Narrow Bounds for the Quantum Capacity of Thermal Attenuators

1, 2, 2 2 Matteo Rosati, ∗ Andrea Mari, and Vittorio Giovannetti 1F´ısica Teòrica: Informaciói FenòmensQuàntics,Departament de F´ısica, Universitat Autònomade Barcelona, 08193 Bellaterra, Spain 2NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56127 Pisa, Italy.

ABSTRACT As usual in information theory, although a formula exists for this quantity, it is quite difficult to compute for Thermal attenuator channels model the decoherence of general channels, due for example to the necessary reg- quantum systems interacting with a thermal bath, e.g., a ularization that takes into account the use of entangled two-level system subject to thermal noise and an electro- inputs over multiple channel uses [27]. The problem sim- magnetic signal travelling through a fiber or in free-space. plifies considerably for the so called degradable channels Hence determining the quantum capacity of these chan- [19, 28–31], but it also exhibits some striking features in nels is an outstanding open problem for quantum compu- other cases [32–36]. An important class of channels is tation and communication. Here we derive several upper that of Thermal Attenuators (TAs), which describe the bounds on the quantum capacity of qubit and bosonic effect of energy loss due to the interaction with a ther- thermal attenuators. We introduce an extended version mal environment. Common examples of this class are the of such channels which is degradable and hence has a qubit thermal attenuator or generalized amplitude damp- single-letter quantum capacity, bounding that of the orig- ing channel [7, 19] and its infinite-dimensional counter- inal thermal attenuators. Another bound for bosonic at- part given by the bosonic Gaussian thermal attenuator tenuators is given by the bottleneck inequality applied to [3, 4, 37]. Physically, the former is a good model for a particular channel decomposition. With respect to pre- the thermalization process of a two-level system (e.g. a viously known bounds we report better results in a broad single-qubit quantum memory) in contact with a ther- range of attenuation and noise: we can now approximate mal bath, while the latter is the standard description for the quantum capacity up to a negligible uncertainty for optical-fiber and free-space communications in the pres- most practical applications, e.g., for low thermal noise. ence of thermal noise. Notice that thermal noise at room temperature is not negligible for low-frequency electromagnetic signals like, e.g., infra-red lasers, microwaves, INTRODUCTION radio waves, etc.. For example, a crude estimate of the thermal noise in a real communication channel can be The study of information transmission between distant obtained by using the Bose-Einstein distribution at the parties has attracted much theoretical attention since the desired wavelength to estimate the excess contribution to the mean photon number of the transmitted signal: seminal work of Shannon [1, 2], who gave birth to the 14 field of information theory by determining the ultimate this grows from O(10− ) at telecom wavelengths, i.e., limits for compression and transmission rate. The lat- 1550nm, and O(10) at microwave wavelengths, i.e., 1mm ter is called information capacity and it depends on the and above. Accordingly, although the thermal noise may specific channel that is used to model a physical trans- be negligible at telecom wavelengths, it becomes increas- mission process. Hence, since the information carriers ingly important as one spans the whole optical domain are ultimately governed by the laws of quantum physics, and it is a crucial parameter in the microwave domain. in more recent years there has been interest in analyz- Hence the study of information transmission on TAs is of ing the communication problem in a quantum setting, particular relevance for future quantum communication giving birth to the field of quantum communication and networks where hybrid interfaces will be employed, e.g., superconducting qubits connected by microwave commu- arXiv:1801.04731v5 [quant-ph] 11 Dec 2018 information theory [3–7]. Several results have been obtained so far, such as: an expression for the capacity of nication lines, see Refs. [38, 39]. In both the qubit and a quantum channel for the transmission of classical in- bosonic cases, the corresponding quantum capacity is formation [8–12] and its explicit value for some classes known [19, 40] only for a zero-temperature environment, of channels [5, 6, 13–19]; the use of entanglement [20] since in this limit both channels are degradable [28–30]. as a resource for communication [21, 22]; the capacity However the more general finite-temperature case breaks of a quantum channel for the transmission of quantum the degradability property and it has not been success- information, i.e., of states preserving quantum coher- fully tackled so far, apart from establishing some upper ence [23–26]. The latter is called quantum capacity of and lower bounds [41–43]. the channel and constitutes the main focus of this work. In this article we introduce a general method to compute new bounds on the capacity of any quantum channel which is weakly degradable in the sense of [29, 31]. It is based on the purification of the environment by ∗ [email protected] an additional system, which is then included in the 2 output of an extended version of the original channel. Degradability is restored for such extended channel and its quantum capacity can be easily computed, providing E E an upper bound on the capacity of the original channel, which is tight in the limit of small environmental noise. e This result applies to any weakly degradable channel ,N 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 ,N the Gaussian attenuator, we provide an additional bound ˆ () based on the bottleneck inequality applied to a twisted A UAE BF B 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 [42, 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 ¯ ˜ e ,N ,N within an error bar which is so narrow to be irrelevant F for many practical applications.

Figure 1. Unitary representation of a thermal attenuator RESULTS channel, its extended channel and their complementaries. The system A interacts with the environment E, which is Thermal Attenuator channels in a thermal state of mean energy N, via a linear coupling, ˆ (η) UAE→BF. The parameter η determines the fraction of energy dispersed from the system into the environment, while the Let us consider a communication channel that connects total energy is preserved. The output of the channel Φη,N is two parties. The sender, Alice, wants to transmit the recovered by discarding the output environment F, while that quantum stateρ ˆ S( ), represented by a positive ¯ A ∈ HA of its weak complementary Φη,N by discarding the output sys- density operator of unit trace on the Hilbert space of the tem B. The two channels are called weakly complementary to system, A. At the other end of the channel Bob re- each other because E is in a mixed state and hence this uni- ceives a transformedH quantum state on the Hilbert space tary representation is not a proper Stinespring dilation. By B. Any physical transformation applied to the state purifying the input environment via an ancilla E’, entangled H e during the transmission can be represented by a quan- with E, we obtain the extended channel Φη,N with output ˜ e tum channel Φ, i.e., a linear, completely-positive and BE’ and its strong complementary Φη,N with output F. Note ¯ trace-preserving map, which evolves the initial state as that the latter coincides with Φη,N . ρˆB = Φ (ˆρA). The class of channels studied in this article is that of mode. Both situations are quite important for practical thermal attenuators, Φη,N , originating from the following physical representation (see Fig. 1). An energy- applications since they model the effect of damping and thermal noise on common information carriers, typically preserving interaction Uˆ (η) , parametrized by a trans- AE BF used in experimental implementations of quantum η →, missivity parameter [0 1], couples the input system information and quantum communication protocols. In ρ ∈ Â with an environment described by a thermal state the following we introduce in detail these two kinds of τ Hˆ Hˆ Ê exp( E), where E is the bath hamiltonian, of physical systems. dimension∝ − equal to that of the system, and the state has h i ˆ mean energy N = Tr HEτÊ 0 in dimensionless units. In order to describe a two-level system, we fix as a basis ≥ The total output state can be written as 0 and 1 corresponding to the ground and excited states respectively.| i | i In this basis we can represent a generally ˆ (η) ˆ (η) ρˆBF = UAE BF (ˆρA τÊ) UAE† BF, (1) mixed quantum state with a density matrix: → ⊗ → while the action of the channel is obtained by tracing out 1 p γ the output environmental system F: ρˆ = − , (3) γ∗ p Φ (ˆρ ) = Tr [ˆρ ] . (2) η,N A F BF where p [0, 1] is the mean population of the excited ∈ 2 This general framework is particularly relevant in the state and γ C is a complex coherence term, with γ ∈ | | ≤ two paradigmatic cases in which the system is repre- p(1 p). The thermal attenuator channel Γη, with η sented by a two-level system or by a single bosonic [0, 1]− and N [0, 1/2], also known in the literatureN as∈ ∈ 3 generalized amplitude damping channel [7, 19], acts on context of Gaussian quantum information [3, 27, 37] and the matrix elements in the following way: its action on the first and second moments is given by:

Γη,N η,N p p ηp η N, m E m0 = √η mm, (10) 0 = + (1 ) (4) −−−→ −−−→ − η,N Γη,N V E V ηV η N 1 , γ γ0 = √ηγ, (5) 0 = + (1 )(2 + 1) 2 (11) −−−→ −−−→ − where η [0, 1] and N 0. As for the qubit case, also and admits a representation according to the general this infinite-dimensional∈ ≥ channel can be represented us- structure given in Eqs. (1) and (2). In this case the ing the general picture of Eqs. (1) and (2). Indeed, intro- environment is given by a thermal two-level system: ducing the single-mode Gaussian thermal stateτ ˆ charac- m 1 N 0 terized by mτˆ = 0 and Vτˆ = (2N + 1)1, the thermal τˆ = , (6) −0 N attenuator channel can be generated by a passive unitary interaction Uˆ that acts on the bosonic operators of where N [0, 1/2] represents the dimensionless mean en- the system,a ˆ, and of the environment,a Ê, as a beam ∈ splitter: ergy and the bath hamiltonian is HÊ 1 1 . The unitary interaction is instead given by an∝ energy-preserving | ih | p U †aUˆ = √ηaˆ + 1 ηaÊ, (12) rotation on the subspace 01 , 10 of the joint Hilbert p − {| i | i} U †aˆ U = 1 ηaˆ + √ηaˆ . (13) space of the system: E − − E  1 0 0 0  It is easy to check that, tracing out the environmental 0 √η √1 η 0 mode, one recovers the definition of the single-mode ther- Uˆ (η) =  −  , (7)  0 √1 η √η 0  mal attenuator given in Eqs. (10) and (11). 0− 0− 0 1 physically inducing the hopping of excitations between Known bounds for the Quantum Capacity of Thermal Attenuators 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 The quantum capacity Q(Φ) of a channel Φ is defined defined in (4) and (5). as the maximum rate at which quantum information can be transferred by using the channel N times with vanish- ing error in the limit N . It is well known [23–26] A single mode of electromagnetic radiation instead is → ∞ formally equivalent to an infinite-dimensional quantum that this quantity can be expressed as: harmonic oscillator. It can be described in terms of 1 N bosonic annihilation and creation operatorsa ˆ anda ˆ†, Q(Φ) = lim max J ρ,ˆ Φ⊗ , (14) N ⊗N N ρˆ S( A ) obeying the bosonic commutation relation [â, aˆ†] = 1 or, →∞ ∈ H √ equivalently, in terms of the quadraturesq ˆ = (â+â†)/ 2 where andp ˆ = i(â† aˆ)/√2, such that [ˆq, pˆ] = i. Given n modes and introducing− the vector of quadrature opera- J (ˆρ, Φ) = S(Φ(ˆρ)) S(Φ(ˆ˜ ρ)) (15) − torsr ˆ = (ˆq1, pˆ1, ..., qˆn, pˆn)>, one can define the characteristic function [37, 45] of a bosonic stateρ ˆ: is the coherent information [27, 46] and S(ˆρ) = tr ρˆlog ρˆ is the Von Neumann entropy; note that all − { 2 } h T T i logarithms henceforth are understood as base-2. Finally, χ ξ ρ eξ Ω rˆ , ( ) = Tr ˆ (8) Φ˜ is the complementary channel [28], obtained from the Stinespring dilation of Φ by tracing out the system in- n 0 1 ⊕ stead of the environment, as detailed in the next sub- where ξ R2n and Ω = is the symplectic ∈ 1 0 section. Because of the peculiar super-additivity phe- form. The most common bosonic− states are Gaussian nomenon [32–36], the so called single-letter capacity states, i.e. those whose characteristic function is Gaus- Q1(Φ) = max J(ˆρ, Φ) (16) sian: ρˆ A ∈H 1 χ (ξ) = exp ξT ΩT V Ω ξ + ξT ΩT m , (9) is in general smaller than the actual capacity of the chan- G −4 nel. This fact directly gives a lower bound for the quantum uniquely determined by the first and second moments of capacity of the qubit thermal attenuator: the stateρ ˆ , given respectively by the vector of mean G values m and by the symmetric covariance matrix V of 1 p 0 Q(Γη,N ) Q1(Γη,N ) = max J ρˆ = − , Γη,N the quadrature operators. The single-mode thermal at- ≥ p 0 p tenuator channel has been extensively studied in the (17) Eη,N 4 where the maximization is only with respect to p [0, 1], that we have previously defined in Eqs. (1, 2). If, instead since one can check numerically that for this channel∈ op- of tracing out the environment as done in Eq. (2), we timal states are diagonal; this quantity can be easily nu- trace out the system, we get what has been defined in merically computed for all values of η and N. Similarly, [29, 31] as the weakly-complementary channel: a lower bound can be obtained also for the bosonic coun- terpart of the thermal attenuator by restricting the opti- Φ¯ η,N (ˆρA) = TrB [ˆρBF] , (20) mization over the class of Gaussian input states [41]: physically representing the flow of information from the Q( η,N ) Q1( η,N ) max J(ˆρG, η,N ) system into the environment. Notice that this is different E ≥ E ≥ ρˆG E form the standard notion of complementary channel: η = max 0, log2 g(N) , 1 η − Φ˜ η,N (ˆρA) = TrB [ˆρBFE0 ] , (21) − (18) where where ρG varies over the set Gaussian states and g(N) = ˆ (η) ˆ (η) (N +1) log2(N +1) N log2 N corresponds to the entropy ρˆBFE0 = UAE BF (ˆρA τ τ EE0 ) UAE† BF (22) of the thermal state− of the environment. → ⊗ | ih | → Whether the right-hand-sides of (17) and (18) are and τ 0 is a purification of the environment, i.e.τ ˆ = | iEE E equal or not to the true quantum capacity of the as- TrE0 [ τ τ EE0 ]. The weak and standard complementary sociated channels is still an important open problem in channels| ih become| equivalent (up to a trivial isometry) quantum information. It can be shown [19, 28, 40], that only in the particular case in which the environment is for a zero-temperature environment this is the case, i.e. initially pure (zero-temperature limit). Finally, we also for N = 0 all inequalities in (17) and (18) are saturated, remark that the two different types of complementarity giving the quantum capacity of both the qubit and the induce different definitions of degradability. A generic Gaussian attenuators. For N > 0 instead, the capacity channel Φ is degradable [28, 30], if there exists another is still unknown, apart from some upper bounds. For the quantum channel ∆ such that Φ˜ = ∆ Φ. Similarly, a qubit case we are not aware of any upper bounds pro- generic channel Φ is weakly degradable◦ [29, 31], if there posed in the literature, while for the Gaussian thermal exists another quantum channel ∆ such that Φ¯ = ∆ Φ. attenuator the best bounds at the moment are those re- For degradable channels, the capacity is additive◦ and cently introduced in [42, 43], which can be combined to much easier to determine. Unfortunately, typical models get the following expression: of quantum attenuators, as the qubit and the bosonic examples considered in this work, are degradable only for Q( ) min Q ( ),Q ( ) , (19) Eη,N ≤ { PLOB Eη,N SWAT Eη,N } N = 0 but become only weakly degradable for N > 0 and Q ( ) = max 0, log [(1 η)ηN ] g(N) , this is the main reason behind the hardness in computing PLOB Eη,N { − 2 − − } their quantum capacity. QSWAT( η,N ) = max 0, log2[η/(1 η)] log2[N + 1] . E { − − } In order to circumvent this problem, we define the ex- Let us note that QPLOB is actually a bound on the quan- tended version of a thermal attenuator channel as tum capacity assisted by two-way classical communica- e tion [42] and thus trivially bounds also the simpler unas- Φη,N (ˆρA) = TrF [ˆρBFE0 ] , (23) sisted capacity discussed in this article (strong-converse bounds for the two-way capacity where derived, e.g., whereρ ˆBFE0 is the global state defined in (22). In other e in [47, 48]). The other bound instead, QSWAT, is it- words, Φη,N represents a situation in which Bob has ac- self a bound on the unassisted capacity and it has been cess not only to the output system B but also to the shown [43] to beat other possible bounds based on - purifying part E’ of the environment, see Fig. 1. A re- degradability [49]. In the next sections we are going to markable fact is that, locally, the auxiliary system E’ derive new upper bounds which are significantly closer remains always in the initial thermal state because it is to the lower limits (17) and (18), especially in the low unaffected by the dynamics; however E’ can be correlated temperature regime. 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 al- The extended channel e ways has Q(Φη,N ) Q(Φη,N ). ≤ e The advantage of dealing with the extended channel Φη,N In this subsection we first review the notions of degrad- is that it is degradable whenever the original channel ability and weak degradability and then we introduce Φη,N is weakly degradable. This fact follows straight- an extended version of thermal attenuator maps, whose forwardly from the observation that the complementary e quantum capacity is easier to compute and can be used channel of Φη,N is the weakly-complementary channel of as a useful upper bound. Φη,N , i.e.: Let us come back to the description of a generic attenu- ˜ e ¯ ator map Φη,N , valid both for qubit and bosonic systems, Φη,N (ˆρA) = TrBE0 [ˆρBFE0 ] = Φη,N (ˆρA) . (24) 5

e The degradability of Φη,N signiﬁcantly simpliﬁes the e evaluation of Q(Φη,N ) and provides a very useful upper bound for the quantum capacity of thermal attenuators:

Q(Φ ) Q(Φe ) = Q (Φe ), (25) η,N ≤ η,N 1 η,N 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:

J ρ, Φe = J ρ, Φ¯ . (26) η,N − η,N 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

τ = √1 N 00 + √N 11 . (27) | i − | i | i

The channel Γη,N can be weakly degraded to Γ¯η,N by the composite map ∆ = Ψ Γ 0 , where η0 = (1 η)/η. µ ◦ η ,N − Here Ψµ 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 γ µγ while leaving the population p constant. Hence the7→ extended e channel Γη,N defined as in (23) is degradable and from (25) we get Figure 2. Bounds of the quantum capacity of a qubit thermal attenuator. e 1 p 0 e Plot (log-linear scale) of the lower bound Q1(Γη,N ) (green Q(Γη,N ) Q1(Γη,N ) = max J ρˆ = − , Γη,N , e ≤ p 0 p line), Eq. (17), and upper bound Q1(Γη,N ) (red line), Eq. (28), (28) of the quantum capacity of the qubit thermal attenuator chan- where the optimization over the single parameter p nel as a function of the attenuation parameter η for two values can be efficiently performed numerically for all values of noise: (a) N = 0.01, (b) N = 0.1. The lower bound is given of η and N, giving the result presented in Fig. 2. In by the single-letter capacity of the channel, whereas the upper bound by the capacity of the extended channel, which equals this case the fact that we can reduce the optimization its single-letter expression because of degradability. The two over diagonal input states follows from the symmetry bounds are close for small noise and weak attenuation. of the coherent information under the matrix-element flipping γ γ and from the concavity of the coherent → − information for degradable channels [50]. where σ = diag(1, 1) is the third Pauli matrix. The 3 − We note that, by construction, the gap between the thermal attenuator η,N is weakly degradable [29, 31], lower (17) and the upper (28) bounds closes in the limit since its weakly complementaryE channel can be expressed N 0, where we recover the zero-temperature capacity as ¯ = ∆ where ∆ = 0 , with η = (1 η)/η. → η,N η,N η ,N 0 of the amplitude damping channel consistently with [19]. HenceE e ◦Eis degradable andE we can apply the− gen- Eη,N [41]. eral upper bound (25). Moreover, as shown in [40, 51], the quantum capacity of a degradable Gaussian channel In the case of bosonic systems instead, the purification is maximized by Gaussian states with fixed second mo- of the thermal environmental modeτ ˆ with first moments ments, so that we can write mτˆ = 0 and covariance matrix Vτˆ = (2N + 1)1 is a e e Gaussian two-mode squeezed state τ [37], characterized Q ( η,N ) Q1 η,N = max J(ˆρG, η,N ) (30) | i E ≤ E ρˆG E by m τ = 0 and | i reducing the problem to a tractable Gaussian optimiza- p 2 Vτˆ σ3 Vτˆ 12 tion. Since the coherent information of the extended V τ = p − , (29) | i σ V 2 1 V channel is concave and symmetric with respect to phase- 3 τˆ − 2 τˆ 6 space rotations and translations, for a fixed energy n it is maximized by the thermal stateτ ˆn with covariance matrix Vτˆn = (1 + 2n)12, see Ref. [52]. Therefore, without energy constraint we have

e e Q ( η,N ) Q1 η,N = lim J(ˆτn, η,N ). (31) n E ≤ E →∞ E 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 Indeed, from the property given in Eq. (26), we have

e ¯ J(ˆτn, η,N ) = J(ˆτn, η,N ) = J(ˆτn, 1 η,N ), (32) E − E − E − but the last term is simply the negative of the coherent information of a thermal attenuator of transmissivity η0 = 1 η and a thermal input state, a quantity which has been already− computed in [41]. In the limit of n , we get our desired upper bound: → ∞

η Q ( ) Q e = max 0, log + g(N) , Eη,N ≤ 1 Eη,N 2 1 η − (33) where g(N) = (N + 1) log2(N + 1) N log2 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 Figure 3. Bounds of the quantum capacity of a bosonic ther- consistently with the previous results of Ref. [40]. mal attenuator. Plot (log-linear scale) of the lower bound Q1(Eη,N ) (green line), Eq. (18), and several upper bounds of the quantum ca- Twisted decomposition of Gaussian attenuators pacity of the bosonic Gaussian thermal attenuator channel as a function of the attenuation parameter τ for two values In this section, through a completely different method, of noise: (a) N = 0.1, (b) N = 0.5. The upper bounds are given by: the capacity of the extended channel Q (E e ) (red we derive a bound for the quantum capacity which is 1 η,N dashed line), Eq. (33); the capacity of the attenuator chan- tighter than Q Φe but applies only to the bosonic η,N nel of the twisted decomposition Qtwist (red line), Eq. (40); version of thermal attenuators. the upper bounds QPLOB of Eq. (19) derived from Ref. [42] Let us first introduce a second kind of thermal Gaus- (blue dashed line) and QSWAT of Ref. [43] (light-blue dashed sian channel: the single-mode amplifier κ,N [37], which line). Note that the best upper bound at small noise val- A combines the input state and the usual thermal stateτ ˆ ues is provided by our Qtwist. As the noise increases, QPLOB of energy N through a two-mode squeezing interaction starts beating the former for weak attenuation, while QSWAT with gain κ > 1. Tracing out the environment, the first remains always strictly larger than our bound. and second moments of the quantum state transform in the following way:

κ,N [15–17]. In this work we introduce a twisted version of m A m0 = √κmmm, (34) −−−→ this decomposition in which the order of the attenuator κ,N and of the amplifier is inverted, which is quite useful for V A V 0 = κV + (κ 1)(2N + 1)1. (35) −−−→ − bounding the quantum capacity of thermal attenuators. In the particular case in which the environment is at zero Lemma 1. Every thermal attenuator that is temperature, i.e. for N = 0, the channel is called η,N κ,0 not entanglement-breaking can be decomposedE as a quantum-limited amplifier. A quantum-limited amplifier followed by a quantum-limited It can be shown that all phase-insensitive Gaussian attenuator: channels can be decomposed as a quantum-limited attenuator followed by a quantum-limited amplifier [31, 44], with important implications for their classical capacity = 0 0 , (36) Eη,N Eη ,0 ◦ Aκ ,0 7 with attenuation and gain coefficients given by 5

η0 = η N(1 η), κ0 = η/η0 (37) − − 1.0 The proof can be obtained by direct substitution and 0.9 4 using the fact that non-entanglement-breaking attenua- 0.8 tors are characterized by the condition N < η/(1 η) 0.7 − 0.6 [27, 53] and so both coefficients in (37) are positive and 0.5 well defined. As shown in the Methods Section, the 3 0.4 previous decomposition can be generalized to all phase- 0.3 insensitive Gaussian channels including thermal ampli- 0.2 0.1 fiers and additive Gaussian noise channels. It is important to remark that, differently from the decomposition 2 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 1 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), 0 we make use of the twisted decomposition (36) obtaining− 0.5 0.6 0.7 0.8 0.9 1.0

Q(Φη,N ) = Q(Φη0,0 κ0,0) Q(Φη0,0) (38) ◦ A ≤ η0 = max 0, log2 , 1 η0 Figure 4. Best-approximation accuracy of the quantum ca- − (39) pacity of a bosonic thermal attenuator. Contour plot of the difference between Qtwist of Eq. (40), i.e., where we used the “bottleneck” inequality Q(Φ Φ ) the twisted-decomposition upper bound of the quantum ca- 1 ◦ 2 ≤ min Q(Φ1),Q(Φ2) and the exact expression for the capacity of the bosonic Gaussian thermal attenuator, and the pacity{ of the quantum-limited} attenuator [40]. Substi- lower bound Q1(Eη,N ) of Eq. (18), i.e., the single-letter capacity of the channel, as a function of the attenuation parameter tuting the value of η0 of Eq. (37) into (38), we get our desired upper bound η ∈ [0.5, 1] and noise values N ∈ [0, 5]. The white region corresponds to zero capacity. Observe that the approxima- η N(1 η) tion is tight in the small-noise region and, at higher values of Q(Φ ) Q = max 0, log − − . noise, in the strong-attenuation region. Note that, as shown η,N ≤ twist 2 (1 + N)(1 η) − in Fig. 3b, in the opposite regime of high values of N and η, (40) the quantum capacity is better upper bounded by (19). One can easily check that this last bound is always better than the one derived in Eq. (33) and the bound QSWAT of Eq. (19). Moreover, for sufficiently small η or for suf- admits a physical dilation with a mixed environmental ficiently small N, it outperforms also the bound QPLOB state, not necessarily thermal. For example one can ap- of Eq. (19), see Fig. 3. By combining our result, Qtwist, ply this method straightforwardly to the bosonic thermal with QPLOB and with the lower bound Q1(Φη,N ), the amplifier, though in this case the previously known upper quantum capacity is now constrained within a very small bound [42] is very tight and cannot be improved in this uncertainty window. Figure 4 shows the tiny gap existing way. The second method we employed is less general, between our new upper bound (40) based on the twisted since it relies on a specific decomposition of thermal at- decomposition and the lower bound (18). tenuators, but provides a better upper bound. Moreover, the twisted decomposition of Gaussian channels that we introduced in this work is an interesting result in itself DISCUSSION which could find application in other contexts. Our methods are of general interest for computing also In this article we computed some upper bounds on the other information capacities. Indeed, the channels that quantum capacity of thermal attenuator channels, mak- we introduce, i.e., the extended channel and those con- ing use of an extended channel whose degradability prop- stituting the twisted decomposition, are by construction erties are preserved when the environment has non-zero less noisy than the TA, in the sense that the latter can mean energy. This method gave interesting bounds in be obtained by any of the former via concatenation with both the qubit and bosonic case, which are tight in the another channel. This key property allows in principle to low temperature limit. Our method is quite general since upper-bound any information capacity of a TA with that it can be applied to any weakly-degradable channel that of any of the channels we introduced. Of course, this may 8 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 strong-converse quantum capacities, the former because of the lack of a closed expression and the latter because of the difficult regularization in- volved 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. 9

METHODS and we have deﬁned for simplicity of notation 1 p γ e Computation of the upper bound for Qubit TA Jη,N (p, γ) = J ρˆ = − , Γη,N , (42) γ∗ p

e The capacity of the extended qubit attenuator Γη,N depending on the parameters p, γ of the input qubit and can be computed by maximizing the coherent informa- on the channel parameters η, N. Recall that the extended e tion of the channel, which is additive, as discussed in the channel Γη,N maps states of the system A to states of main text: the joint system BE’ that includes the purifying part of ˜e the environment. Conversely, its complementary Γη,N h i e e ˜e maps states of A to states of the interacting part of the Q1(Γη,N ) = max S Γη,N (ˆρ) S Γη,N (ˆρ) ρˆ − environment F. Therefore to compute their entropies we start by writing the joint state of the system AEE’, in = max Jη,N (p, γ) (41) p,γ 3 the basis 0 , 1 ⊗ : {| i | i}

 (1 − N)(1 − p) 0 0 p(1 − N)N(1 − p) γ(1 − N) 0 0 p(1 − N)Nγ   0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0   p p   (1 − N)N(1 − p) 0 0 N(1 − p) (1 − N)Nγ 0 0 Nγ  ρˆAEE0 =  ∗ p ∗ p  . (43)  (1 − N)γ 0 0 (1 − N)Nγ p − Np 0 0 (1 − N)Np     0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0  p(1 − N)Nγ∗ 0 0 Nγ∗ p(1 − N)Np 0 0 Np (η) Next, we compute its evolvedρ ˆBFE0 under the action of AE BF IE0 and the marginals with respect to the bipartition BE’-F, which correspond to the output states of the twoU channels:→ ⊗ √  (1 − N)(1 − pη) p(1 − N)N(1 − η)ηγ∗ γ(1 − N) η (2p − 1)p(1 − N)N(1 − η)  √ γp(1 − N)N(1 − η)η N(1 − p)η 0 Nγ η Γe (ˆρ) =  √  , η,N  (1 − N) ηγ∗ 0 p(1 − N)η −p(1 − N)N(1 − η)ηγ∗  √ (2p − 1)p(1 − N)N(1 − η) N ηγ∗ −γp(1 − N)N(1 − η)η (p − 1)ηN + N (44) √ −p(1 − η) − Nη + 1 (1 − 2N)γ 1 − η ˜e ρ √ . Γη,N (ˆ) = (1 − 2N) 1 − ηγ∗ −ηp + p + Nη

The entropies of these states can be computed numeri- Twisted decomposition of phase-insensitive Gaussian cally and it can be checked that they are invariant under channels phase-flip, i.e., γ γ. Hence we obtain an expression → − of the coherent information that is an even function of γ Here we generalize the twisted decomposition intro- and write, following [5]: duced in the main text for bosonic thermal attenuators to the more general class of phase-insensitive Gaussian J (p, γ) + J (p, γ) J (p, γ) = η,N η,N − J (p, 0), channels τ,y, defined by the following action on the first η,N 2 ≤ η,N and secondG moments of single-mode Gaussian states [37]: (45) τ,y where the inequality follows from the concavity of the m G m0 = √τ mm, (47) −−−→ mutual information as a function of the input state, τ,y which holds since the channel is degradable [50]. Hence V G V 0 = τV + y12, (48) −−−→ we have restricted the optimization to diagonal states in where τ 0 is a generalized transmissivity and y 1 τ the chosen basis, i.e., on the single parameter p for fixed is a noise≥ parameter [27]. This family includes the≥ | ther-− | η, N: mal attenuator for 0 τ < 1, the thermal amplifier ≤ e for τ > 1, and the additive Gaussian noise channel for Q1(Γη,N ) = max Jη,N (p, 0), (46) p τ = 1. If y = 1 τ , the channel introduces the minimum noise allowed| − by| quantum mechanics and is said for η > 1/2 and 0 otherwise. The latter expression can to be quantum-limited. On the other hand, it can be be easily solved numerically, as shown by the plots in the shown [27, 53] that a phase-insensitive Gaussian chan- main text. nel is entanglement-breaking if and only if y 1 + τ, ≥ 10 which determines a noise threshold above which the The proof follows by direct substitution of the param- channel has trivially zero quantum capacity. Below the eters (50) into (49) and from the application of Eqs. (47, entanglement-breaking threshold, the following decom- 48). Moreover, the hypothesis y < 1+τ is necessary since position holds. it ensures the positivity of both the attenuation and the gain parameters η0 and κ0. Lemma 2. Every phase-insensitive Gaussian channel τ,y which is not entanglement-breaking (y < 1 + τ), canG be decomposed as a quantum-limited amplifier followed by a quantum-limited attenuator:

τ,y = η0,1 η0 κ0,κ0 1 = η0,0 κ0,0, (49) DATA AVAILABILITY G G − ◦ G − E ◦ A with attenuation and gain coeﬃcients given by No datasets were generated or analysed during the cur- η0 = (1 + τ y)/2, κ0 = τ/η0. (50) rent study. −

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