If y ¼ 1 À τ j j, the channel introduces the minimum noise allowed by quantum mechanics and is said to be quantum-limited. On the other hand, it can be shown 27,53 that a phase-insensitive Gaussian channel is entanglement-breaking if and only if y ≥ 1 + τ, which determines a noise threshold above which the channel has trivially zero quantum . The white region corresponds to zero capacity. Observe that the approximation is tight in the small-noise region and, at higher values of noise, in the strongattenuation region. Note that, as shown in Fig. 3b, in the opposite regime of high values of N and η, the quantum capacity is better upper bounded by (19) capacity. Below the entanglement-breaking threshold, the following decomposition holds.Lemma 2. Every phase-insensitive Gaussian channel G τ;y which is not entanglement-breaking (y < 1 + τ), can be decomposed as a quantum-limited amplifier followed by a quantum-limited attenuator: G τ;y ¼ G η′;1Àη′ G κ′;κ′À1 ¼ E η′;0 A κ′;0 ; ð49Þ with attenuation and gain coefficients given by η′ ¼ ð1 þ τ À yÞ=2; κ′ ¼ τ=η′: The proof follows by direct substitution of the parameters (50) into (49) and from the application of Eqs. (47) and (48). Moreover, the hypothesis y < 1 + τ is necessary since it ensures the positivity of both the attenuation and the gain parameters η′ and κ′.

Data availability
No datasets were generated or analyzed during the current study.