text:
\section*{Abstract}
{\color{red}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 travelling 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.}

\section*{Introduction}
The study of information transmission between distant parties has attracted much theoretical attention since the seminal work of Shannon~\cite{shannonSeminal,shannonSeminal1}, who gave birth to the field of information theory by determining the ultimate limits for compression and transmission rate. The latter is called information capacity and it depends on the specific channel that is used to model a physical transmission process. Hence, since the information carriers are ultimately governed by the laws of quantum physics, in more recent years there has been interest in analyzing the communication problem in a quantum setting, giving birth to the field of quantum communication and information theory~\cite{holevoBOOK,cavesDrum,wildeBOOK,hayashiBOOK,nChuangBOOK}. Several results have been obtained so far, such as: an expression for the capacity of a quantum channel for the transmission of classical information~\cite{holevo1,holevo2,holevo3,schumawest1,schumawest2} and its explicit value for some classes of channels~\cite{wildeBOOK,hayashiBOOK,kingUnital,depolarizingCCap,gaussOpt,gaussMaj,gaussCap,LOSSY,amplitudeDamping}; the use of entanglement~\cite{horodecki}  as a resource for communication~\cite{bennetEntAss1,bennetEntAss2};
the capacity of a quantum channel for the transmission of quantum information, i.e., of states preserving quantum coherence~\cite{lloydQuant,shorQuantum,qCap1,qCap2}.
The latter is called quantum capacity of the channel and constitutes the main focus of this work. As usual in information theory, although a formula exists for this quantity, it is quite difficult to compute for general channels, due for example to the necessary regularization that takes into account the use of entangled inputs over multiple channel uses~\cite{holGiovRev}. The problem simplifies considerably for the so called {\it degradable} channels ~\cite{deg1,antiDeg,deg2,amplitudeDamping,weakDeg}, but it also exhibits some striking features in other cases~\cite{DiVincenzo1998,Smith2007,zeroQCap,hastings2009,supAddQCap}.
An important class of channels is that of Thermal Attenuators (TAs), which describe the effect of energy loss due to the interaction with a thermal environment. Common examples of this class are the qubit thermal attenuator or  generalized amplitude damping channel~\cite{nChuangBOOK,amplitudeDamping} and its infinite-dimensional counterpart given by the bosonic Gaussian thermal attenuator \cite{cavesDrum, holevoBOOK, RevGauss}. Physically, the former is a good model for the thermalization process of a two-level system (e.g. a single-qubit quantum memory) in contact with a thermal bath, while the latter is the standard description for optical-fiber and free-space communications in the presence of thermal noise. {\color{red} Notice that thermal noise at room temperature is not negligible for low-frequency electromagnetic signals like, e.g., infra-red lasers, microwaves, radio waves, {\it etc.}. For example, a crude estimate of the thermal noise in a real communication channel can be obtained by using  the Bose-Einstein distribution at the desired wavelength to estimate the excess contribution to the mean photon number of the transmitted signal: this grows from $O(10^{-14})$ at telecom wavelengths, i.e., $1550 nm$, and $O(10)$ at microwave wavelengths, i.e., $1 mm$ and above. Accordingly, although
the thermal noise may be negligible at telecom wavelengths, it becomes increasingly important as one spans the whole optical domain and it is a crucial parameter in the microwave domain. Hence the study of information transmission on TAs is of particular relevance for future quantum communication networks where hybrid interfaces will be employed, e.g., superconducting qubits connected by
microwave communication lines, see Refs.~\cite{Xiang2017,Kurizki2015}.}
In both the qubit and bosonic cases, the corresponding quantum capacity is known \cite{amplitudeDamping,qCapLossy} only for a zero-temperature environment, since in this limit both channels are degradable~\cite{deg1,antiDeg,deg2}.
However the more general finite-temperature case breaks the degradability property and it has not been successfully tackled so far, apart from establishing some upper and lower bounds \cite{holWer,plob,wildeDecDirettaBound}.

 In this article we introduce a general method to compute new bounds on the capacity of any quantum channel which is {\it weakly degradable} in the sense of~\cite{weakDeg,antiDeg}. It is based on the purification of the environment by an additional system, which is then included in the 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 an upper bound on the capacity of the original channel, which is tight in 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~\cite{deco2,weakDeg,gaussOpt}. Eventually we compare all our bounds with those already known in the literature and in particular with the recent results of~\cite{plob,wildeDecDirettaBound}. 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.   \\

\section*{Results}

\subsection*{Thermal Attenuator channels}\label{sec:thAttExt}
Let us consider a communication channel that connects two parties. The sender, Alice, wants to transmit the quantum state $\hat{\rho}_{\mathrm{A}}\in\mathfrak{S}(\mathcal{H}_{\mathrm{A}})$, represented by a positive density operator of unit trace on the Hilbert space of the system, $\mathcal{H}_{\mathrm{A}}$. At the other end of the channel Bob receives a transformed quantum state on the Hilbert space $\mathcal{H}_{\mathrm{B}}$. Any physical transformation applied to the state during the transmission can be represented by a quantum channel $\Phi$, i.e., a linear, completely-positive and trace-preserving map, which evolves the initial state as $\hat{\rho}_{\mathrm{B}}=\Phi\left(\hat{\rho}_{\mathrm{A}}\right)$. \\
The class of channels studied in this article is that of thermal attenuators, $\Phi_{\eta,\ensuremath{N}}$, originating from the following physical representation (see Fig.~\ref{fg1}). An energy-preserving interaction $\hat{U}^{(\eta)}_{\mathrm{AE}\rightarrow \mathrm{BF}}$, parametrized by a transmissivity parameter $\eta\in[0,1]$, couples the input system $\hat{\rho}_{\mathrm{A}}$ with an environment described by a thermal state $\hat{\tau}_{\mathrm{E}}\propto \exp(-\hat{H}_\mathrm{E})$, where $\hat{H}_\mathrm{E}$ is the bath hamiltonian, of dimension equal to that of the system, and the state has mean energy $N=\tr{}{\hat{H}_\mathrm{E}\hat{\tau}_\mathrm{E}} \geq0$ in dimensionless units. The total output state can be written as
\begin{equation}\label{phiDil}
\hat{\rho}_{\mathrm{BF}}=\hat{U}^{(\eta)}_{\mathrm{AE}\rightarrow \mathrm{BF}}\left(\hat{\rho}_{\mathrm{A}}\otimes\hat{\tau}_{\mathrm{E}}\right)\hat{U}_{\mathrm{AE}\rightarrow \mathrm{BF}}^{(\eta)\dag},
\end{equation}
while the action of the channel is obtained by tracing out the output environmental system F:
\begin{equation}\label{generalTA}
\Phi_{\eta,\ensuremath{N}}\left(\hat{\rho}_{\mathrm{A}}\right)=\tr{F}{\hat{\rho}_{\mathrm{BF}}}.
\end{equation}
This general framework is particularly relevant in the two paradigmatic cases in which the system is represented by a two-level 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.
\begin{figure}[t!]
\includegraphics[scale=.37,trim={5.5cm 2.5cm 7.5cm 2.5cm},clip]{fig1.pdf}
{\color{red}\caption{Unitary representation of a thermal attenuator channel, its extended channel and their complementaries.\\ The system A interacts with the environment E, which is in a thermal state of mean energy $N$, via a linear coupling, $\hat{U}^{(\eta)}_{\mathrm{AE}\rightarrow \mathrm{BF}}$. The parameter $\eta$ determines the fraction of energy dispersed from the system into the environment, while the total energy is preserved. The output of the channel $\Phi_{\eta,N}$ is recovered by discarding the output environment F, while that of its weak complementary $\bar{\Phi}_{\eta,N}$ 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 $\Phi_{\eta,N}^\mathrm{e}$ with output BE' and its strong complementary $\tilde{\Phi}_{\eta,N}^\mathrm{e}$ with output F. Note that the latter coincides with $\bar{\Phi}_{\eta,N}$. }\label{fg1}}
\end{figure}\\

In order to describe a two-level system, we fix as a basis $\ket{0}$ and $\ket{1}$ corresponding to the ground and excited states respectively. In this basis we can represent a generally mixed quantum state with a density matrix:
\begin{equation}\label{qubSt}
\hat{\rho}=\left[ \begin{array}{c c}
1-p & \gamma \\
\gamma^{*} & p \\
\end{array}\right],
\end{equation}
where $p\in[0,1]$ is the mean population of the excited state and $\gamma\in\mathbb{C}$ is a complex coherence term, with $\abs{\gamma}^{2}\leq p(1-p)$. The thermal attenuator channel $\Gamma_{\eta,\mathcal{N}}$ with $\eta \in [0,1]$ and $N\in[0,1/2]$, also known in the literature as generalized amplitude damping channel ~\cite{nChuangBOOK,amplitudeDamping}, acts on the matrix elements in the following way:
 \begin{align}
p  \xrightarrow{\Gamma_{\eta,\ensuremath{N}}} & \; p'=  \eta p + (1-\eta) \ensuremath{N},   \label{GAD1} \\
\gamma  \xrightarrow{\Gamma_{\eta,\ensuremath{N}}} &\; \gamma'=  \sqrt{\eta} \gamma,  \label{GAD2}
\end{align}
and admits a representation according to the general structure given in Eqs. \eqref{phiDil} and \eqref{generalTA}.
In this case the environment is given by a thermal two-level system:
\begin{equation}\label{thQub}
\hat{\tau}=\left[\begin{array}{c c}
1-\ensuremath{N} & 0 \\
0 & \ensuremath{N} \\
\end{array}\right],
\end{equation}
where $\ensuremath{N} \in [0,1/2]$ represents the dimensionless mean energy  and the bath hamiltonian is $\hat{H}_\mathrm{E}\propto\ketbra{1}$.  The unitary interaction is instead given by an energy-preserving rotation on the subspace $\{\ket{01},\ket{10}\}$ of the joint Hilbert space of the system:
\begin{equation}
\hat{U}^{(\eta)}=\left[\begin{array}{cccc}
1&0&0&0\\
0&\sqrt{\eta}&\sqrt{1-\eta}&0\\
0&-\sqrt{1-\eta}&\sqrt{\eta}&0\\
0&0&0&1
\end{array}\right],
\end{equation}
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 \eqref{GAD1} and \eqref{GAD2}.\\

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 operators $\hat a$ and $\hat a^\dag$, obeying the bosonic commutation relation $[\hat a, \hat a^\dag]=1$ or, equivalently, in terms of the quadratures $\hat q=(\hat a+\hat a^\dag)/\sqrt{2}$ and $\hat p=i (\hat a^\dag- \hat a)/\sqrt{2}$, such that $[\hat{q},\hat{p}]=i$. Given $n$ modes and introducing the vector of quadrature operators $\vecc{\hat{r}}=\left(\hat{q}_1, \hat{p}_1,..., \hat q_n , \hat p_n\right)^{\top}$,  one can define the characteristic function~\cite{RevGauss,serafiniBOOK} of a bosonic state $\hat{\rho}$:
\begin{equation}
\chi\left(\vecc{\xi}\right)=\tr{}{\hat{\rho}~e^{\vecc{\xi}^{T}\Omega^{T}\vecc{\hat{r}}}},
\end{equation}
where $\vecc{\xi}\in\mathbb{R}^{2n}$ and $\Omega=\left[\begin{array}{cc}0& 1 \\- 1&0 \end{array}\right]^{\oplus n}$ is the symplectic form.
The most common bosonic states are Gaussian states, i.e. those whose characteristic function is Gaussian:
\begin{equation}
\chi_{\mathrm{G}}\left(\vecc{\xi}\right)=\exp\left\{-\frac{1}{4}\vecc{\xi}^{T}\Omega^{T}V\Omega\;\vecc{\xi}+\vecc{\xi}^{T}\Omega^{T}\vecc{m} \right\},
\end{equation}
uniquely determined by the first and second moments of the state $\hat{\rho}_{\mathrm{G}}$, given respectively by the vector of mean values $\vecc{m}$ and by the symmetric covariance matrix $V$ of the quadrature operators.
 The single-mode thermal attenuator channel $\mathcal E_{\eta,\ensuremath{N}}$  has been extensively studied in the context of Gaussian quantum information~\cite{RevGauss,holevoBOOK,holGiovRev}
 and its action on the first and second moments is given by:
 \begin{align}
\vecc{m}  \xrightarrow{\mathcal{E}{\eta,\ensuremath{N} } }& \; \vecc{m}'=  \sqrt{\eta} \,\vecc{m},  \label{GTA1} \\
V  \xrightarrow{\mathcal{E}{\eta,\ensuremath{N} }} &\; V'=  \eta V + (1-\eta) (2 \ensuremath{N} + 1)\mathbf{1}_2, \label{GTA2}
\end{align}
where $\eta\in[0,1]$ and $N\ge0$.
As for the qubit case, also this infinite-dimensional channel can be represented using the general picture of Eqs. \eqref{phiDil} and \eqref{generalTA}. Indeed, introducing the single-mode Gaussian
thermal state $\hat{\tau}$ characterized by $\vecc{m}_{\hat{\tau}}=0$ and $V_{\hat{\tau}}=(2 \ensuremath{N}+1)\mathbf{1}$, the thermal attenuator channel can be generated by a passive unitary interaction $\hat{U}$ that acts on the bosonic operators of the system, $\hat{a}$, and of the environment, $\hat{a}_{\mathrm{E}}$, as a beam splitter:
 \begin{align}
U^\dag \hat a U &=  \sqrt{\eta} \hat{a} + \sqrt{1- \eta} \hat{a}_{\mathrm{E}}, \\
U^\dag \hat a_\mathrm{E} U &=  -\sqrt{1-\eta} \hat{a} + \sqrt{\eta}  \hat{a}_{\mathrm{E}}.
\end{align}
It is easy to check that, tracing out the environmental mode, one recovers the definition of the single-mode thermal attenuator given in Eqs. \eqref{GTA1} and \eqref{GTA2}.

\subsection*{Known bounds for the Quantum Capacity of Thermal Attenuators}
The quantum capacity $Q(\Phi)$ of a channel $\Phi$ 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 $N\rightarrow\infty$. It is well known~\cite{lloydQuant,shorQuantum,qCap1,qCap2} that this quantity can be expressed as:
\begin{equation}\label{qCap}
Q(\Phi)=\lim_{N\rightarrow\infty}\frac{1}{N}\max_{\hat{\rho}\in\mathfrak{S}\left({\mathcal{H}_{\mathrm{A}}}^{\otimes N}\right)}J\left(\hat{\rho},\Phi^{\otimes N}\right),
\end{equation}
where
\begin{equation}\label{cohInfo}
J\left(\hat{\rho},\Phi\right)= S(\Phi(\hat \rho))-S(\tilde \Phi(\hat\rho))
\end{equation}
is the coherent information~\cite{cohInfo,holGiovRev} and $S(\hat{\rho})= -{\rm tr}\{ \hat{\rho} \log_{\mathrm{2}}  \hat{\rho}\} $ is the Von Neumann entropy; note that all logarithms henceforth are understood as base-2. Finally, $\tilde \Phi$ is the complementary channel~\cite{deg1}, obtained from the Stinespring dilation of $\Phi$ by tracing out the system instead of the environment, as detailed in the next subsection.
Because of the peculiar super-additivity phenomenon \cite{DiVincenzo1998,Smith2007,zeroQCap,hastings2009,supAddQCap}, the so called single-letter capacity
\begin{align}\label{q1}
Q_{1}(\Phi)=\max_{\hat{\rho}\in\mathcal{H}_{\mathrm{A}}}J(\hat{\rho},\Phi)
\end{align}
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:
\begin{equation}\label{qubLowBound}
Q(\Gamma_{\eta,N}) \ge  Q_{1}(\Gamma_{\eta,N})=\max_{p} J \left ( \hat{\rho}=\left[ \begin{array}{c c}
1-p & 0 \\
0& p \\
\end{array}\right] ,\Gamma_{\eta,N} \right)
\end{equation}
where the maximization is only with respect to  $p\in [0,1]$, since one can check numerically that for this channel optimal states are diagonal; this quantity can be easily numerically computed for all values of $\eta$ 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 \cite{holWer}:
\begin{align}\label{GaussLowBound}
Q(\mathcal E_{\eta,N}) \ge  Q_{1}(\mathcal E_{\eta,N})  &\ge \max_{\hat \rho_\mathrm{G}} J (\hat \rho_\mathrm{G} ,\mathcal E_{\eta,N})  \nonumber \\
& =\max \left\{0, \log_{\mathrm{2}} \frac{\eta}{1-\eta}-g(\ensuremath{N}) \right\},
\end{align}
where $\rho_\mathrm{G}$ varies over the set Gaussian states and $g(\ensuremath{N})=(\ensuremath{N}+1)\log_{\mathrm{2}} (\ensuremath{N}+1)-\ensuremath{N}\log_{\mathrm{2}} \ensuremath{N}$ corresponds to the entropy of the thermal state of the environment.

Whether the right-hand-sides of \eqref{qubLowBound} and \eqref{GaussLowBound} 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 \cite{deg1,amplitudeDamping,qCapLossy}, that
for a zero-temperature environment this is the case, i.e. for $N=0$ all inequalities in \eqref{qubLowBound} and \eqref{GaussLowBound} 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 \cite{plob,wildeDecDirettaBound}, which can be combined to get the following expression:
{\color{red}\begin{eqnarray}\label{boundPLOB}
&&Q(\mathcal E_{\eta,N}) \le \min \left\{Q_{\mathrm{PLOB}}(\mathcal E_{\eta,N}), Q_{\mathrm{SWAT}}(\mathcal E_{\eta,N})\right\},\\
&&Q_{\mathrm{PLOB}}(\mathcal E_{\eta,N})=\max\{0,- \log_{\mathrm{2}} [(1-\eta)\eta^N]-g(\ensuremath{N})\},\nonumber\\
&&Q_{\mathrm{SWAT}}(\mathcal E_{\eta,N})=\max\{0,\log_{\mathrm{2}} [\eta/(1-\eta)]-\log_{\mathrm{2}} [N+1]\}.\nonumber
\end{eqnarray}
Let us note that $Q_{\mathrm{PLOB}}$ is actually a bound on the quantum capacity assisted by two-way classical communication~\cite{plob} and thus trivially bounds also the simpler unassisted capacity discussed in this article (strong-converse bounds for the two-way capacity where derived, e.g., in ~\cite{wilde2017,Christandl2017}). The other bound instead, $Q_{\mathrm{SWAT}}$, is itself a bound on the unassisted capacity and it has been shown~\cite{wildeDecDirettaBound} to beat other possible bounds based on $\epsilon$-degradability~\cite{Sutter2017}.}
In the next sections we are going to derive new upper bounds which are significantly closer to the lower limits \eqref{qubLowBound} and \eqref{GaussLowBound}, especially in the low temperature regime.

\subsection*{The extended channel}\label{sec:extCh}
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 $\Phi_{\eta,\ensuremath{N}}$,  valid both for qubit and bosonic systems, that we have previously defined in Eqs.~(\ref{phiDil}, \ref{generalTA}).
If, instead of tracing out the environment as done in Eq. \eqref{generalTA}, we trace out the system, we get what has been defined in \cite{weakDeg,antiDeg} as the weakly-complementary channel:
\begin{equation} \label{wComp}
\bar \Phi_{\eta,\ensuremath{N}}\left(\hat{\rho}_{\mathrm{A}}\right)=\tr{B}{\hat{\rho}_{\mathrm{BF}}},
\end{equation}
physically representing the flow of information from the system into the environment. Notice that this is different form the standard notion of complementary channel:
\begin{equation}
\tilde \Phi_{\eta,\ensuremath{N}}\left(\hat{\rho}_{\mathrm{A}}\right)=\tr{B}{\hat{\rho}_{\mathrm{BFE'}}},
\end{equation}
where
\begin{equation}\label{phiExtDil}
\hat{\rho}_{\mathrm{BFE'}}=\hat{U}^{(\eta)}_{\mathrm{AE}\rightarrow \mathrm{BF}}\left(\hat{\rho}_{\mathrm{A}}\otimes\ketbra{\tau}_{\mathrm{EE'}}\right)\hat{U}_{\mathrm{AE}\rightarrow \mathrm{BF}}^{(\eta) \dag}
\end{equation}
and $| \tau \rangle_{\mathrm{EE'}}$ is a purification of the environment, i.e.  $\hat{\tau}_{\mathrm{E}}=\tr{E'}{ \ketbra{\tau}_{\mathrm{EE'}} }$.
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 (zero-temperature limit).
Finally, we also remark that the two different types of complementarity induce different definitions of degradability. A generic channel $\Phi$ is degradable \cite{deg1,deg2},  if there exists another quantum channel $\Delta$ such that $\tilde \Phi= \Delta \circ  \Phi$. Similarly, a generic channel $\Phi$ is weakly degradable \cite{weakDeg,antiDeg}, if there exists another quantum channel $\Delta$ such that $\bar \Phi= \Delta \circ  \Phi$. 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
\begin{equation} \label{extended}
\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}\left(\hat{\rho}_{\mathrm{A}}\right)=\tr{F}{\hat{\rho}_{\mathrm{BFE'}}},
\end{equation}
where $\hat{\rho}_{\mathrm{BFE'}}$ is the global state defined in \eqref{phiExtDil}. In other words, $\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}$ 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.~\ref{fg1}. 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 $Q(\Phi_{\eta,\ensuremath{N}}) \le Q(\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}})$.\\
The advantage of dealing with the extended channel $\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}$ is that it is degradable whenever the original channel $\Phi_{\eta,\ensuremath{N}}$ is weakly degradable. This fact follows straightforwardly from the observation that the complementary channel of $\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}$ is the weakly-complementary channel of $\Phi_{\eta,\ensuremath{N}}$, i.e.:
\begin{equation}\label{property1}
\tilde{\Phi}_{\eta,\ensuremath{N}}^{e}\left(\hat{\rho}_A\right)= \tr{BE'}{\hat{\rho}_{\mathrm{BFE'}}}= \bar{\Phi}_{\eta,\ensuremath{N}}\left(\hat{\rho}_A\right).
\end{equation}
The degradability of $\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}$ significantly simplifies the evaluation of $Q(\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}})$ and provides a very useful upper bound for the quantum capacity of thermal attenuators:
\begin{equation}\label{generalBound}
Q(\Phi_{\eta,\ensuremath{N}}) \le Q(\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}})=Q_1(\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}),
\end{equation}
where in the last step we used the additivity property valid for all degradable channels \cite{deg1,deg2}.\\
Another useful property of the extended channel, which follows from Eq.\ \eqref{property1} and the definition of coherent information, \eqq{cohInfo}, is the following:
\begin{equation}\label{property2}
J\left(\rho, \Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}\right) = - J\left(\rho, \bar{\Phi}_{\eta,\ensuremath{N}}\right).
\end{equation}
relating the coherent information of the extended and of the weakly-complementary channels.\\
Below we compute more explicitly the previous bound \eqref{generalBound} 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 \eqq{thQub} is
\begin{equation}
\ket{\tau}=\sqrt{1-\ensuremath{N}}\ket{00}+ \sqrt{\ensuremath{N}}\ket{11}.
\end{equation}
The channel $\Gamma_{\eta,\ensuremath{N}}$ can be weakly degraded to $\bar \Gamma_{\eta,\ensuremath{N}}$ by the composite map $\Delta=\Psi_{\mu}\circ\Gamma_{\eta',\ensuremath{N}}$, where $\eta'=(1-\eta)/\eta$. Here $\Psi_{\mu}$ is a phase-damping channel~\cite{nChuangBOOK} of parameter $\mu=1-2\ensuremath{N}$, which acts on the generic qubit state of \eqq{qubSt} by damping the coherence matrix element as $\gamma\mapsto\mu\gamma$ while leaving the population $p$ constant. Hence the extended channel $\Gamma_{\eta,\ensuremath{N}}^{\mathrm{e}}$ defined as in \eqref{extended} is degradable and from \eqref{generalBound} we get
\begin{equation}\label{qubUpBound}
 Q(\Gamma_{\eta,N}) \le Q_1(\Gamma_{\eta,N}^\mathrm{e}) = \max_{p} J \left ( \hat{\rho}=\left[ \begin{array}{c c}
1-p & 0 \\
0& p \\
\end{array}\right] ,\Gamma^\mathrm{e}_{\eta,N} \right),
\end{equation}
where the optimization over the single parameter $p$ can be efficiently performed numerically for all values of $\eta$ and $N$, giving the result presented in Fig.~\ref{fig:plotQub}. 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 matrix-element flipping $\gamma \rightarrow - \gamma$ and from the concavity of the coherent information for degradable channels \cite{Yard2005}.\\
We note that, by construction, the gap between the lower  \eqref{qubLowBound}  and the upper \eqref{qubUpBound}  bounds closes in the limit $\ensuremath{N} \rightarrow 0$, where we recover the zero-temperature capacity of the amplitude damping channel consistently with \cite{amplitudeDamping}.
\cite{holWer}.
\begin{figure}[t!]
\includegraphics[scale=.55,trim={3cm 4cm 1.5cm 2cm},clip]{fig2.pdf}
{\color{red}\caption{Bounds of the quantum capacity of a qubit thermal attenuator. \\ Plot (log-linear scale) of the lower bound $Q_1(\Gamma_{\eta,N})$ (green line), \eqq{qubLowBound}, and upper bound $Q_1(\Gamma_{\eta,N}^\mathrm{e})$ (red line), \eqq{qubUpBound}, of the quantum capacity of the qubit thermal attenuator channel as a function of the attenuation parameter $\eta$ for two values of noise: (a) $N=0.01$, (b) $N=0.1$. The lower bound is given by the single-letter capacity of the channel, whereas the upper bound by the capacity of the extended channel, which equals its single-letter expression because of degradability. The two bounds are close for small noise and weak attenuation.}\label{fig:plotQub}}
\end{figure}\\

In the case of bosonic systems instead, the purification of the thermal environmental mode $\hat\tau$ with first moments $\vecc{m}_{\hat{\tau}}=0$ and covariance matrix $V_{\hat{\tau}}=(2 \ensuremath{N}+1)\mathbf{1}$ is a Gaussian two-mode squeezed state $\ket{\tau}$  ~\cite{RevGauss}, characterized by $\vecc{m}_{\ket{\tau}}=0$ and
\begin{equation}
V_{\ket{\tau}}=\left(\begin{array}{cc} V_{\hat\tau}&\sigma_{3} \sqrt{ V_{\hat\tau}^2-\mathbf{1}_2} \\ \sigma_{3}\sqrt{V_{\hat\tau}^2-\mathbf{1}_2}&V_{\hat\tau}\end{array}\right),
\end{equation}
where $\sigma_{3}=\operatorname{diag}(1,-1)$ is the third Pauli matrix.
The thermal attenuator $\mathcal E_{\eta,\ensuremath{N}}$ is weakly degradable \cite{weakDeg,antiDeg}, since its weakly complementary channel can be expressed as $\bar{\mathcal E}_{\eta,N}=\Delta\circ \mathcal E_{\eta,N}$ where $\Delta=\mathcal E_{\eta',\ensuremath{N}}$, with $\eta'=(1-\eta)/\eta$.
Hence $\mathcal E_{\eta,\ensuremath{N}}^\mathrm{e}$ is degradable and we can apply the general upper bound \eqref{generalBound}.  Moreover, as shown in~\cite{wolf2006,qCapLossy}, the quantum capacity of a degradable Gaussian channel is maximized by Gaussian states with fixed second moments, so that we can write
\begin{align}\label{upBos0}
Q\left(\mathcal E_{\eta,\ensuremath{N}}\right) \le Q_1\left(\mathcal E_{\eta,\ensuremath{N}}^\mathrm{e}\right)&= \max_{\hat{\rho}_\mathrm{G}} J (\hat{\rho}_\mathrm{G} ,\mathcal E_{\eta,N}^\mathrm{e})
\end{align}
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 state $\hat{\tau}_n$ with covariance matrix $V_{\hat{\tau}_n}=(1+2n)\mathbf{1}_2$, see Ref.~\cite{wildeThermOpt}. Therefore,
without energy constraint we have
\begin{align}\label{upBos0}
Q\left(\mathcal E_{\eta,\ensuremath{N}}\right) \le Q_1\left(\mathcal E_{\eta,\ensuremath{N}}^\mathrm{e}\right)&=\lim_{n\rightarrow \infty} J (\hat{\tau}_n ,\mathcal E_{\eta,N}^\mathrm{e}).
\end{align}
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.\ \eqref{property2}, we have
\begin{equation}
J(\hat{\tau}_n ,\mathcal E_{\eta,N}^\mathrm{e})=-J(\hat{\tau}_n ,\bar{\mathcal E}_{\eta,N})=-J(\hat{\tau}_n ,\mathcal E_{1-\eta,N}),
\end{equation}
but the last term is simply the negative of the coherent information of a thermal attenuator of transmissivity $\eta'=1-\eta$ and a thermal input state, a quantity which has been already computed in \cite{holWer}. In the limit of $n \rightarrow \infty$, we get our desired upper bound:
\begin{align}\label{upBos}
Q\left(\mathcal E_{\eta,\ensuremath{N}}\right) \le Q_1\left(\mathcal E_{\eta,\ensuremath{N}}^\mathrm{e}\right)&= \max \left\{0, \log_{\mathrm{2}} \frac{\eta}{1-\eta}+g(\ensuremath{N}) \right\},
\end{align}
where $g(\ensuremath{N})=(\ensuremath{N}+1)\log_{\mathrm{2}} (\ensuremath{N}+1)-\ensuremath{N}\log_{\mathrm{2}} \ensuremath{N}$, shown in Fig.~\ref{fig:plotBos}.
Comparing the upper bound \eqref{upBos} with the lower bound \eqref{GaussLowBound} we observe that we can determine the quantum capacity of the thermal attenuator up to an uncertainty of $2g(\ensuremath{N})$, which vanishes in the limit of small thermal noise. For the special case $\ensuremath{N}=0$, the gap closes and we recover the capacity
of the pure lossy channel consistently with the previous results of Ref.\ \cite{qCapLossy}.

\subsection*{Twisted decomposition of Gaussian attenuators}\label{sec:altBound}

In this section, through a completely different method, we derive a bound for the quantum capacity which is tighter than $Q\left(\Phi_{\eta,\ensuremath{N}}^{\mathrm{e}}\right)$ 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 $\mathcal{A}_{\kappa,\ensuremath{N}}$ \cite{RevGauss}, which combines the input state and the usual thermal state $\hat{\tau}$ of energy $\ensuremath{N}$ through a two-mode squeezing interaction with gain $\kappa>1$. Tracing out the environment, the first and second moments of the quantum state transform in the following way:
 \begin{align}\label{GAD}
\vecc{m}  \xrightarrow{\mathcal{A}_{\kappa,\ensuremath{N}}} & \; \vecc{m}'=  \sqrt{\kappa} \,\vecc{m},  \\
V  \xrightarrow{\mathcal{A}_{\kappa,\ensuremath{N}}} &\; V'=  \kappa V + (\kappa-1) (2\ensuremath{N} + 1)\mathbf{1}.
\end{align}
In the particular case in which the environment is at zero temperature, i.e. for $\ensuremath{N}=0$, the channel $\mathcal A_{\kappa,0}$ is called quantum-limited amplifier.
\begin{figure}[t!]
\includegraphics[scale=.48,trim={2cm 1.5cm 2cm 2cm},clip]{fig3.pdf}
{\color{red}\caption{Bounds of the quantum capacity of a bosonic thermal attenuator. \\ Plot (log-linear scale) of the lower bound $Q_{1}(\mathcal{E}_{\eta,N})$ (green line), \eqq{GaussLowBound}, and several upper bounds of the quantum capacity of the bosonic Gaussian thermal attenuator channel as a function of the attenuation parameter $\tau$ for two values of noise: (a) $N=0.1$, (b) $N=0.5$. The upper bounds are given by: the capacity of the extended channel $Q_{1}(\mathcal{E}_{\eta,N}^{\mathrm{e}})$ (red dashed line), \eqq{upBos}; the capacity of the attenuator channel of the twisted decomposition $Q_{\mathrm{twist}}$ (red line), \eqq{twistBound}; the upper bounds $Q_{\mathrm{PLOB}}$ of \eqq{boundPLOB} derived from Ref.~\cite{plob} (blue dashed line) and $Q_{\mathrm{SWAT}}$ of Ref.~\cite{wildeDecDirettaBound} (light-blue dashed line). Note that the best upper bound at small noise values is provided by our $Q_{\mathrm{twist}}$. As the noise increases, $Q_{\mathrm{PLOB}}$ starts beating the former for weak attenuation, while $Q_{\mathrm{SWAT}}$ remains always strictly larger than our bound.}\label{fig:plotBos}}
\end{figure}

It can be shown that all phase-insensitive Gaussian channels can be decomposed as a quantum-limited attenuator followed by a quantum-limited amplifier \cite{deco2,weakDeg}, with important implications for their classical capacity \cite{gaussOpt,gaussCap,gaussMaj}.
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.

\theoremstyle{remark}
\newtheorem{lemma}{Lemma}
\begin{lemma}
Every thermal attenuator $\mathcal E_{\eta,\ensuremath{N}}$ that is not entanglement-breaking can be decomposed as a quantum-limited amplifier followed by a quantum-limited attenuator:
\begin{equation}\label{twist}
\mathcal E_{\eta,\ensuremath{N}}=  \mathcal E_{\eta',0}  \circ \mathcal A_{\kappa',0} ,
\end{equation}
with attenuation and gain coefficients given by
\begin{equation}\label{coeff}
\eta'=\eta - \ensuremath{N} (1-\eta) , \quad \kappa'= \eta/\eta'
\end{equation}
\end{lemma}
The proof can be obtained by direct substitution and using the fact that non-entanglement-breaking attenuators are characterized by the condition $\ensuremath{N} < \eta/(1-\eta)$ \cite{holGiovRev,holevo2008} and so both coefficients in \eqref{coeff} 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. \cite{deco2} and employed in Ref.~\cite{wildeDecDirettaBound}, 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 $\ensuremath{N} < \eta/(1-\eta)$, we make use of the twisted decomposition \eqref{twist} obtaining
\begin{align} \label{bottle}
Q(\Phi_{\eta,\ensuremath{N}})= Q( \Phi_{\eta',0}  \circ \mathcal A_{\kappa',0} ) &\le Q( \Phi_{\eta',0}) \\
                                                                       &=\max \left\{0,\log_{\mathrm{2}}   \frac{\eta'}{1-\eta'}\right\},
\end{align}
where we used the ``bottleneck" inequality $Q(\Phi_1 \circ \Phi_2 )\le \min\{ Q(\Phi_1), Q(\Phi_2)  \}$ and the exact expression for the capacity of the quantum-limited attenuator \cite{qCapLossy}.
Substituting the value of $\eta'$ of Eq. \eqref{coeff} into \eqref{bottle}, we get our desired upper bound
\begin{align} \label{twistBound}
Q(\Phi_{\eta,\ensuremath{N}}) \le Q_{\mathrm{twist}}=\max \left\{0, \log_{\mathrm{2}}  \frac{\eta-\ensuremath{N} (1-\eta)}{ (1+\ensuremath{N}) (1-\eta)} \right\}.
\end{align}
One can easily check that this last bound is always better than the one derived in Eq.\ \eqref{upBos} and the bound $Q_{\mathrm{SWAT}}$ of Eq.~\eqref{boundPLOB}. Moreover, for sufficiently small $\eta$ or for sufficiently small $\ensuremath{N}$, it outperforms also the bound $Q_{\mathrm{PLOB}}$ of Eq.~\eqref{boundPLOB}, see Fig.~\ref{fig:plotBos}. {\color{red} By combining our result, $Q_{\mathrm{twist}}$, with $Q_{\mathrm{PLOB}}$ and with the lower bound $Q_{1}(\Phi_{\eta,\ensuremath{N}})$, the quantum capacity is now constrained within a very small uncertainty window.}
Figure~\ref{fig:contBos} shows the tiny gap existing between our new upper bound \eqref{twistBound} based on the twisted decomposition and the lower bound \eqref{GaussLowBound}.

\begin{figure}[t!]
\includegraphics[scale=1,trim={6.5cm 17.5cm 6.5cm 1.5cm},clip]{fig4Full.pdf}
{\color{red} \caption{Best-approximation accuracy of the quantum capacity of a bosonic thermal attenuator. \\ Contour plot of the difference between $Q_\mathrm{twist}$ of \eqq{twistBound}, i.e., the twisted-decomposition upper bound of the quantum capacity of the bosonic Gaussian thermal attenuator, and the lower bound $Q_{1}(\mathcal{E}_{\eta,N})$ of \eqq{GaussLowBound}, i.e., the single-letter capacity of the channel, as a function of the attenuation parameter $\eta\in[0.5,1]$ and noise values $N\in[0,5]$. 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 strong-attenuation region. Note that, as shown in Fig.~\ref{fig:plotBos}b, in the opposite regime of high values of $N$ and $\eta$, the quantum capacity is better upper bounded by \eqref{boundPLOB}.\label{fig:contBos}}
}
\end{figure}

\section*{Discussion}\label{sec:conc}
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 \cite{plob} 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 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 strong-converse 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.

\cleardoublepage
\section*{Methods}

\subsection*{Computation of the upper bound for Qubit TA}
The capacity of the extended qubit attenuator $\Gamma_{\eta,N}^\mathrm{e}$ can be computed by maximizing the coherent information of the channel, which is additive, as discussed in the main text:
\begin{align}
Q_{1}(\Gamma_{\eta,N}^\mathrm{e})&=\max_{\hat{\rho}}\left[S\left(\Gamma_{\eta,N}^\mathrm{e}\left(\hat{\rho}\right)\right)-S\left(\tilde{\Gamma}_{\eta,N}^\mathrm{e}\left(\hat{\rho}\right)\right)\right] \nonumber \\
&=\max_{p,\gamma}J_{\eta,N}(p,\gamma)
\end{align}
and we have defined for simplicity of notation
\begin{equation}
J_{\eta,N}(p,\gamma)=J \left ( \hat{\rho}=\left[ \begin{array}{c c}
1-p & \gamma \\
\gamma^*& p \\
\end{array}\right] ,\Gamma^\mathrm{e}_{\eta,N} \right),
\end{equation}
depending on the parameters $p,\gamma$ of the input qubit and on the channel parameters $\eta,N$.
Recall that the extended channel $\Gamma_{\eta,N}^\mathrm{e}$ maps states of the system A to states of the joint system BE' that includes the purifying part of the environment. Conversely, its complementary $\tilde{\Gamma}_{\eta,N}^\mathrm{e}$ 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 $\{\ket{0},\ket{1}\}^{\otimes 3}$:
\begin{widetext}
\begin{equation}\small
\hat{\rho}_\mathrm{AEE'}=\left(
\begin{array}{cccccccc}
 (1-N) (1-p) & 0 & 0 & \sqrt{(1-N) N} (1-p) & \gamma (1-N) & 0 & 0 & \sqrt{(1-N) N} \gamma  \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 \sqrt{(1-N) N} (1-p) & 0 & 0 & N(1-p) & \sqrt{(1-N) N} \gamma  & 0 & 0 & N \gamma  \\
 (1-N) \gamma ^* & 0 & 0 & \sqrt{(1-N) N} \gamma ^* & p-N p & 0 & 0 & \sqrt{(1-N) N} p \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
 \sqrt{(1-N) N} \gamma ^* & 0 & 0 & N \gamma ^* & \sqrt{(1-N) N} p & 0 & 0 & N p \\
\end{array}
\right).
\end{equation}
Next, we compute its evolved $\hat{\rho}_{\mathrm{BFE'}}$ under the action of $\mathcal{U}^{(\eta)}_{\mathrm{AE}\rightarrow \mathrm{BF}}\otimes\mathbf{I}_\mathrm{E'}$ and the marginals with respect to the bipartition BE'-F, which correspond to the output states of the two channels:
\begin{align}
&\small\Gamma_{\eta,N}^\mathrm{e}\left(\hat{\rho}\right)=\left(
\begin{array}{cccc}
 (1-N) (1-p \eta ) & \sqrt{(1-N) N (1-\eta ) \eta } \gamma ^* & \gamma  (1-N)\sqrt{\eta } & (2 p-1) \sqrt{(1-N) N (1-\eta )} \\
 \gamma  \sqrt{(1-N) N (1-\eta ) \eta } & N (1-p)\eta  & 0 & N \gamma  \sqrt{\eta } \\
 (1-N) \sqrt{\eta } \gamma ^* & 0 & p (1-N) \eta  & -\sqrt{(1-N) N (1-\eta ) \eta } \gamma ^* \\
 (2 p-1) \sqrt{(1-N) N (1-\eta )} & N \sqrt{\eta } \gamma ^* & -\gamma  \sqrt{(1-N) N (1-\eta ) \eta } & (p-1) \eta  N+N \\
\end{array}
\right),\nonumber\\
\\
&\small\tilde{\Gamma}_{\eta,N}^\mathrm{e}\left(\hat{\rho}\right)=\left(
\begin{array}{cc}
 -p (1-\eta )-N \eta +1 & (1-2 N) \gamma  \sqrt{1-\eta } \\
 (1-2 N) \sqrt{1-\eta } \gamma ^* & -\eta  p+p+N \eta  \\
\end{array}
\right).\nonumber
\end{align}
\end{widetext}
The entropies of these states can be computed numerically and it can be checked that they are invariant under phase-flip, i.e., $\gamma\rightarrow-\gamma$. Hence we obtain an expression of the coherent information that is an even function of $\gamma$ and write, following \cite{wildeBOOK}:
\begin{equation}
J_{\eta,N}(p,\gamma)=\frac{J_{\eta,N}(p,\gamma)+J_{\eta,N}(p,-\gamma)}{2}\leq J_{\eta,N}(p,0),
\end{equation}
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~\cite{Yard2005}. Hence we have restricted the optimization to diagonal states in the chosen basis, i.e., on the single parameter $p$ for fixed $\eta,N$:
\begin{equation}
Q_{1}(\Gamma_{\eta,N}^\mathrm{e})=\max_{p}J_{\eta,N}(p,0),
\end{equation}
for $\eta>1/2$ and $0$ otherwise. The latter expression can be easily solved numerically, as shown by the plots in the main text.

\subsection*{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 $\mathcal G_{\tau,y}$, defined by the following action on the first and second moments of single-mode Gaussian states \cite{RevGauss}:
 \begin{align}
\vecc{m}  \xrightarrow{\mathcal{G}{\tau,y } }& \; \vecc{m}'=  \sqrt{\tau} \,\vecc{m},  \label{G1} \\
V  \xrightarrow{\mathcal{G}{\tau,y }} &\; V'=  \tau V + y \mathbf{1}_2, \label{G2}
\end{align}
where $\tau\ge 0$ is a generalized transmissivity and $y\ge |1-\tau| $ is a noise parameter \cite{holGiovRev}. This family includes the thermal attenuator for $0\le\tau<1$, the thermal amplifier for $\tau>1$, and the additive Gaussian noise channel for $\tau=1$. If $y=|1-\tau|$, 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 \cite{holGiovRev,holevo2008} that a phase-insensitive Gaussian channel is entanglement-breaking if and only if $y\ge 1+\tau$,
which determines a noise threshold above which the channel has trivially zero quantum capacity.
Below the entanglement-breaking threshold, the following decomposition holds.

\begin{lemma}
Every phase-insensitive Gaussian channel $\mathcal G_{\tau,y}$ which is not entanglement-breaking ($y< 1+\tau$), can be decomposed as a quantum-limited amplifier followed by a quantum-limited attenuator:
\begin{equation}\label{twist2}
\mathcal G_{\tau,y}=  \mathcal G_{\eta',1-\eta'} \circ \mathcal G_{\kappa',\kappa'-1} =\mathcal E_{\eta',0}  \circ \mathcal A_{\kappa',0} ,
\end{equation}
with attenuation and gain coefficients given by
\begin{equation}\label{coeff2}
\eta'=(1+\tau - y)/2 , \quad \kappa'= \tau/\eta'.
\end{equation}
\end{lemma}
The proof follows by direct substitution of the parameters \eqref{coeff2} into \eqref{twist2} and from the application of Eqs.~(\ref{G1}, \ref{G2}).
Moreover, the hypothesis $y< 1+\tau$ is necessary since it ensures the positivity of both the attenuation and the gain parameters $\eta'$ and $\kappa '$.

\section*{Data availability}
No datasets were generated or analysed during the current study.

redpajama_set_name: RedPajamaArXiv


Theology on Tap is an opportunity for young adults to ask questions and learn about their faith. This month, Darby Young will be presenting on Mary, the Mother of God. Darby is the Assistant Director of Marriage & Family Life for the Diocese of Sioux City. A social will begin at 6:30 pm with the presentation starting at 7:00 pm at River Hops Brewing in Fort Dodge. Appetizers will be provided. Theology on Tap is for parishioners ages 19-39. Darby Young's presentation is made possible by DAA funds.


The People Operations team strives to create and deliver a seamless and positive employee experience across the entire employee life cycle. Leading with care and empathy, we develop efficient and simple people processes, systems, policies, and programs. We deliver data that is useful and accurate to create People-related strategies across the business. We manage people knowledge and build project management capability across our company. Our work also ensures Facebook remains protected and compliant. We strive to ensure employees feel heard and given the resources and support they need during the most important moments during their careers.
We are looking for a project manager who can help lead a number of process and continuous improvement related projects as well as holistically look at our compliance in the region. We need someone with expertise in this area who can thrive managing multiple projects and working with a global team to drive improvements. We are looking for someone who enjoys building and figuring out the best way to do things and to be able to communicate and work with the rest of the team to achieve this.
Support on the creation of quarterly business reviews that can accurately capture performance by using SLAs and KPIs.
Project manage complex cross functional projects for the region while partnering closely with our global counterparts.
Spread knowledge among the team relating to project management best practices and process improvement.


These example sentences are selected automatically from various online news sources to reflect current usage of the word 'spa.' Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. Send us feedback.
: a place where people go to improve their health and appearance by exercising, relaxing, etc.
What made you want to look up spa? Please tell us where you read or heard it (including the quote, if possible).


When it comes to first-person shooters, post-apocalyptic adventures are a dime a dozen. But when S.T.A.L.K.E.R. was first released in the spring of 2007, it dared to be different. How? By basing the game off of a real-world tragedy, the Chernobyl nuclear disaster, which occurred way back in 1986 near the city of Prypiat in the Ukraine. Despite the disaster happening so long ago, people are still unable to live in the surrounding area, and will be unable to for at least another 150 years.
In addition to the games real-world ties, S.T.A.L.K.E.R. happened to be one of the grittiest, realistic (aside from the problematic AI) and expansive games we've seen on the PC in a while. Having the ability to roam as you like is a huge benefit and really helped make the game feel real. Clear Sky further delivers on what made the original so great, but at the same time, adds support for DX10.
It might be difficult to judge from the screenshot, but Clear Sky (like the original) is one of the most demanding games on the PC today, especially if you wish to play using DX10. To help push all of our GPUs to their breaking-point, we stick to that mode while using the "High" quality setting.
It's all too common to hear someone say, "But, will it run Crysis?", but in all reality, it should be S.T.A.L.K.E.R. we're talking about. With DX10 lighting, the game is a true bog on any rig. It looks good, but when it's not that playable without a HD 4870 X2 and 1680×1050 resolution, all we can do is laugh.
Like all the other low-end cards in our roundup, this game is only playable at a maximum of 1920×1200 and with a render setting of Full Dynamic Lighting. It does manage to up the quality setting of the 9600 GT though, to High, from Medium.


text:
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    android:layout_width="match_parent"
    android:layout_height="?attr/actionBarSize"
    xmlns:tools="http://schemas.android.com/tools">

    <Button
        android:id="@+id/nav_btn"
        android:layout_width="30dp"
        android:layout_height="30dp"
        android:layout_marginLeft="10dp"
        android:layout_alignParentLeft="true"
        android:layout_centerVertical="true"
        android:background="@drawable/ic_home"/>

    <TextView
        android:id="@+id/title_city"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:layout_centerInParent="true"
        android:textColor="#fff"
        android:textSize="20sp"
        tools:text="城市标题"/>

    <TextView
        android:id="@+id/title_update_time"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
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        android:textSize="16sp"
        android:textColor="#fff"
        tools:text="标题更新时间"/>

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Wills - ElDeiry & ElDeiry, P.A.
A Will is a legal document that outlines how you would like your assets divided and who will inherit them after you pass away. That's the most basic definition, but your Will can do a lot more for you and your family.
When carefully prepared by a qualified, licensed attorney in your state of residence, a Will can address several other important issues.
1. A Will lets you appoint guardians for minor children. This is important because oftentimes, courts don't have the time or ability to select a guardian who is best suited to care for your minor children. We know it's a tough topic to consider, and odds are you will never need it, but planning ahead is always far better than the alternative.
2. A Will helps you make funeral arrangements. You get to plan the "who, what, when, where, and how" surrounding your funeral. The more details you provide, the fewer matters your loved ones will have to handle while they grieve.
4. Personal property is distributed under the terms of your Will. Oftentimes a letter attached to the Will can properly bequest personal items, jewelry, heirlooms, clothes, cars, or any personal property so that everyone in the family understands your wishes. Don't leave them guessing. Another tip: a nice note with the item will go a long way to creating a great family tradition.
5. It specifies who can act on your behalf. Upon the proper execution of a Power of Attorney – you immediately give someone else the right to act on your behalf. The rights can be as broad or limited as you like, but it legally gives someone else the authority to act as you have directed. Power of Attorneys can be tricky, and some third-parties are reluctant to honor them, but they are a critical piece of the estate planning process, and you must confer with an attorney to make sure these documents are properly executed to give yourself proper and adequate protection as needed.
Essentially, the primary benefits of a Power of Attorney are to indicate how you would like things to be handled, in the event you become disabled, unavailable or incapacitated. If you are available and capable of handling your affairs, the Power of Attorney does not need to be utilized by the person you have entrusted. Whether you are traveling away from the area, or undergoing a medical procedure, this is a great document to address and handle your personal and business affairs as needed.
6. It specifies what medical decisions you want made on your behalf. A Medical Power of Attorney is a legal document that gives someone else the authority to make decisions on your behalf. Often times, next of kin will communicate with the medical providers, but the more details you establish in writing the less confusion will be created. This helps ensure you get the best medical attention you desire, without conflicts or issues with the medical providers.
Ultimately, a Will is a critical document because it gives guidance on many factors to the loved ones you leave behind. For more specific guidance on how to create your Will and make it work with your overall estate plan, contact ElDeiry & ElDeiry, P.A. Our personable attorneys would be happy to walk you through the process.


text:
Q: What is the equivalent word for "compile" in an interpreted language? (I was encouraged to ask this question here.)
In C, we say:

GCC compiles foo.c.

For interpreters (such as Lua), what is the equivalent verb?

The Lua interpreter ____________ foo.lua.

When I write instructions for users of my Lua script, I often say:

Run the interpreter on foo.lua.

I think this can be said more succinctly:

Interpret (or Translate) foo.lua.

but that sounds awkward for some reason (perhaps because I'm unsure of its correctness). I can't really say compile because users may confuse it with the usage of the Lua compiler when I actually mean the Lua interpreter.

A: "Interprets" or "runs" are, I think, the two most natural verbs. Maybe "executes".

A: I have always thought along these lines:

A compiler compiles code.
An interpreter interprets code.

So I would say interprets.

A: In addition to the obvious interprets, you could say that the interpreter runs or executes some code.

A: It sounds awkward because compilers and interpreters are really different things.
You can say "run the compiler" to get the binary and maybe deploy the application but you cannot say "run the interpreter" to produce binary and deploy it. So you cannot compare them.
Interpret is the right word.

redpajama_set_name: RedPajamaStackExchange


text:
Ethics and Governance Framework
< Back to contents
1.1 Background and context
1.2 What is new or different about Our Future Health?
1.3 Boundaries between research and clinical care
Improvements in early detection and prevention of disease will enable better provision of care, improve outcomes and reduce costs to the health service. The Our Future Health (Our Future Health) programme aims to enrol up to five million people to a research cohort to help address this need. A major role will be to enable the recruitment of targeted sub-populations for trials of diagnostics and therapeutics. The success of the Our Future Health cohort depends on building and maintaining public trust and confidence. This will require the programme to demonstrate high ethical and governance standards across all its activities.
The starting point for an ethical framework for Our Future Health is the three key principles underpinning most research involving human participants: respect for autonomy, beneficence and justice.1 Building on these principles, Our Future Health can learn from previous cohorts where appropriate, but some aspects need new deliberation. These include the practicalities of recruiting such a large and diverse cohort, and the proposal to provide individuals with information about their risk categorisation. Our Future Health must make the most of the opportunity to become an exemplar, using new digital technologies in a way that sets the bar high for care and sensitivity.
Ethics and Feedback Advisory Group
Very early in its development, Our Future Health established an independent Ethics and Feedback Advisory Group (EFAG) to provide strategic advice on the development of ethical guidelines and principles for the Our Future Health cohort, and to develop an Ethics and Governance Framework to guide its operations (see Annex A for a list of members of EFAG). This Framework provides advice to the Our Future Health Board and Executive, and will be publicly available for funders, partners, researchers, participants and the general public. EFAG, an independent group which reports to the Our Future Health Board, will continue as part of the long-term governance of the cohort and will be responsible for monitoring the implementation of the Framework, and for reviewing and updating it as appropriate.
It will be important to ensure close interaction between EFAG and the emerging scientific strategy as the operating principles and discussions develop. Extensive public involvement and significant piloting will be crucial to help answer some of the questions that have been raised and to provide evidence to inform the Framework and the Our Future Health programme.
We envisage the Framework is a living document. It will initially need to be reviewed frequently, as experience accumulates from initial piloting and the impact of the COVID-19 pandemic on health services is clearer. After that, the Framework should be reviewed on a regular basis, at least every five years, to ensure that it remains relevant to current scientific and ethical standards.
1 The Declaration of Helsinki sets out the defining principles for research involving human participants. https://www.wma.net/policies-post/wma-declaration-of-helsinki-ethical-principles-for-medical-research- involving-human-subjects/
The Our Future Health cohort will learn from and build on best practice established by other cohorts, particularly UK Biobank. However, there are some aspects of Our Future Health which are new or different, and these need particular thought.
The size of Our Future Health: Recruiting a cohort of 5 million people has never been attempted before. While the scale does not necessarily raise new ethical issues, it will be important to consider the practicalities of recruiting such a large and diverse cohort. Our Future Health will be interacting with more than 10% of the adult population of the UK, and it will be particularly important to consider how to ensure appropriate support is available for participants who need it (discussed further in Section 3.1 and 4.4).
Digital platform: Our Future Health will make extensive use of a digital platform to keep in contact with participants and collect information. It will be important to make sure the platform is used transparently and responsibly, with careful and sensitive communications. Our Future Health must also work to ensure this approach does not exclude any groups, making alternative arrangements available if necessary.
The proposal to frequently use the cohort to recruit participants for further trials: Our Future Health will have two elements – Phase 1 will include the recruitment and ongoing follow-up of 5 million people; Phase 2 will involve selected sub-sets of participants being invited to take part in additional studies. It will be important to be very clear about the different consent required for each phase. The need for initial broad consent, with further detailed supplementary consent for Phase 2 studies is discussed further in Section 3.2.2.
Feedback: Our Future Health is still considering making certain health related information available to participants. The nature of some of the additional studies which can be envisaged will make it unavoidable to disclose some individual information of possible clinical relevance, to participants. Return of clinically relevant information could be of benefit to participants, if it is correct and leads to better health management or leaves them better informed about their health. But it can also be harmful, if it is confusing or misleading or leaves them with anxieties and concerns which are not properly managed. This means it will be particularly important for Our Future Health to take a careful and responsible approach to decisions about feedback, as discussed in Section 3.4.
Clinical care involving patients has the primary purpose of providing a direct benefit to the patient through diagnosis, prevention, care or treatment. People are also sometimes asked to participate in projects which are purely research, with the primary purpose to test a hypothesis or to generate new generalisable knowledge. These two activities are often seen as separate, with different governance processes and expectations. However, the boundaries can be blurred. For example, a clinical trial of a new treatment involves both research and care; a genome analysed for diagnostic purposes might also provide evidence for research on the association of other genomic variations with particular diseases. Where research is undertaken in a clinical context with an individual patient it can be easily explained, but when it is scaled to involve large numbers of people, it becomes important to be clear what is involved.
There is therefore a spectrum between care and research, and the scope of any related duty of care tracks that spectrum. The patient / clinician relationship creates a duty of care from the clinician to the patient which has been the subject of many years of legal precedent and covers many different facets of the interaction, from consultation to information, consent, treatment and follow up care. At the other end of the spectrum, the participant / researcher duty of care is still a legal duty of care although its nature and scope are less well defined by legal precedent and more influenced by context, expectations and normal practice of biomedical research.
The scope of the duty depends on, amongst other things:
the nature of the research project;
the basis on which participants are recruited (do they or should they expect any benefit from participating?); and
what the participants should be and are told about the project.
For example, a researcher owes the participant a duty of care which includes:
the obligation to be clear, transparent and fair about the process;
the obligation to make the process as safe as possible;
the obligation to make the process relevant for effective research;
the obligation to provide certain information back to participants. This is more nuanced and will depend on the nature of the information and how expectations have been set.
For example, UK Biobank was established as a research project. The information provided to participants at the outset made it clear that participants should not expect any personal benefit, including clinical feedback, from taking part. The emphasis was on creating a research resource with the objective of generating discovery about disease in a generalisable manner.
By contrast, the 100,000 Genomes Project was designed as a hybrid between research and clinical care. This is reflected in the information provided to participants, where there is an emphasis on the individual benefit that participants may receive as a result of taking part, including the potential of receiving an individual diagnosis, in addition to the research potential of the genome sequencing information. Participants were recruited as part of their NHS care, and any findings are provided by the clinical team, to ensure that appropriate clinical care and support is available.
Our Future Health is different again. It is intended primarily as a research resource for generalisable discoveries, but participants may be invited to take part in additional trials, to test out diagnostics, treatments or behavioural interventions. This means that participants might receive different clinical care, or have more interaction with the health service, as a result of their involvement. From a participant's perspective, Our Future Health is therefore further along the research-care spectrum than UK Biobank, but it is not as far along as 100,000 Genomes.
In order to ensure clarity in its relationship with participants, we recommend that Our Future Health should be approved and regulated as a research programme. Participants should not expect to receive individual clinical care as a result of taking part. However, Our Future Health should recognise that its relationship with some participants may go beyond that of pure research. Where a project includes both research and clinical care, the legal situation and the scope of the duty inevitably becomes more complicated.  Our Future Health should ensure this is appropriately considered and any clinical duty of care required, (for example if an individual needs clinical assessment, screening, preventive measures or treatment as a result of information discovered through participation), will need to be appropriately resourced and supported.
The implications are particularly relevant when considering consent (Section 3.2), feedback (Section 3.4) and ongoing support for participants (Section 4.4, Box 2).
2.0 Guiding Principles >

redpajama_set_name: RedPajamaCommonCrawl


High deworming efficiency with small dose of drug.
Suitable for medical feed mix preparation by any available method.
Essential for combined rehabilitation of fish-breeding unit.
1 g of Fenomix contains 80 mg of phenasalum as an active ingredient, as well as excipients. It is an antihelmintic drug.
Phenasalum (a component of the drug) has cestodocidic effect on all development phases of tapeworms parasitizing in fish intestine.
Mechanism of action includes phosphorylation damage in worm tissues, blocking of main energy processes, neuromuscular system paralysis causing worm fixation loss on intestinal mucosa and elimination from gastrointestinal tract.
Use to treat and prevent cyprinid bothriocephalosis and khawiasis.
Do not use in other types of fish. It is not allowed to use in half drained ponds.
Use with feed stuff. Single group administration in dose of 0.5 g/kg of fish body weight.
Mixture of feed stuff and the drug is prepared in feed stuff factory or in farm by adding Fenomix to feed stuff (10 kg per 1 ton of feed stuff).
Feed fish with obtained feeding without previous starvation diet according to approved feeding technology. Daily dose is 5% of total weight of fish to be dewormed (5 kg of mixture per 100 kg of fish body weight).
If water resistance of mash bolus is <30 minutes, deworming efficiency will be decreased.
Commercial fish can be caught and eaten not earlier than 16 days after the drug administration.
Adverse events and complications are not generally observed in case of administration as per package insert.
Store carefully in original package. Protect from moisture. Keep out of the reach of children and animals. Store separately from food and animal feeding at <25°С. Shelf life is 2 years from manufacturing date.


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Ebay's Weirdest Items
Indeed, E-bay is today's most sought-after shopping portal in the Internet. It has created in itself a worldwide phenomenon that people get inflicted with. The joy and the thrill of bidding online create a certain diagrammatic representation of life's drama.
Consequently, due to its underlying nature, the success of E-bay can be well attributed to its wide range of market reach. Hence, a lot of people are going in or out of the arena with a number of diverse products at hand.
Because of its diversity, people all over the world resort to e-bay to sell just about anything, from the basic necessities to computer techies, and even weird stuffs.
The most common case of weird auctions on E-bay was about virginity being on sale. There were about a dozen of auctions that put one's virginity on the limelight. There was even one auction that sold for $10,000,000 with 17 bidders at the most. Though, there was no clear implication if the report was true enough.
Next, there were cases of auctioned letters from a serial killer being sold at 1500 pounds each.
Then, there was one auction about a man who is selling his "fully functioning" kidneys. The reported bid amount was $5.7 million. There was no way to verify the truth, but reports said that the sale did not push through.
There is much other weird stuff like one night stand with someone's wife, ghost in a jar, ghost cane, and air guitar, which were all sold, just for the heck of getting a sale on E-bay.
But among all, the report about a 10-year old Virgin Mary grilled cheese sandwich that sold for $28,000 was the most sensationalized and the weirdest stuff ever sold on E-bay. The seller was an old woman from Miami, Florida.
According to MSNBC, (this means it's for real) E-bay had deliberately post back the auction last November 2004 after it inadvertently pulled the auction out from the list, thinking it was just a hoax.
The GoldenPalace.com, an online casino, had won the auction. Executives of the company said that $28,000 was nothing compared to the privilege of getting to own this 10-year old half-bitten sandwich bearing the face of Virgin Mary.
Consequently, the fact that the Virgin Mary grilled cheese sandwich was already 10 years old and still it did not bear any traces of mold. That made those who read and have seen the auction go "Huh?"
Previous PostPrevious Ebay's History And Beginnings
Next PostNext EBay Scams To Watch Out For Or You May Be Next


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PNE vs Liverpool: Carabao Cup TV details, confirmed rule change and quarter final draw info
Carabao Cup fourth round takes place this week
Preston North End head coach Frankie McAvoy
Get the best PNE updates sent directly to your inbox
Preston North End face Liverpool tonight in the Carabao Cup with a quarter final place up for grabs.
Jurgen Klopp's side thrashed Manchester United in the Premier League on Sunday and although the Reds are set to make a host of changes to their side they will be heavy favourites to progress.
PNE have beaten three lower league opponents to reach this stage, overcoming Oxford, Cheltenham and Morecambe, and now face the daunting prospect of taking on Liverpool.
Here's all you need to know about the contest.
READ MORE: Carabao Cup quarter final draw details
Is it on TV?
Indeed it is, the game was one of the games selected for live TV coverage by Sky Sports.
Kick off is 7.45pm and coverage will be on Sky Sports Main Event from 7pm although the best way to follow the action will be with us in our live blog!
Highlights will be available on Quest at 10.30pm on Wednesday night.
Is VAR in use?
It is not, it will be later in the competition but at this stage, as in previous rounds, there is no technology assistance for the officials.
What's the subs rule?
The five substitution rule has been re-implemented after it was introduced for the semi-final and final last season.
In the previous rounds this season, teams were only allowed to make three substitutions but after discussions between the EFL board and the Premier League, a U-turn has been announced.
A statement via the EFL website read: "Following a proposal from 14 of the 16 clubs remaining in this season's Carabao Cup, the EFL can confirm Clubs will be permitted to make five substitutions from the nine they nominate for the remainder of the 2021/22 competition.
"After consultation between the EFL Board, the Premier League and the two non-signatory Clubs, an amendment has been made to Competition Rule 10.1.
"The amendment comes into effect ahead of the next round of ties which will take place during the week commencing Monday 25 October 2021.
"The change was made after consideration of the proposal which highlighted concerns of a demanding fixture schedule over the coming months."
When's the last eight draw?
The quarter final draw is on Saturday live on Soccer AM, which is on Sky Sports from 10.30am-12noon on Sky Sports Main Event and Sky Sports Premier League.
A set time for the draw has not been confirmed but show hosts Jimmy Bullard and John Fendley will be pulling out the numbers.
The draw is also likely to be shown for free on the Sky Sports You Tube channel.
Preston North End hit with FA charge following fracas in derby defeat to Blackpool
PNE decisions, Liverpool rotation and the predicted line-ups for Carabao Cup tie


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The Toniebox Is A Screen-Free Audio Toy That Plays Stories, Songs, & So Much More
by Ellen Schmidt
Once in a while, a super cool invention comes along for kids, and I just really want it for myself. Today's example is the magical world of Toniebox, a screen-free audio entertainment system designed for children ages 3 and up. Developed by two dads looking for ways to keep their kids entertained without relying on screens, Tonies offers a digital listening experience that plays stories, songs and more while fostering imagination and independent play. And luckily, the Toniebox Starter Set ($99.99) is now available here in the United States.
The starter set includes a Toniebox, a soft-to-the-touch and shockproof audio cube designed specifically for small hands, and a Creative Tonie, a hand-painted figurine that magnetically (and magically!) attaches to your Toniebox, which then begins playing up to 90 minutes of your own unique content. Whether its favorite books recorded in your own voice on the Tonie app, parents singing their child's favorite songs, or preserving family memories for the future, Creative Tonies put a custom spin on your child's experience with their Toniebox. My favorite part: the recording option is particularly great for grandparents who might not live close enough for regular story time activities together.
Additional Tonies available include collectible favorites from the Disney playbook, like The Lion King, Cinderella, and The Little Mermaid, to name a few. So your child can switch from grandma reading a story to singing "Hakuna Matata" from The Lion King by simply taking their Creative Tonie off the top of the Toniebox and replacing him with Simba. Magic. (Bonus: these extra Tonies are priced nicely from $11.99 to $14.99.)
We only include products that have been independently selected by Romper's editorial team. However, we may receive a portion of sales if you purchase a product through a link in this article.
Toniebox Audio Player Starter Set
See on Target
Which brings me to my next point: I love that after initial set-up, it looks like parents aren't really needed to help operate this device for their child, unlike many supposedly child-friendly tech gadgets. It has a durable design and uncomplicated controls, so the idea is that kids can simply use it on their own by changing out the figurines on top and clicking through what they want to hear or do with the Toniebox.
And for those worried about privacy issues, the Toniebox also sounds like a pretty safe tech option for your child's entertainment. After initial set-up, your home WiFi will no longer be required for use. Unlike YouTube or Amazon's Alexa, the Toniebox and tonies figurines work on a closed ecosystem utilizing RFID technology, not Bluetooth. And because it's not connected to the internet beyond the very initial set-up, the Toniebox can be used by your child anywhere.
The Toniebox opens up a world of stories and songs to explore without staring at a screen, and I am so excited that it's now available in the U.S. If you want to see a Toniebox in action, check out the brand's introductory video on its YouTube page.


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Q: Java InetAddress failed I am trying to check if some host is reachable using the "isReachable" method.
line 113: oaiBaseURL = "http://www.cnn.com";//////////////////////////////////////
line 114: boolean res = InetAddress.getByName(oaiBaseURL).isReachable(10000);
line 115: System.out.println("------reachable:"+res);

and get the following error message (in eclipse):
java.net.UnknownHostException: http://www.cnn.com
    at java.net.Inet6AddressImpl.lookupAllHostAddr(Native Method)
    at java.net.InetAddress$1.lookupAllHostAddr(Unknown Source)
    at java.net.InetAddress.getAddressFromNameService(Unknown Source)
    at java.net.InetAddress.getAllByName0(Unknown Source)
    at java.net.InetAddress.getAllByName(Unknown Source)
    at java.net.InetAddress.getAllByName(Unknown Source)
    at java.net.InetAddress.getByName(Unknown Source)
    at com.irWizard.web.validator.WizardValidator.validateForm(WizardValidator.java:114)

Does anyone understand what might be the reason for this error?
Thanks in advance!

A: You need to remove the http:// prefix.
As far as I know the InetAddress.getByName() method takes a hostname not a URL.
You can change the code as follows:
   URL url = new URL("http://www.cnn.com");
   boolean res = InetAddress.getByName(url.getHost()).isReachable(10000);
   System.out.println("------reachable:"+res);

However keep in mind the mechanisms the method isReachable() uses to determine whether it is reachable or not. It uses mostly ICMP techniques, which a lot of websites or intermediate firewalls might block.

A: Quote from documentation:

The host name can either be a machine name, such as "java.sun.com", or a textual representation of its IP address. If a literal IP address is supplied, only the validity of the address format is checked.

You don't need to specify scheme. Just remove "http://" and it should work.

A: did you parsed the url as it should ? you need to get the host name only from URL
URL oaiBaseURL = new URL("http://www.cnn.com");
boolean res = InetAddress.getByName(oaiBaseURL.getHost()).isReachable(10000);
System.out.println("------reachable:"+res);

A: I tried this on my laptop by creating hotspot and it worked.
Sometimes there may be problem in giving the subnet input to the program at that time you may face this problem. Hope it will work for you.
subnet = subnet.trim();
int timeout = 1500;
for(int i=1;i<254;i++)
    {
    try
      {
        String host = subnet +"."+i;
        if (InetAddress.getByName(host).isReachable(timeout))
          {
            Check = Check+host+"\n";
            System.out.println(host);
          }

       }
    catch (UnknownHostException ex) {
        Logger.getLogger(WiFi.class.getName()).log(Level.SEVERE, null, ex);
    } catch (IOException ex) {
        Logger.getLogger(WiFi.class.getName()).log(Level.SEVERE, null, ex);

A: Adding the entry in /etc/hosts with relevant hostname & ip address resolved in my case.


The mission of Daystar University is to "develop managers, professionals, researchers and scholars to be effective Christian servant-leaders through the integration of Christian faith and holistic learning for the transformation of Church and society". This mission guided the development of science degree courses in Daystar and is in line with the United Nations Millennium Development Goals, which gives clear guidelines for the improvement of lives of the world's poorest.
The objective is to equip graduates with skills that would make them agents of change and enable them do their work in line with the Christ-centred training background. The School of Science, Engineering and Health of Daystar University has as its Motto " Science for service, a simple statement but one that reminds each of our science graduates that they are servants even though they be leaders at work or service points.
Kenya entered the 21st century with a blast in the field of technology. In a span of a few years, the type writer was replaced by the computer, the telephone booth by the handset, phone lines are no longer fixed, cameras and tape recorders for news collection were replaced by tablets, bank halls are smaller and empty as banking went mobile or online. Exchange of information is at real time reducing distances and need for face to face meeting. The most life changing is the money transfer platforms like MPESA that has become a global phenomenon.
To be part of this ever-changing field, Daystar offers a degree in Applied Computer Science. Daystar University graduates in Computer science are pacesetters in the markets and have gone on to influence IT teaching up to University level, industry and specifically in the development of IT hubs and innovation centres. The university ensures that the curriculum keeps abreast with the changes in the field through continuous review of the curriculum.
Daystar University offer undergraduate degrees in the fields of Health: Daystar's degree in Nursing trains both at generic and upgrading levels. Trained under the motto " Christ-Centred nursing service", the nurses do not just gain the skills but also the heart of serving, a character confirmed by the institutions where they work or go for clinical practicums. It is sobering to note that most of the Daystar Nursing graduates are absorbed by the Ministry of Health immediately after graduation.
Daystar also trains graduates in Actuarial Science. With the growth of technology and ensuring enlargement of businesses space, many Kenyans have taken to investing in existing businesses or starting their own business. The risks of doing business has increased even as knowledge has grown. This creates a need for people with skill to access risks involved in the business as well as explain, educate and inform business community.
Research and Innovation is rapidly becoming the driver of development as people look for solutions to problems such as employment and emerging disease. Daystar University has a programme in Biomedical sciences, a research-based degree. The graduates of the biomedical program carryout innovative research, and look for solutions in medical field.
Thus, Daystar University has ably continued to contribute to the national development through providing skilled labor in the field of sciences, to the ever growing job market.


A future classic. The MALA bench offers a winning combination of clear functionality and curved lines. Minimalist in form, it is compatible with any interior style and comes in a range of coloured matt finishes. Standing on a solid wood frame, MALA is ideal as a handy place to put things, or as a seat. It can be used in virtually anywhere, from entrance areas to all kinds of other spaces. A multi-tasker with a clear team focus, MALA is the perfect complement to the slim BUREAU desk of the same design. Sleek and absolutely timeless.
The Earnest Studio explores themes of materiality and modularity and is the brainchild of the American-born designer Rachel Griffin. Griffin studied at Washington University in St Louis and the Design Academy in Eindhoven. She then went on to establish her own studio in Rotterdam and is also a lecturer at the Willem de Kooning Academy. Her creative spectrum is broad but consistently centres on her distinctive handling of materials. For Schönbuch, the Earnest Studio has created the BUREAU writing desk and MALA bench, both of which are elegant and purist in their designs.


We are pleased to offer a double room to rent in this beautifully furnished property opposite Moat Park in Withington. The house is decorated to a high standard throughout, has modern furnishings and is within walking distance of lots of takeaways, retails shops, off licences, Greggs, post office and The Victoria Pub. There are also nearby bus links for Christies, West Didsbury, Oxford Road hospitals, Manchester University and Piccadilly Plaza.
Ground Floor Front room (£435) is individually decorated to a high standard and comes complete with king size bed, wardrobe, chest of drawers, bedside cabinet and chair. The rent includes council tax, water, electric, gas, wifi, a TV license in the shared areas and a weekly cleaner in the shared areas.
The shared areas include a living room with comfy leather style sofas, coffee table and book shelves and is the perfect place to relax after a busy day. There is a large fully fitted basement kitchen with dishwasher, washing machine,oven/hob, fridge, freezers plenty of cupboard space and a white dining table with 4 chairs. There is also a small secondary kitchen, two shower rooms and a private garden. On street parking is available.
Current housemates are 3 males and 1 female aged mid 20's to mid 30's. Interests include surfing, reading, gaming, tv series, movies, travel, yoga, socialising and boxing.


Incantation. This demo is actually sold-out.
Thanks to the Cryptic Soul Productions, Enthralment realised the 7″ "Accursed In Divinity". This contained 2 trx. A true masterpiece of brutality. But the sound contained in the new trx were defenitely more brutal oriented and strange. As the musical direction was no longer the same as the beginning, it was useless to keep the old name. The one choosen for this new form of madness was "Corpsegod". With a different name but same line-up, they recorded in '95 the mini cd "Opinion About Sorrow" for Torture Records. The mini cd contained 5 trx of original and strange brutal death-metal. Ultrabrutal riffing with techno and industrial parts, splatter movies and a sound to the limit of human endurance make "Opinion About Sorrow" a really vanguard record compared to the others on the death-metal scene.
In the meantime the passion for the ultrabrutal death-metal in the band is getting' higer and higer. The old line-up dies to create a new band totally devoted to the brutal death-metal without compromises. A new band was born then, Corpsefucking Art. From the old line-up Andrea and Max are still in. After recording a 2 trx promo with the new guys (Massimo – guitar, Marcello – drums, Edo – bass) and the trx "Molesting The Corpse" for the underground splatter movie "Sick-o-pathic" soundtrack, and have played as support for Cannibal Corpse and Immolation for a show in Rome, Corpsefucking Art (aka C.F.A.) record their debut cd named "A Puzzle Of Flesh " for the italian label Sterminio Productions. Nine trx of blasting ultrabrutal death-metal without mercy. An inhuman voice and a blasting riffing make "A Puzzle Of Flesh" a cd that every U.S. brutal death-metal maniac must get.
In 2001 the line-up changes again. Andrea stills in and works with Marco – bass, Stuart – drums. They recorded a new split CD with Goretrade: Zombiecronomicon for the Coagulated Records. In april 2002 United Guttural Records took the license by Coagulated records and restamped the split cd with Goretrade, released with a completely new artwork. It was also released a tape version of the split by the Dark Angel Records from Thailandia.
After being offered a contract by Forever Underground Corpsefucking Art entered the studio to record their new full length "Splatter Deluxe".
After this they realized another studio album for Goregiastic Records titled "War of the toilet gear" and two album for Savared Records titled "Zombiefuck" and "Quel Cimitero Accanto alla Villa".
In July 2014 they had their first Tour in U.S.A. around many States and playing in alot of festivals, including the big Las Vegas Deathfest; 3 weeks coast to coast to promote the last album "Quel Cimitero Accanto Alla Villa".
Actually the band is focused on the songwriting of the new album, and for the second half of 2016 is planned the release of the official DVD "BEVERLY HILLS CORPSE".


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You are here: Home / Sports Memorabilia News / Babe Ruth Cap Famously Worn By David Wells For Sale Again
Babe Ruth Cap Famously Worn By David Wells For Sale Again
July 9, 2018 By Bob D'Angelo
David Wells was a big fan of Babe Ruth. So big, when he joined the New York Yankees and realized he could not wear the Bambino's No. 3 uniform, the left-handed pitcher asked for No. 33.
Never mind that Wells grew up around the Hell's Angels, loved metal bands like Metallica and AC/DC and sported several tattoos. Underneath that uniform beat the heart of a baseball fanatic. He threw a perfect game on May 17, 1998, but the title of his 2004 autobiography sums up his life perfectly: "Perfect I'm Not."
A year before his gem, Wells pulled off one his more memorable stunts. On June 28, 1997, at Yankee Stadium, Wells started against the Cleveland Indians and sported a baseball cap worn by Ruth. He wound up getting fined by manager Joe Torre, who made Wells take the cap off.
"It turned out," James Buckley Jr. wrote in 2002, "that Wells was as much of a baseball history geek as you or I."
"I didn't know he had it on until he was out in the middle of the diamond," Torre explained to writers after the game. "It wasn't the standard uniform."
Wells paid that $2,500 fine during the next home stand—in the form of 2,500 one dollar bills.
Wells had originally purchased the cap for $35,000 and sold it through SCP Auctions five years later for $537,278.40 in 2012. Now, it's been consigned once more as part of the company's Spring 2018 sale. This time, auction officials estimate it could bring $600,000-$800,000.
In 1997, David Wells wore a cap used by Babe Ruth during the 1930s
The cap dates to the 1930-1934 period, when Ruth was winding down his career with the Yankees. The navy-blue flannel cap features the now familiar interlocking "NY" logo embroidered on the front. Spalding's manufacturer's stamp appears in the cap's interior leather band. In script embroidered lettering is the cap size — 7 3/8 — followed by the player designation, "G. Ruth."
SCP's description notes that the cap has "substantial usage wear," including perspiration-induced soiling, but there is no deterioration and the cap remains "structurally sound." If one accepts the idea that Ruth probably wore the cap between 1930 and 1934, then that cap saw the Sultan of Swat hit 102 home runs, drive in 630 runs and bat .336.
Bidding will open July 25 and conclude August 11 at SCPAuctions.com.
The cap includes a letter of authenticity and a letter of provenance from Wells. It's a slice of history and a unique piece of memorabilia; after all, it's rare that a baseball cap sees action in a baseball game six decades apart.
Treasures from Career of Roger Peckinpaugh Headline RR Auction Catalog Large Yankees Card, Memorabilia Collection Headlines 1,000+ Lot Clean Sweep Auction V61 Neilson's Chocolate Cards Brought E120 Issue to Canada Babe Ruth's Will Coming to Auction Collector Loans Ruth Cap to Complete Little League Museum Display Newly Discovered Gehrig Bat Sells for $436,970 1924 Yankees Team Photo Shows Young Gehrig Babe Ruth's 1923 World Series Watch Coming to Auction
About Bob D'Angelo
Bob has been a sportswriter and copy editor for more than 35 years and a blogger for a decade. He is celebrating his 50th year of card collecting, and still counts his 1965 Topps Mickey Mantle as his favorite. You can reach him at [email protected].
Filed Under: Sports Memorabilia News Tagged With: babe ruth, Babe Ruth cap, baseball, game worn cap, new york yankees


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Perrie from Little Mix tells us her favourite One Direction songs!
This week Little Mix released their album Salute in the US, so the girls decided to do a Q&A for their fans to ask questions about it.
The whole ship went a bit off course though with Perrie at the helm, and ended up telling everyone what her favourite One Direction song from the boys' new album Midnight Memories!
Perrie said her favourite was: "You and I, Zayns adlibs are outta this world!". One clever questioner then asked if Zayn and Perrie adlib around the house and apparently Perrie's too shy to! How about that?
What would you have asked Perrie? Let us know in the comment box below.
Remember, we're always playing loads of Little Mix on Fun Kids. Tune in to catch it.
I love little mix and 1D they were ace in concert cannot wait to see little mix in shefield!


text:
Lighthouse at Grand Marais, Minnesota
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Date Note
Date based on the earliest and latest dates of the Lindquist collection.
4 1/4 x 5 1/4 in.
Annotation on Front
Lighthouse at Grand Marais, Minn.
Annotation on Back
Light House at Garnd Marais, Minn.
Grand Marais (Minn.)
States & Provinces
Lighthouses; Landscapes; Winter
Folder and Group Title
Minnesota Indians; Grand Marais, Minnesota
Box and Folder Number
43: 732
Chicago:
Lighthouse at Grand Marais, Minnesota, MRL 10: G.E.E. Lindquist Papers, 43, 460, The Burke Library Archives (Columbia University Libraries) at Union Theological Seminary, New York. Can be viewed at http://lindquist.cul.columbia.edu:443/catalog/burke_lindq_043_0460. Web accessed 18 Jul 2019.


A few pieces of the bank debt of Horizon Natural Resources, formerly known as AEI Resources, was said to have traded in the mid- to high 50s last week after the company revised its EBITDA guidance downward by $60-70 million for the remainder of 2002. The revision puts future covenant compliance into question, and the company therefore has admittedly begun to speak with its banks to resolve the issue. Horizon has been hurt by operating difficulties at its Evergreen mine in West Virginia and by the unavailability of Section 404 permits in certain mines in Kentucky. Michael Nemser, cfo, could not be reached by press time.


Home / Sport / Every Super Bowl Winner & MVP From Past 52 Years!
I really felt this total should have been at least 60 with the game being played indoors at Mercedes-Benz Stadium in Atlanta, and I'm actually loving that early wagering has pushed it lower.
But both teams made the Super Bowl largely because they played excellent defense during the conference championship round two weeks ago. This year, they've largely coasted through the NFC under the guidance of wunderkind coach Sean McVay, who's turned quarterback Jared Goff and running back Todd Gurley into MVP candidates and helped L.A. resurrect a long-mediocre franchise.
At +2.5, the Rams have the third-shortest odds of any underdog going into the Super Bowl since the turn of the Millennium. The Patriots and Los Angeles Rams were playing Sunday in Atlanta.
Seattle thrashed the two-point favorite Denver Broncos 43-8 in one of the biggest upsets in the history of the game to clinch Super Bowl XLVIII. Let us know in the poll below.
Super Bowl LIII will be televised by CBS and streamed live right here. However, I can't get onboard with the Rams this Sunday.