text:
\section{Introduction}
In quantum computation and communication, quantum error correction (QEC) is necessary for preserving coherent states from noise and other unexpected interactions. Based on classic schemes using redundancy, Shor~\cite{1} has championed a strategy where a bit of quantum information is stored in an entanglement state of nine qubits. This scheme permits one to correct any error incurred by any of the nine qubits. For the same purpose, Steane~\cite{2} has proposed a protocol that uses seven quits. The five-qubit code was discovered by Bennett, DiVincenzo, Smolin and Wootters~\cite{3}, and independently by Laflamme, Miquel, Paz and Zurek~\cite{4}. The QEC conditions were proved independently by Bennett and co-authors~\cite{3}, and by Knill and Laflamme ~\cite{5}. The protocols above with different quantum error correction codes (QECCs) can be viewed as active error correction. There are passive error avoiding techniques such as the decoherence-free subspaces~{\cite{6,7,8} and noiseless subsystem~\cite{9,10,11}. Recently, it has been proved that all the active and passive QEC methods can be unified~\cite{12,13,14}.

With the known codes constructed in Refs.~\cite {2,3,4}, the standard QEC procedure is designed according to the principle of perfect correction for arbitrary single-qubit errors, where one postulates that single-qubit errors are the dominant terms in the noise process~\cite{15}. Recently, an optimization-based approach to QEC was explored. In each case, rather than correcting for arbitrary single-qubit errors, the error recovery scheme was adapted to model for the noise, with the goal to maximize the fidelity of the operation~\cite{16,17,18,19}. Robust channel-adapted QEC protocols, where the uncertainty of the noise channel is considered, have also been developed~\cite{20,21,22}.

For some tasks such as storing or transferring a qubit of information, a near-idealized channel is usually required. If the fidelity obtained from error correction is not high enough, the further increase in levels of concatenation is necessary. In the previous works~\cite{Poulin2,Rahn,Kersting}, the application of QEC with the concatenated code was discussed for the Pauli channel which includes the depolarizing channel as the most important example. In general, a QEC  protocol contains three steps: encoding, error evolution, and decoding. When the concatenated code is used for encoding, there are two known methods for decoding: the widespread blockwise hard decoding technique and the optimal decoding using a message-passing algorithm~\cite{Poulin2}. As shown by Poulin, the Monte Carlo results using  the five-qubit and Steane's code on depolarizing channel reveal significant advantages of message-passing algorithms. For the depolarizing channel, the concatenated five-qubit code is also more efficient than the concatenated seven-qubit code.

In the present work, we shall focus on the following questions:  For an arbitrary noise model, instead of finding the optimal QEC protocol adapted to it, is it possible for us to construct a near-perfect channel with a error below $10^{-5}$ by performing a QEC procedure with the concatenated five-qubit code? In order to answer this question, two important methods developed in the previous works are applied here. At first, following the idea in Ref.~\cite{3}, we shall show that the blockwise decoding can be carried out in a way without performing the error syndrome, and the realization needs some additional quantum resources, required in the message-passing algorithms. The other method comes from the recent works where several  general schemes have been developed for describing the exact performance of the QEC procedure based on a $N^l$-qubit concatenated code~\cite{Rahn,Kersting}. The main ideas in these schemes can be summarized in the following: First, the exact performance of the QEC with a $N^l$-qubit code is denoted by an effective Choi matrix; Secondly, the Choi matrix in the $l$-th level of concatenation is obtained by simulating the standard quantum process tomography (SQPT)~\cite{15,23,24,25,26,27}. Based on these results, the Choi matrix in the $(l+1)$-th level can be obtained in a similar way. The advantage of introducing the effective Choi matrix is clear: Instead of directly working in the $2^{N^{l}}$-dimensional Hilbert space, one could always simulate the error correction in each level of concatenation in a $2^N$-dimensional system.

For a general noise model, with five free parameters, our numerical simulation indicates that the concatenated five-qubit code is general: The QEC protocol with the concatenated five-qubit code, which is able to construct a near-perfect effective channel from the noise depolarizing channels, is also sufficient to complete the same task for other types of noise channels with the same channel fidelity.

The content of present work is organized as follows. In Sec.~\ref{sec2}, we construct an error correction protocol with the five-qubit code~\cite{3}, and a chosen unitary transformation is shown to be sufficient for correcting the errors of the principle system. In Sec.~\ref{sec3}, an explicit scheme is designed to obtain the effective Choi matrix. For three types of channels, the depolarizing channel, the amplitude damping channel and the bit-flip channel, the effective Choi matrices obtained by performing QEC with the concatenated five-qubit code is shown in Sec.~\ref{sec4}. In Sec.~\ref{section5}, it is shown that the concatenated seven-qubit is not general. In Sec.~\ref{sec6}, a noise model containing five free parameters is constructed, and we argue that the concatenated five-qubit code is general. Finally, we end our work with a short discussion.

\section{unitary realization of quantum error correction}
\label{sec2}
The $N$-qubit code concatenated with itself $L$ times yields a $N^{L}$-qubit code, providing a better error resistance with increasing $L$. The direct simulation of the quantum dynamics, coding and encoding procedure require massive computation resources. By following the idea in Refs.~\cite{Rahn,Kersting}, the concept that the exact performance of QEC can be described by an effective channel, will make the calculation simplified.

Let us consider the QEC protocol with code in Ref.~\cite{3},
\begin{eqnarray}
\vert 0_\mathcal{L}\rangle =&&\frac{1}{4}[\vert 00000\rangle+\vert 10010\rangle+\vert 01001\rangle+\vert 10100\rangle\nonumber\\
&&+\vert 01010\rangle-\vert 11011\rangle-\vert 00110\rangle-\vert 11000\rangle\nonumber\\
&&-\vert 11101\rangle-\vert 00011\rangle-\vert 11110\rangle-\vert 01111\rangle\nonumber\\
&&-\vert 10001\rangle-\vert 01100\rangle-\vert 10111\rangle+\vert 00101\rangle],\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\vert 1_\mathcal{L}\rangle=&&\frac{1}{4}[\vert 11111\rangle+\vert 01101\rangle+\vert 10110\rangle+\vert 01011\rangle\nonumber\\
&&+\vert 10101\rangle-\vert 00100\rangle-\vert 11001\rangle-\vert 00111\rangle\nonumber\\
&&-\vert 00010\rangle-\vert 11100\rangle-\vert 00001\rangle-\vert 10000\rangle\nonumber\\
&&-\vert 01110\rangle-\vert10011\rangle-\vert 01000\rangle+\vert 11010\rangle].\nonumber
\end{eqnarray}

We use $H_{\mathcal{S}}$  for the two-dimensional principle system, where the basis vectors are denoted by $\vert 0\rangle$ and $\vert 1\rangle$, while the ancilla system system lies in a $2^4$-dimensional Hilbert space $H_{\mathcal{A}}$ with the basis $\{\vert a_m\rangle\}_{m=0,...,15}$.
The standard way to get the effective noise channel is depicted in Fig.~\ref{fig1}(a). It contains the following steps.

(i) The encoding procedure can be realized with a unitary transformation $U$,
\begin{equation}
U\vert a_0\rangle\otimes \vert 0\rangle\rightarrow\vert 0_\mathcal{L}\rangle,
U\vert a_0\rangle\otimes \vert 1\rangle\rightarrow\vert 1_\mathcal{L}\rangle.
\end{equation}

(ii) The noise evolution is denoted by $\Lambda$.  The five-qubit code above is designed to correct the set of single-qubit errors, $\{E_m\}_{m=0,1,...,15}$. Usually, $E_0$ is fixed to be identity operator $\hat{\texttt{I}}$, and each $E_m(m\neq 0)$ is one of the Pauli operators $\hat{\sigma}^i_j (i=1,...,5, j=x,y,z)$. With the logical codes, one could introduce a set of normalized states
\begin{equation}
\label{norsta}
\vert m,+\rangle=E_m\vert 0_\mathcal{L}\rangle, \vert m,-\rangle=E_m\vert
1_\mathcal{L}\rangle.
 \end{equation}
Since the set of errors, $\{E_m\}_{m=0,...,15}$, could be perfectly corrected, there should be $\hat{P}_{\mathcal{C}}E^{\dagger}_mE_n\hat{P}_{\mathcal{C}}=\delta_{mn}\hat{P}_{\mathcal{C}}$,
where $\hat{P}_{\mathcal{C}}=\vert 0_\mathcal{L}\rangle\langle 0_\mathcal{L}\vert+ \vert
1_\mathcal{L}\rangle\langle
 1_\mathcal{L}\vert$. Therefore, one may easily verify that $\{\vert m, \pm\rangle\}_{m=0}^{15}$ form an orthogonal basis.

 (iii) With the denotation $\vert a_m , i\rangle=\vert a_m\rangle\otimes \vert i\rangle $,  the recovery operation can be described by a process $\mathcal{R}$ such that $\mathcal{R}(\rho^{\mathcal{SA}})=
\sum_{m=0}^{15}R_{m}\rho^{\mathcal{SA}}R_{m}^{\dagger}$, where the Kraus operators $R_m$ are~\cite{Rahn},
\begin{equation}
 R_m=\vert a_m, 0\rangle \langle m, +\vert+\vert a_m, 1\rangle\langle m, -\vert.
 \end{equation}

(iv) The decoding is realized by $U^{\dagger}$, the Hermite conjugate of $U$, $UU^{\dagger}=\mathrm{I}^{\otimes 5}$, and the effective channel $\bar{\varepsilon}$ is
\begin{equation}
\tilde{\varepsilon}(\rho^{\mathcal{S}})=\mathrm{Tr}_{\mathcal{A}}[\mathcal{U}^{\dagger}\circ \mathcal{R}\circ\Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \rho^{\mathcal{S}})].
\end{equation}

\begin{figure}
\centering
\includegraphics[scale=0.37]{fig1.eps}
\caption{(a) The way of getting the effective channel from the standard QEC protocol including encoding, noise evolution, recovery and decoding. (b) The two processes, $\mathcal{R}\circ \mathcal{U}^{\dagger}$ and $\mathcal{U}^{\dagger}\circ \mathcal{\tilde{R}}$, are equivalent.
(c) When  $U^{\dagger}$ for decoding is fixed  according Eq. (6),  the process   $\mathcal{ \tilde{R}}$ can be decomposed as $\mathcal{\tilde{R}}_{\mathcal{SA}}=\mathcal{\tilde{R}}_{\mathcal{A}}\otimes \mathbb{I}_{\mathcal{S}}$. Certainly, it can be moved away.  (d) Our  protocol where the chosen unitary transformation is sufficient to correct the errors of the principle system. The errors of the ancilla system is left to be uncorrected.}
\label{fig1}
\end{figure}

The above is the standard QEC protocol, and the unitary transformation $U$ ($U^{\dagger}$) used for encoding (decoding) is not unique. In this work, however, the unitary transformation is fixed as
\begin{equation}
U\vert a_m\rangle\otimes \vert 0\rangle\rightarrow \vert m, +\rangle, U\vert a_m\rangle\otimes \vert 1\rangle\rightarrow \vert m, -\rangle,\end{equation}
or in the equivalent form
\begin{equation}
\label{udag}
U^{\dagger}\vert m, +\rangle\rightarrow\vert a_m\rangle\otimes \vert 0\rangle,U^{\dagger}\vert m, -\rangle\rightarrow\vert a_m\rangle\otimes \vert 1\rangle,
\end{equation}
Certainly, $\vert 0_\mathcal{L}\rangle =\vert 0,+\rangle, \vert 1_\mathcal{L}\rangle =\vert 0,-\rangle$. [It should be emphasized that the $U^{\dagger}$ introduced  above is nothing else but the $U_2$ used  in the Eq. (87) of the original work in Ref.~\cite {3}.] As it has been argued in Ref.~\cite{3}, the recovery process $\mathcal{R}$ is not a necessary step, since the $U^{\dagger}$ defined in Eq.~(\ref{udag}) is sufficient for correcting the errors of the principle system. One can observe that, the following two processes are equivalent
\begin{equation}
\mathcal{R}\circ \mathcal{U}^{\dagger}\equiv \mathcal{U}^{\dagger}\circ \mathcal{\tilde{R}},
\end{equation}
where the process $\mathcal{\tilde{R}}$ is defined as
\begin{equation}
\mathcal{\tilde{R}}=\mathcal{U}\circ \mathcal{R} \circ \mathcal{U}^{\dagger}.
\end{equation}
Furthermore, it can be expressed with a more explicit way, $\mathcal{\tilde{R}}(\rho^{\mathcal{SA}})=\sum_{m=0}^{15}\tilde{R}_m(\rho^{\mathcal{SA}})\tilde{R}_m^{\dagger}$,
where the Kraus operators $\tilde{R}_m$ take the form
$\tilde{R}_m=UR_mU^{\dagger}$.
By some simple algebra, one may get
$\tilde{R}_m=\vert a_0\rangle\langle  a_m\vert\otimes \mathrm{I}$,
and one can easily verify that: After an arbitrary state $\rho^{\mathcal{SA}}$ of the jointed system is subjected to the process $\mathcal{\tilde{R}}$, the state of the principle system remains unchanged, say,
\begin{equation}
\rho^{\mathcal{S}}=\mathrm{Tr} _{\mathcal{A}}[\rho^{\mathcal{SA}}]=\mathrm{Tr} _{\mathcal{A}}[\mathcal{\tilde{R}}(\rho^{\mathcal{SA}})].
\end{equation}
According to  this  analysis,  the process  $\mathcal{\tilde{R}}$ can be moved away. It has been shown in Fig.~\ref{fig1}(d) that a simplified protocol to obtain the effective channel is defined as,
 \begin{equation}
 \label{vare}
  \tilde{\varepsilon}(\rho^{\mathcal{S}})=\mathrm{Tr} _{\mathcal{A}}[\mathcal{U}^{\dagger}\circ\Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \rho^{\mathcal{S}})].
  \end{equation}

\section{the standard quantum process tomography}
\label{sec3}
To get the complete information about the effective channel, we shall introduce a convenient tool where a bounded matrix in $H_{2}$ is related to a vector in the enlarged Hilbert space $H_{2}^{\otimes 2}$. Let $A$ be a bounded matrix in the $2$-dimensional Hilbert space $H_2$, with $A_{ij}=\langle i\vert  A\vert j\rangle$
the matrix elements for it, and an isomorphism between $A$ and a $2^2$-dimensional vector $\vert A\rangle\rangle$ is defined as
 \begin{equation}
 \vert A\rangle\rangle =\sqrt{2} A\otimes \mathrm{I}_{2}\vert S_+\rangle=\sum_{i,j=0}^1 A_{ij}\vert ij\rangle,
 \end{equation}
 where $\vert S_+\rangle$ is the maximally entangled state for $H_{2}^{\otimes 2}$, and $\vert S_+ \rangle =\frac{1}{\sqrt{2}}\sum_{k=0}^{1}\vert kk\rangle$ with $\vert ij\rangle=\vert i\rangle\otimes \vert j\rangle$.
This   isomorphism provides a one-to-one mapping between the matrix and its vector form.

For a quantum process $\varepsilon$, the Kraus operators $\{A_m\}$ can be described by a corresponding Choi matrix,
\begin{equation}
\label{choi}
\chi(\varepsilon)=\sum_{m}\vert A_m\rangle\rangle\langle\langle A_m\vert.
\end{equation}
Via the isomorphism above,  this matrix can also be rewritten as
$ \chi(\varepsilon)=2\bar{\rho}$, with $\bar{\rho}=\varepsilon\otimes \mathrm{I}(\vert S_+\rangle\langle S_+\vert)$.
For the normalized state $\bar{\rho}$, Schumacher's entangling  fidelity is defined as $F=\langle S_+\vert\bar{\rho}\vert S_+\rangle$, and it provides a measure for how well the entanglement is preserved by the quantum process $\varepsilon$~\cite{29}. Certainly, one may calculate the entangling fidelity
\begin{equation}
\label{fed}
F(\varepsilon)=\frac{1}{2} \langle S_+\vert\chi(\varepsilon)\vert S_+\rangle.
 \end{equation}

With $\chi$ known, one can derive the Kraus operators of $\varepsilon$. This can be completed through the following simple protocol: The eigenvalues $\lambda^m$ and the corresponding eigenvectors $\vert \Phi^m\rangle$ of $\chi$ can be easily calculated, say,
$\chi=\sum_m\lambda^m\vert \Phi^m\rangle\langle \Phi^m\vert$. Suppose that $\vert\Phi^m\rangle$ can be expanded as $\vert \Phi^m\rangle=\sum_{ij}c^m_{ij}\vert ij\rangle$, with $c^m_{ij}=\langle ij\vert\Phi^m\rangle$ the expanding coefficients. Then, the Kraus operators $A_m$ can be expressed as
\begin{equation}
\label{operator}
A_m=\sqrt{\lambda^m} \sum_{i,j=0}^1 c^m_{ij}\vert i\rangle\langle j\vert.
\end{equation}
With these operators, one may verify that the relation in Eq.~(\ref{choi}) is recovered.

The effective channel can be obtained via the performing the SQPT~\cite{15}. Here, it should be mentioned that the way of performing SQPT is not limited. In the present work, we shall apply the protocol presented in Ref.~\cite{Wu}. For convenience, a brief review of this protocol is organized in following: Introducing the set of operators, say, ${E}_{cd}=\vert c\rangle\langle d\vert (c,d=0,1)$, we take them as the inputs for the principle system, and for a given $E_{cd}$, the corresponding output is
\begin{equation}
\label{varep}
\tilde{\varepsilon}({{E}}_{cd })=\mathrm{Tr}_{\mathcal{A}}[\mathcal{U}^{\dagger}\circ \Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \vert c\rangle\langle d\vert)].
\end{equation}
Then, one may introduce the coefficients
\begin{equation}
\label{lambda}
\tilde{\lambda}_{ab;cd}=\langle a\vert \tilde{\varepsilon}({{E}}_{cd })\vert b\rangle,
\end{equation}
and $\tilde{\varepsilon}({{E}}_{cd })$ can be expanded as
${\tilde{\varepsilon}}({{E}}_{cd })=\sum_{a,b=0}^{1}\tilde{\lambda}_{ab;cd}\vert a\rangle\langle b\vert$.
The Choi matrix of the effective channel $\tilde{\varepsilon}$ in Eq.~(\ref{vare}), can be expanded as
\begin{equation}
\chi(\tilde{\varepsilon})=\sum_{a,b,c,d=0}^1\tilde{\chi}_{ab;cd} \vert ab\rangle\langle cd \vert,
\end{equation}
with $\tilde{\chi}_{ab;cd}$ its matrix elements,
\begin{equation}
\label{tchi}
\tilde{\chi}_{ab;cd}=\langle ab\vert \chi(\tilde{\varepsilon})\vert cd\rangle.
\end{equation}
It has been shown in Ref.~\cite{Wu} that the $\tilde{\chi}_{ab;cd}$ can be obtained in a simple way,
\begin{equation}
\label{mapping}
\tilde{\chi}_{ab;cd}=\tilde{\lambda}_{ac;bd}.
\end{equation}

\section{exact performance of the concatenated five-qubit code}
\label{sec4}
Now, we shall restrict our attention to the uncorrelated errors. We use $\varepsilon:\{A_m\}$ to denote the quantum process of each two-dimensional  subsystem. Formally, $\Lambda=\varepsilon^{\otimes 5}$. Let us consider the lowest level of error correction for the depolarizing channel $\varepsilon_{\mathrm{DP}}$,
 \begin{equation}
 A_0=\sqrt{F_0}\hat{\texttt{I}}_2, A_i=\sqrt{\frac{1-F_0}{3}}\hat{\sigma}_i
 \end{equation}
 with $i=1,2,3$ and   $F_0$ its channel fidelity. We shall take it as
an explicit example to show how the SQPT is completed.

(a) Let $\vert 0\rangle\langle 0\vert$ the input of the principle system. The corresponding output  is denoted by $\tilde{\varepsilon}(\vert 0\rangle\langle 0\vert)$. With the equation
 \[\tilde{\varepsilon}(\vert 0\rangle\langle 0\vert)=\mathrm{Tr} _{\mathcal{A}}[\mathcal{U}^{\dagger}\circ\Lambda\circ \mathcal{U}(\vert a_0\rangle\langle a_0\vert\otimes \vert 0\rangle\langle 0\vert],\]
one has
\begin{equation}
\tilde{\varepsilon}(\vert 0\rangle\langle 0\vert)=\left(
                                                     \begin{array}{cc}
                                                       a & 0 \\
                                                       0 & 1-a \\
                                                     \end{array}
                                                   \right),
                                                   \nonumber
                                                   \end{equation}
where $a=\frac{1}{81}(1+2F_0)^2(37-108F_0+144F_0^2-64F_0^3)$. Based on the definition in Eq.~(\ref{lambda}),  the four matrix elements, $\tilde{\lambda}_{ab;00}$ ($a,b=0,1$), are
\begin{equation}
\tilde{\lambda}_{00;00}=a, \tilde{\lambda}_{10;00}=\tilde{\lambda}_{01;00}=0, \tilde{\lambda}_{11;00}=1-a.\nonumber
\end{equation}

(b) Similarly with step (a), one may also obtain
\begin{eqnarray}
\tilde{\varepsilon}(\vert 0\rangle\langle 1\vert)&=&\left(
                                                     \begin{array}{cc}
                                                       0 & 2a-1 \\
                                                       0 & 0 \\
                                                     \end{array}
                                                   \right),\nonumber\\
                                                   \tilde{\varepsilon}(\vert 1\rangle\langle 0\vert)&=&\left(
                                                     \begin{array}{cc}
                                                       0 & 0 \\
                                                       2a-1 & 0 \\
                                                     \end{array}
                                                   \right),\nonumber\\
                                                   \tilde{\varepsilon}(\vert 1\rangle\langle 1\vert)&=&\left(
                                                     \begin{array}{cc}
                                                       1-a & 0 \\
                                                       0 & a \\
                                                     \end{array}
                                                   \right),  \nonumber
\end{eqnarray}
and therefore,
\begin{eqnarray}
\tilde{\lambda}_{00;01}= \tilde{\lambda}_{10;01}=\tilde{\lambda}_{11;01}=0, \tilde{\lambda}_{01;01}=2a-1,\nonumber\\
\tilde{\lambda}_{00;10}= \tilde{\lambda}_{01;10}=\tilde{\lambda}_{11;10}=0, \tilde{\lambda}_{10;10}=2a-1,\nonumber\\
\tilde{\lambda}_{00;11}=1-a, \tilde{\lambda}_{10;11}=\tilde{\lambda}_{01;11}=0, \tilde{\lambda}_{11;11}=a.\nonumber
\end{eqnarray}

(c) With all the matrix elements, $\tilde{\lambda}_{ab;cd}$, the so-called matrix $\tilde{\lambda}$ can be orgnized as
 \begin{equation}
 \tilde{\lambda}=\left(
           \begin{array}{cccc}
             a & 0 & 0 & 1-a \\
             0 & 2a-1& 0 & 0 \\
             0 & 0 &  2a-1& 0 \\
             1-a & 0 & 0 & a \\
           \end{array}
         \right).\nonumber
         \end{equation}
 According to the one-to-one relation in  Eq.~(\ref{mapping}), the effective Choi matrix $\tilde{\chi}$ is
  \begin{equation}
\label{varchi}
\tilde{\chi}(\varepsilon_{\mathrm{DEP}})=\left(
               \begin{array}{cccc}
                a& 0 & 0 & 2a-1 \\
                0 & 1-a & 0 & 0 \\
                0 & 0 & 1-a & 0 \\
               2a-1 & 0 & 0 & a \\
                               \end{array}
                                            \right).
  \end{equation}

(d) The entangling fidelity of the effective channel $\tilde{\chi}$ is denoted by $F_1$, and now Eq.~(\ref{fed}) can be rewritten as
\begin{equation}
F_1=\frac{1}{4}(\tilde{\chi}_{00;00}+\tilde{\chi}_{00;11}+\tilde{\chi}_{11;00}+\bar{\chi}_{11;11}).\nonumber
\end{equation}
Based on it, there should be
\begin{equation}
\label{fed1}
F_1(\varepsilon_{\mathrm{DEP}})=\frac{1}{27}(5+20F_0-70F_0^2+40F_0^3+160F_0^4-128F_0^5).
\end{equation}
A parameter $p$ can be used to characterize the depolarizing channel, say, $A_0=\sqrt{1-3p/4}\hat{\texttt{I}}_2, A_i=\sqrt{p}/2\hat{\sigma}_i$, and with the relation $F_0=1-3p/4$, Eq.~(\ref{fed1}) can be rewritten as
\begin{equation}
F_1(\varepsilon_{\mathrm{DEP}})=1-\frac{45}{8}p^2+\frac{75}{8}p^3-\frac{45}{8}p^4+\frac{9}{8}p^5.
\end{equation}
This is the same as the result by Reimpell and Werner~\cite{16}. Furthermore, by requiring that $F_1\ge F_0$, we have the threshold $p<0.18$ (or $F_0>0.86$), the condition under which the five-qubit code works  for the depolarizing channel.

(e) By some simple algebra, the eigenvalues $\lambda^m$ and the corresponding eigenvectors $\vert \Phi^m\rangle$ can be derived,
\begin{eqnarray}
\lambda^0&=&2F_1, \lambda^1=\lambda^2=\lambda^3=\frac{2(1-F_1)}{3},\nonumber\\
\vert\Phi^0\rangle&=&\frac{1}{\sqrt{2}}(\vert 00\rangle+\vert 11\rangle),\nonumber\\
\vert\Phi^1\rangle&=&\frac{1}{\sqrt{2}}(\vert 01\rangle+\vert 10\rangle),\nonumber\\
\vert\Phi^2\rangle&=&\frac{1}{\sqrt{2}}(\vert 01\rangle-\vert 10\rangle),\nonumber\\
\vert\Phi^3\rangle&=&\frac{1}{\sqrt{2}}(\vert 00\rangle-\vert 11\rangle).\nonumber
\end{eqnarray}
With Eq.~(\ref{operator}), one may verify that the effective Choi matrix can be also expressed by a set of Kraus operators $\tilde{A}_m$,
$\tilde{\chi}=\sum_m\vert \tilde{A}_m\rangle\rangle\langle\langle \tilde{A}_m\vert$, where
\begin{equation}
 \tilde{A}_0=\sqrt{F_1}\hat{\texttt{I}}_2, \tilde{A}_i=\sqrt{\frac{1-F_1}{3}}\hat{\sigma}_i
 \end{equation}
 with $i=1,2,3$ and   $F_1$ the entangling fidelity. Obviously, the effective channel
is also a depolarizing channel ~\cite{Rahn,Kersting}.

(f) With the Kraus operators above, a new supper operator $\tilde{\varepsilon}$: $\tilde{\varepsilon}(\rho)=\sum \tilde{A}_m\rho \tilde{A}_m^{\dagger}$ can be defined. Let $\Lambda=\tilde{\varepsilon}^{\otimes 5}$ and follow the steps from (a) to (e), we can obtain the effective
channel by the $5^2$-qubit concatenated code. By repeating the argument above, we can get the effective channel from the $5^L$-qubit
concatenated code.

The protocol developed above, used for the depolarizing channel $ \tilde{\varepsilon}_{\mathrm{DEP}}$, can be easily generalized for other cases. For instance, the amplitude damping channel $\varepsilon_{\mathrm{AD}}$, has been widely discussed in previous works, and the Kraus operators are now
\begin{equation}
\label{damping1}
A_0=\left(
      \begin{array}{cc}
        1 & 0 \\
        0 & \sqrt{1-\gamma} \\
      \end{array}
    \right),A_1=\left(
                  \begin{array}{cc}
                    0 & \sqrt{\gamma} \\
                    0 & 0 \\
                  \end{array}
                \right),\end{equation}
where $\gamma$ is the damping parameter. The entangling fidelity of it is $F_0=\frac{1}{4}(1+\sqrt{1-\gamma})^2$.
 When the five-qubit code is applied for correcting the amplitude damping errors, the entangling fidelity of the effective channel is
\begin{eqnarray}
\label{damping1}
F_1(\gamma)&=&\frac{1}{4}[1+\frac{1}{4}(1-\gamma)^2(4+8\gamma-3\gamma^2+\gamma^3)\nonumber\\
&&+\frac{1}{2}\sqrt{1-\gamma}(4+2\gamma-11\gamma^2+5\gamma^3)].
\end{eqnarray}

Another important case is the bit-flip channel $\varepsilon_{\mathrm{BF}}$ with
\begin{equation}
A_0=\sqrt{F_0}\left(
                \begin{array}{cc}
                  1 & 0 \\
                  0 & 1 \\
                \end{array}
              \right),
 A_1=\sqrt{1-F_0}\left(
                                      \begin{array}{cc}
                                        0 & 1 \\
                                        1 & 0 \\
                                      \end{array}
                                    \right),
                                    \end{equation}
where $F_0$ is the entangling  fidelity.
For a reason which will be clear soon, here we first introduce the denotations,
$\varepsilon_{\mathrm{DP}}\equiv\varepsilon^{0}(\omega_0,F_0)$, $\varepsilon_{\mathrm{AD}}\equiv\varepsilon^{0}(\mathbb{\omega}_1, F_0)$ and $\varepsilon_{\mathrm{BF}}\equiv
\varepsilon^{0}(\omega_2, F_0)$, where $F_0$ is the fidelity of the initial channel without performing QEC. After performing QEC with the
$5^l$-qubit code, the effective channel is denoted by $\varepsilon^{l}(\omega, F_0)$ with  the setting $(\mathbf{\omega}, F_0)$ indicating that the effective channel is originated from the initial $\varepsilon^0(\mathbf{\omega}, F_0)$.
With the Choi matrix $\chi(\varepsilon^{l}(\omega, F_0))$, the entangling fidelity of the effective channel  can be calculated as
\begin{equation}
\label{Fedl}
F_{l}(\omega,F_0)=\frac{1}{2}\langle S_+\vert \chi(\varepsilon^{l}(\omega, F_0))\vert S_+\rangle.
\end{equation}

In the present work, a quantum channel is called \emph{near-perfect} if the entangling fidelity has a value above $1-10^{-5}$. For a given initial channel $\varepsilon^{0}(\omega, F_0)$, we introduce the quantity $L_{(\omega,F_0)}$ and let $5^{L_{(\omega,F_0)}}$ be the minimum size of the concatenated code sufficient for constructing a near-perfect channel, $F_{L_{(\omega,F_0)}}\ge 1-10^{-5}$.
Correspondingly, a $5^{L_{(\omega,F_0)}}$-qubit code is said to be general if $L_{(\omega,F_0)}$ is independent of $\omega$, which is used for denoting the actual type of the error model.

When the concatenated code is applied, the effective  $\chi(\varepsilon^{l}(\omega, F_0))$ ($\omega\neq \omega_0$) usually does not have an analytical form. Noting that $\chi(\varepsilon^{l}(\omega_0, F_0))$   always represents a depolarizing channel, we suppose that $\chi(\varepsilon^{l}(\omega, F_0))$ can be approximated by  $\chi(\varepsilon^{l}(\omega_0, F_0))$, and the error of the approximation is characterized by the distance measure
 \begin{equation}
 \label{dist}
 D_{l}(\omega, F_0)=\frac{1}{4}\vert \chi(\varepsilon^{l}(\mathbf{\omega}, F_0))-\chi(\varepsilon^{l}(\mathbf{\omega}_0, F_0))\vert,
\end{equation}
with $\vert A\vert=\sqrt{A^{\dagger}A}$ ~\cite {15}.  Especially, if $D_{l}(\omega, F_0)\ll 0$,  the entangling fidelities of the two channels,  $\varepsilon^{l}(\mathbf{\omega}, F_0)$ and $\varepsilon^{l}(\mathbf{\omega}_0, F_0)$, almost have the same value.

Under the condition that the entangling fidelity $F_0$ ($F_0\ge 0.86$) is fixed,  besides the steps from (a) to (f) for getting the effective channel, we added another two ones:

(g) With the Choi matrix $\chi(\varepsilon^{1}(\mathbf{\omega}, F_0))$ corresponding to  the effective channel $\varepsilon^{1}(\mathbf{\omega}, F_0)$, the entangling fidelity $F_1( \mathbf{\omega}, F_0)$ is decided by Eq.~(\ref{Fedl}), and the distance $D_{1}(\omega , F_0)$ is calculated   according to Eq.~(\ref{dist}).

(h) With a simple program, the Kraus operators of the effective
channel can be decided, and the calculation for the effective channel in the second level starts from step (a).
 The calculation for a given $\varepsilon(\mathbf{\omega}, F_0)$ can be terminated if
$F_l(\omega, F_0)\geq 1-10^{-5}$.

Based on the iterative protocol developed above, the effective channels in each level of the concatenation can be worked out. For the typical case
where $F_0=0.92$, as shown in Table \Rmnum{1}, we have two observations that: (I) For all the possible channels, a perfect effective channel (with a error below $10^{-5}$) can be constructed by using the same concatenated five-qubit code;
\begin{table}\caption{The fidelity $F_{l}(\omega, F_0)$}
\begin{tabular}{|c|c|c|c|}
  \hline

    $l$ & $F_l(\omega_0)$ & $F_l({\omega_1})$ & $F_{l}(\omega_2)$\\
    \hline
  0 & ~~0.920~~ & ~~0.920~~ & ~~0.920~~ \\
    \hline
  1 & ~~0.946665~~ & ~~0.946762~~ & ~~0.945639~~ \\
    \hline
  2 & ~~0.974784~~ & ~~0.97487~~ & ~~0.973903~~ \\
    \hline
  3 & ~~0.993991~~ & ~~0.99403~~ & ~~0.993576~~ \\
    \hline
  4 & ~~0.999644~~ & ~~0.999648~~ & ~~0.999593~~ \\
    \hline
  5 & ~~0.999999~~ & ~~0.999999~~ & ~~0.999998~~\\
  \hline
\end{tabular}
\end{table}
(II) Meanwhile, the resulted effective channel can be approximated by the depolarizing channel. As shown in Table \Rmnum{2}, the error  of the approximation approaches to zero when $l$, the level of concatenation, is increased.
\begin{table}\caption{The error of the approximation.}
\begin{tabular}{|c|c|c|}
    \hline

  $l$ & $D_l(\omega_1, 0.92)$ & $D_l(\omega_2, 0.92)$ \\
   \hline
 0 & $8.02\times 10^{-2}$ & $5.33\times 10^{-2}$ \\
    \hline
 1 & $3.68\times 10^{-3}$ & $1.79\times 10^{-2}$ \\
   \hline
 2 & $8.77\times 10^{-5}$ & $2.04\times 10^{-3}$ \\
    \hline
 3 & $5.51\times 10^{-6}$ & $2.92\times 10^{-5}$ \\
   \hline
4& $2.15\times 10^{-10}$ & $6.29\times 10^{-9}$ \\
   \hline
 5 & $6.04\times 10^{-14}$ & $1.46\times 10^{-13}$ \\
  \hline
\end{tabular}
\end{table}

The effective channel for other cases, where $F_0$ takes different values, have also been calculated. Our calculation indicates  that  the two observations, (I) and (II) above, are   independent of the choice of $F_0$.

\section{exact performance of the concatenated seven-qubit code}
 \label{section5}
 In this section, we will show that the concatenated seven-qubit code is not general, or in other word, the number of levels for the concatenation, which is necessary for constructing a near-perfect channel, is dependent on the actual type of the noise. For the seven-qubit code~\cite{2}, we can also define a unitary transformation $V$ in a $2^7$-dimensional Hilbert space for encoding, and its inverse $V^{\dagger}$ is applied for decoding.
 Let $\vert 0_{\mathcal{L}}\rangle$ and $\vert 1_{L}\rangle$ be the logical codes, select a set of correctable errors$\{E_m\}_{m=0}^{63}$ including: The identity operator $ E_0=\mathrm{I}^{\otimes 7}$), all the rank-one Pauli operator $\sigma_i^{n}$ ($i=x,y,z$, $n=1,2,...,7)$, and a number of 44 rank-two operators like $\sigma^1_1\otimes \sigma^2_2$, $\sigma^1_3\otimes \sigma^5_2$, ...,\emph{ etc.}, and with the definition
\[ \vert m,+\rangle=E_m\vert 0_{\mathcal{L}}\rangle, \vert m,-\rangle=E_m\vert 1_{\mathcal{L}}\rangle, \]
we find that the set of normalized vectors $\{\vert m, \pm\rangle\}_{m=0}^{63}$ form the basis of the $2^7$-dimensional Hilbert space.
Use $\{\vert a_m\rangle\}_{m=0}^{63}$ to denote the basis of ancilla system, and with the denotations
$\vert a_m, 0\rangle=\vert a_m \rangle\otimes \vert 0\rangle$ and $\vert a_m, 1\rangle=\vert a_m \rangle\otimes \vert 1\rangle$, the unitary transformation $V$ is
\[V=\sum_{m=0}^{63} \big(\vert m, +\rangle\langle a_m, 0\vert+  \vert m, -\rangle\langle a_m, 1\vert\big).\]
Finally, define $\Lambda=\varepsilon^{\otimes 7}$ and $\vert a_0\rangle=\vert 000000\rangle$, and the effective channel, which is obtained by performing
QEC procedure with the seven-qubit code, can be obtained as
\begin{equation}
  \tilde{\varepsilon}(\rho^{\mathcal{S}})=\mathrm{Tr} _{\mathcal{A}}[\mathcal{V}^{\dagger}\circ\Lambda\circ \mathcal{V}(\vert a_0\rangle\langle a_0\vert\otimes \rho^{\mathcal{S}})].
  \end{equation}

To make sure that our program works in a perfect way, we consider a scenario where the seven-qubit code is applied for the amplitude damping in
Eq.~(\ref{damping1}). The analytical expression for the entangling fidelity of  the
effective channel is
\begin{eqnarray}
\label{damping2}
F_1(\gamma)&=&\frac{1}{4}[1+\sqrt{1-\gamma}(2+\gamma)\nonumber\\
&&+\frac{(1-\gamma)^3}{8}(8+24\gamma-33\gamma^2+21\gamma^3-42\gamma^4)\\
&&+\frac{\sqrt{1-\gamma}}{16}(-150\gamma^2+180\gamma^3-117\gamma^4+39\gamma^5)]\nonumber.
\end{eqnarray}
One can easily check that this result  recovers  the numerical one given in Ref.~\cite{19}, and the entangling fidelities of the effective channel when the five-qubit code and the seven-qubit code are used against the amplitude errors are compared in Fig.~\ref{fig2}.

\begin{figure} \centering
\includegraphics[scale=0.37]{fig2.eps}
\caption{(Color online) The entangling fidelities of the effective channel when the five-qubit code and the seven-qubit code are applied against the amplitude damping errors. The exact function in Eq.~(\ref{damping1}) is shown with the solid line while the one  in Eq.~(\ref{damping2}) for the seven-qubit code,  is in the dash line.}
\label{fig2}
\end{figure}

\begin{table}\caption{The fidelity obtained for the concatenated Steane's  code}
\begin{tabular}{|c|c|c|c|}
  \hline
 1& $F_l(\varepsilon_{\mathrm{DEP}})$ & $F_l(\varepsilon_{\mathrm{AD}})$&  $F_l(\varepsilon_{\mathrm{BF}})$ \\  \hline
 0 &0.94 &0.94 & 0.94 \\  \hline
 1 & 0.952211& 0.943496 & 0.943035 \\  \hline
 2 & 0.968897 &0.950234 & 0.947904 \\  \hline
  3& 0.986173 & 0.960975 & 0.955409 \\  \hline
 4 & 0.997048 & 0.975311 & 0.966146 \\  \hline
  5 & 0.99985 & 0.989674 & 0.979469 \\  \hline
 6 & $\geqslant 1-10^{-5}$ & 0.99811 & 0.99196 \\  \hline
  7 & $\geqslant 1-10^{-5}$ & 0.999935 & 0.998693 \\  \hline
 8 & $\geqslant 1-10^{-5}$ & $\geqslant 1-10^{-5}$ & 0.99964 \\  \hline
  9&$\geqslant 1-10^{-5}$ &$\geqslant 1-10^{-5}$&$\geqslant 1-10^{-5}$\\
   \hline
\end{tabular}
\end{table}

As shown in the above section, one can get the exact performance of the QEC protocol based on the concatenated Steane code. For three types of noise models, the depolarizing channel $\varepsilon_{\mathrm{DEP}}$, the amplitude damping channel $\varepsilon_{\mathrm{AD}}$, and
the bit flip channel $\varepsilon_{\mathrm{BF}}$, the corresponding entangling fidelities $F_l$ have been calculated under the condition that the fidelity of the uncorrected channels is fixed to be $F_0=0.94$. Our results, listed in Table~\Rmnum{3}, demonstrate that the necessary numbers of the concatenation are dependend on the actual types of the error.

\section{the general noise model}
\label{sec6}
In this section, we shall show that the observation, which has been observed  for the amplitude damping and bit flip channels, is  general. Let $\{\bar{A}_m\}$ to be a set of Kraus-operators, introduce an arbitrary $2\times2$ unitary transformation,
\begin{equation}
U_2(\theta,\phi)=\left(
                   \begin{array}{cc}
                     \cos\frac{\theta}{2} & \sin\frac{\theta}{2}\exp\{-i\phi\} \\
                    -\sin\frac{\theta}{2}\exp\{i\phi\} & \cos\frac{\theta}{2} \\
                   \end{array}
                 \right),\nonumber
                 \end{equation}
and another set of operators $\{A_m\}$ can be  defined as
\begin{equation}
\label{Abar}
A_m=U_2(\theta,\phi)\bar{A}_m U^{\dagger}_2(\theta,\phi),
\end{equation}
where $\theta$ and $\phi$ are two free parameters.
Three free parameters, $\alpha$, $\beta$, and $\gamma$, are used to define the following four operators,
\begin{eqnarray}
\label{As}
\bar{A}_0=\left(
            \begin{array}{cc}
              \cos\alpha & 0 \\
             0 & \sin\beta\cos\gamma \\
            \end{array}
          \right), \bar{A_1}=\left(
                               \begin{array}{cc}
                                 0 & 0 \\
                                 \sin\alpha\sin\gamma & 0 \\
                               \end{array}
                             \right),\nonumber\\
\bar{A}_2=\left(
            \begin{array}{cc}
              0 & \sin\beta\sin\gamma \\
            0 & 0\\
                       \end{array}  \right),\bar{A}_3=\left(
                                     \begin{array}{cc}
                                       \sin\alpha\cos\gamma & 0 \\
                                     0 & \cos\beta \\
                                     \end{array}
          \right).\nonumber\\
\end{eqnarray}
Now, let us recall some discussions about using the Bloch sphere representation to  describe the single-qubit channel~\cite{15}. With $\vec{r}$ the Bloch vector for an input state $\rho$, and $\vec{\bar{r}}$ for the output state, $\bar{\varepsilon}(\rho)=\sum_m\bar{A}_m\rho(\bar{A}_m)^{\dagger}$, on can obtain a map
\begin{equation}
\vec{r}\rightarrow\vec{\bar{r}}=M\vec{r}+\vec{\delta},
\end{equation}
where $M$ is a $3\times3$ real matrix, and $\vec{\delta}$ is a constant vector. This is an affine map, mapping the Bloch vector into itself. The set of operators in Eq.~(\ref{As}) can be described as
\begin{equation}
\label{map}
\left(
  \begin{array}{c}
    \bar{r}_x \\
    \bar{r}_y \\
    \bar{r}_z \\
  \end{array}
\right)=\left(
          \begin{array}{ccc}
            \eta_{\bot} & 0 & 0 \\
            0 & \eta_{\bot} & 0 \\
            0 & 0 & \eta_z \\
          \end{array}
        \right)\left(
                 \begin{array}{c}
                   r_x \\
                   r_y \\
                   r_z \\
                 \end{array}
               \right)+\left(
                         \begin{array}{c}
                           0 \\
                           0 \\
                           \delta_z \\
                         \end{array}
                       \right),
\end{equation}
with the coefficients
\begin{eqnarray}
\eta_{\bot}&=&\sin(\alpha+\beta)\cos\gamma,\nonumber\\
\eta_z&=&1-(\sin^2\alpha+\sin^2\beta)\sin^2\gamma,\\
\delta_z&=&(\sin^2\beta-\sin^2\alpha)\sin^2\gamma.\nonumber
\end{eqnarray}
 This affine map can be roughly classified into the following two cases: the centered map  with  $\delta_z=0 $ (if $ \alpha=\beta$ ) and the  non-centered one with  $\delta_z\neq 0$.

The centered    map of Eq.~(\ref{map}) is equivalent with the Pauli-channel,
$\sqrt{p_0}\hat{\texttt{I}_2}, \sqrt{p_x}\hat{\sigma}_x,\sqrt{p_y}\hat{\sigma}_y,\sqrt{p_z}\hat{\sigma}_z$, and the parameters $p_i$ are given by
$p_x=p_y=\frac{1}{2}\sin^2\alpha\sin^2\gamma$,
$p_z=\frac{1}{2}(\cos\alpha-\sin\alpha\cos\gamma)^2$,
and $p_0=1-p_x-p_y-p_z$.
 Certainly, it also contains the following two important situations: If $\alpha$ is fixed as
 \begin{equation}
 \cos\alpha=\frac{\cos\gamma+\sin\gamma}{\sqrt{2+\sin 2\gamma}},
 \sin\alpha=\frac{1}{\sqrt{2+\sin 2\gamma}},\nonumber
  \end{equation}
one can obtain a depolarizing channel, while for $\gamma=0$ and an arbitrary $\alpha$, we have the \emph{phase-flip} channel.

With the  $U_2(\theta,\phi)$ introduced above, our noise model should also contain the \emph{bit-flip} channel and \emph{bit-phase flip} channel. For the non-centered case, let $\alpha=0$ and $\beta=\frac{\pi}{2}$, and we can come to the \emph{amplitude damping} channel,
\begin{equation}
\bar{A}_0=\left(
             \begin{array}{cc}
               1& 0 \\
               0 & \cos\gamma \\
             \end{array}
           \right),\bar{A}_1=\left(
                               \begin{array}{cc}
                                 0 & \sin\gamma \\
                               0 &0 \\
                               \end{array}
                             \right).\nonumber
\end{equation}
For the case where the constraint $\alpha+\beta=\frac{\pi}{2}$ holds, we have the so-called \emph{generalized amplitude damping} channel~\cite{15},
\begin{eqnarray}
\bar{A}_0=\cos\alpha\left(
            \begin{array}{cc}
              1 & 0 \\
              0 & \cos\gamma \\
            \end{array}
          \right), \bar{A}_1=\sin\alpha\left(
                                         \begin{array}{cc}
                                           0 & 0 \\
                                           \sin\gamma & 0 \\
                                         \end{array}
                                       \right),\nonumber\\
                                       \bar{A}_2=\cos\alpha\left(
                                                             \begin{array}{cc}
                                                               0 & \sin\gamma \\
                                                               0 & 0 \\
                                                             \end{array}
                                                           \right),\bar{A}_3=\sin\alpha\left(
                                                                                \begin{array}{cc}
                                                                                \cos\gamma& 0 \\
                                                                                0 & 1 \\
                                                                                \end{array}
                                                                                \right).\nonumber
 \end{eqnarray}
 From the discussions above, it can be seen that nearly all the noise channel listed in Ref.~\cite{15} are included here. Therefore, our noise model, defined in Eqs.~(\ref{Abar}) and~(\ref{As}), is general.

Now, let us consider a typical case where the entangling fidelity of the uncorrected channel is fixed as $ F_0=0.9$. For the depolarizing channel, using
the result in Eq. (\ref{fed1}),
we can get the entangling fidelities in each level of the concatenation:
\begin{eqnarray}
\label{0.9dep}
F_1(\omega_0)&=&0.920491, F_2(\omega_2)=0.947258, \nonumber \\
F_3(\omega_0)&=&0.975308, F_4(\omega_0)=0.9942310,\nonumber\\
F_5(\omega_0)&=&0.999714, F_6(\omega_0)=0.999999.
\end{eqnarray}
Therefore, to construct a near perfect channel from the depolarizing channel with entangling fidelity $F_0=0.9$, the five-qubit code should be concatenated with itself $L=6$ times. From Eq.~(\ref{varchi}) and the known $F_l(\omega)$, the effective choi matrix
 $\chi(\varepsilon^{1}(\omega_0, F_0))$ can be easily calculated.

 Then,    under the condition that  $F_0$  is fixed, $F_0=0.9$,  we can design a program to generate an arbitrary setting for the five free parameters introduced above. The generated channel is denoted by  $\varepsilon^0(\mathbf{\omega}_1, F_0)$. Let $\Lambda= \varepsilon^0(\mathbf{\omega}_1, F_0)^{\otimes 5}$,
the effective channel in each $l$-th level of concatenation can be decided by following the same method as the amplitude channel in Sec.~\ref{sec4}. As the result of this run of calculation, we can get the exact values $F_l(\omega_1, F_0)$ and $D_l(\omega_1, F_0)$ with $l=1,2,...,6$.
After the calculation of the first one is completed, another channel with a fidelity of $0.9$ will be generated and denoted by $\varepsilon^0(\mathbf{\omega}_2, F_0)$. Similarly, we have the results $F_l(\omega_2, F_0)$ and $D_l(\omega_2, F_0)$ with $l=1,2,...,6$.
Usually, in the $m$-th run of calculation, the generated channel is denoted by $\varepsilon^0(\mathbf{\omega}_m, F_0)$
with $F_0=0.9$. After performing QEC with the $5^6$-qubit concatenated code, we can get a series of exact values
$F_l(\omega_m, F_0)$ and $D_l(\omega_m, F_0)$ with $l=1,2,...,6$.
 For a fixed value of $F_0$, there is about  $M$ ($M\ge 10^5$) examples of noise channel that will be generated.  Based on the numerical data in each level of concatenation, a distance measure $D_l^{\mathrm{max}}(F_0)$ can be defined to denote the maximum value of error for the  approximation defined in Eq.~(\ref{dist}),
\begin{equation}
\label{dist1}
D_l^{\mathrm{max}}(F_0)=\max \{ D_{l}(\omega_m, F_0)\}, 1\leq l\leq 6, 0\leq m\leq M,
\end{equation}
and  the minimum fidelity
\begin{equation}
\label{fedelity}
F_l^{\mathrm{min}}(F_0)=\min \{F_l(\omega_m,F_0)\}_{m=0}^{M}, 1\leq l\leq 6, 0\leq m\leq M.
\end{equation}

For the case $F_0=0.9$, our numerical calculation gives
\begin{eqnarray}
\label{fmin}
F_1^{\mathrm{min}}(F_0)=0.918540, F_2^{\mathrm{min}}(F_0)=0.945006,\nonumber\\
 F_3^{\mathrm{min}}(F_0)=0.973293,
F_4^{\mathrm{min}}(F_0)=0.993281,\nonumber\\ F_5^{\mathrm{min}}(F_0)=0.999555, F_6^{\mathrm{min}}(F_0)=0.999998.
\end{eqnarray}

\begin{figure} \centering
\includegraphics[scale=0.37]{fig3.eps}
\caption{(Color online) Numerical results for the maximum distance measure $D_l^{\mathrm{max}}(F_0)$ defined in Eq.~(\ref{dist1}).}
\label{fig3}
\end{figure}

The numerical results for $D^{\mathrm{max}}_l(F_0)$ are given in Fig.~\ref{fig3}, where the initial fidelity take different values, 0.9, 0.92, 0.945, 0.97, 0.97, 0.992, and 0.9993. For each given initial fidelity, the values of the distance measure $D^{\mathrm{max}}_l(F_0)$ are given in the domain $0\leq l\leq L_{(\omega_0, F_0)}$. Our numerical results indicate that: In each level of the concatenation, the effective channel can be approximated by a corresponding depolarizing channel. The error of the approximation, which is characterized by the distance measure $D^{\mathrm{max}}_l(F_0)$, approaches zero as the level of concatenation is increased.

Based on the results in Eq.~(\ref{0.9dep}) and Eq.~(\ref{fmin}), one can observe that: (a) In each level of concatenation, $F_l(\omega_0)-F^{\mathrm{min}}_l\ge 0$. When $l$ is increased, this difference will approach to zero. (b) Since that $F_6^{\mathrm{min}}(F_0)\ge 1-10^{-5}$, the concatenated five-qubit code, which is able to construct a near perfect channel from the depolarizing channel, is also suitable for the arbitrary channel with the same initial fidelity as the depolarizing channel. For the other cases where the initial fidelity takes different values,  0.92, 0.945, 0.97, 0.97, 0.992,
our numerical results, which are depicted in Fig.~\ref{fig4}, show that the properties (a) and (b) can be also observed.

Therefore, as a direct consequence of the observations,  we argue that the concatenated five-qubit is general: Under the condition that the fidelity is fixed,     a  $5^{L_{(\omega_0, F_0)}}$-qubit code, which  is the minimal-sized one for constructing a near-perfect qubit-channel from the  depolarizing channels,  can be used to complete the same task  for the  general noise channels with the same initial fidelity $F_0$.

\begin{figure} \centering
\includegraphics[scale=0.37]{fig4.eps}
\caption{(Color online) (a) For the general noise error model, the numerical results of the minimum fidelity $F_l^{\mathrm{min}}(F_0)$ defined in Eq.~(\ref{fedelity}) are depicted; (b) Numerical results for the entangling fidelity when the initial channels are fixed to be the depolarizing ones. The results show the generality of the five-qubit code. Though the numeral results look similar in the two viewgraphs, they are not strictly identical. More specifically, for the case $F_0=0.9$, the entangling fidelities and minimum fidelities in each level of the concatenation given in Eq.~(\ref{0.9dep}) and Eq.~(\ref{fmin}) respectively, are not strictly the same.}
\label{fig4}
\end{figure}

\section{Discussions and remarks}
Our present work is based on the assumption that the noise of quantum channel  comes from the interaction between the physical qubit and the environment. The apparatus, which are used for state  preparation, encoding and decoding, are supposed to be perfect.
The main results of this work are: (I) A simple and explicit scheme is developed to get the effective Choi matrix resulted by the QEC procedure with the concatenated five-qubit code. (II)  Based on this scheme, we  have shown that the QEC procedure, which is optimal for the noise depolarizing channels, also offers an efficient way to construct a near-perfect channel, where the noise of the physical qubit is a general one including the amplitude damping. Within a general noise model, though not a strict proof, our numerical results indicate that the concatenated five-qubit code is general: To construct an  effective channel with a error below $10^{-5}$, the necessary number of the levels for concatenation  is decided by the fidelity of the initial channels and it does not depend on the actual  types of noise models.

Naturally, there is  still an open question: Is the concatenated five-qubit general for an arbitrary noise channel? For the single-qubit case, such a trace-preserving model requires about twelve parameters. Under the constraint that the fidelity is fixed, how to effectively generate an arbitrary setting for all these parameters is still an unsolved problem. From the results of the present work, we guess that the concatenated five-qubit is general. Our  argument is based on the following two facts.

First, a general QEC protocol does exist, and in the work of Horodeckis'~\cite{30}, the so-called twirling procedure has been introduced. Performing twirling on an arbitrary channel $\varepsilon(\omega, F_0)$, one may obtain a depolarizing channel $\varepsilon(\omega_0, F_0)$ with the unchanged channel fidelity, and before performing QEC with the concatenated five-qubit code, all the initial  (unknown) channels may be transferred into the depolarizing channels through twirling.  We may call the so-constructed QEC protocol, which is general for the arbitrary noise models, the \emph{term twirling-assisted QEC scheme}. As a basic property, the effective channel in each level of concatenation should always be a depolarizing channel.

Second, within the general noise model, our numerical results indicate that \emph{the QEC protocol based on the concatenated five-qubit code can be viewed as  an approximate realization of the twirling-assisted QEC scheme}. As shown, the effective channel in each level can be approximated by the depolarizing channel while the error of the approximation approaches to zero as the level of concatenation is increased.

The blockwise decoding protocol, which is suboptimal, is used in our work. From the work of Poulin~\cite{Poulin2}, it is known that the message-passing   decoding algorithm is optimal, and by jointing the twirling stage and the message-passing decoding protocol together, one can have an optimal way to construct a near perfect channel for the arbitrary noise. However, from the results in present work, it is shown that twirling is not a necessary step if the blockwise decoding is applied. Actually, one may guess that the twirling is also not necessary for the QEC with the message-passing decoding algorithm. In other word, the optimal QEC protocol for the depolarizing channel, which has been developed by Poulin, may also be general. We expect this guess could lead to further theoretical or experimental consequences.

\section*{acknowledgements}
The authors are grateful to the referee for very helpful comments. This work was partially supported by the National Natural Science Foundation of China under the Grant No.~11405136, and the Fundamental Research Funds for the Central Universities of China A0920502051411-56.

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text:
Google announces new developer policies to keep data safe
Published: October 9th, 2018
Data is king in today's software world. The insights a business gains from its data can set it up for success. However, there is a dark side to the data world, and if a business misuses, abuses or doesn't protect the data inside its company, it can result in a loss of reputation or trust.
Google is making a number of improvements and announcements to try to ensure they never fall down the wrong data path. As part of its Project Strobe effort, an initiative to review third-party developer access to its services, the company has decided to shut down Google+ for consumers. As part of the review, the company found a bug within its Google+ People APIs that granted users access to profile data, but also gave apps access to data.
"The review did highlight the significant challenges in creating and maintaining a successful Google+ that meets consumers' expectations. Given these challenges and the very low usage of the consumer version of Google+, we decided to sunset the consumer version of Google+," Ben Smith, Google fellow and vice president of engineering, wrote in a post.
Google+ for consumers will shut down over a 10-month period and is expected to complete by next August. The Google+ enterprise version will continue to operate. Google plans on launching new features for businesses.
Other results being made in response to Project Strobe include more granular Google Account permissions for fine-grained data control; a limit of the types of use cases permitted to access Gmail data; and limitations for apps to be able to receive call log and SMS permissions on Android devices.
On the developer side, Google announced strong controls and policies focused around Gmail APIs. These new updates will go into effect January 9, 2019. New policies around providing appropriate access, guidelines on how data may not be used, how to secure data, and accessing only the necessary information. Applications will be required to go through a review on January 9, 2019. If the review is not submitted by the following month, apps will be disabled.
In addition, the company recommends developers migrate to Gmail Add-ons, a solution launched earlier this year for developers to integrate their applications in Gmail. "Gmail Add-ons also offer a stronger security model for users because email data is only shared with the developer when a user takes action," Andy Wen, group product manager for Google, wrote in a post. "We've continually strengthened these controls and policies over the years based on user feedback."
Lastly, the company announced an updated to Google Play Developer policies. "Our Google Play Developer policies are designed to provide a safe and secure experience for our users while also giving developers the tools they need to succeed. For example, we have always required developers to limit permission requests to only what is needed for their app to function and to be clear with users about what data they access," Paul Bankhead, director of product management for Google Play wrote in a post.
Starting November 1, 2018, Google Play will require any updates to existing apps to target API level 26 or higher. This is to ensure apps are built using the latest APIs and therefore provide the latest security and performance capabilities, according to the company.
In addition, the company is making changes to SMS and call log permissions. Google Play will now limit the types of applications that are able to ask for permission to access a user's phone. "The trust of our users is critical and together we'll continue to build a safe and secure Android ecosystem," Bankhead wrote.
Android, APIs, developer, Gmail API, Google, Project Strobe
SD Times news digest: GitHub for Android beta now available, Windows Terminal preview 0.8, and WhiteSource announces support for GitLab Ultimate
SD Times news digest: Visa acquires API startup Plaid, Git 2.25, and Nintex's RPA capabilities
Google files its opening Supreme Court brief in the Oracle Java lawsuit
SD Times news digest: Facebook fined again for Cambridge Analytica scandal, Google's privacy commitment and Visual Studio 2019 for Mac version 8.4 preview 4


Every summer, Mama, Sister, and Sarah Marie take the bus down south to visit Grandmama. The three of them sit in the back of the bus, because, as Mama says, it is the best seat Throughout the summer, Aunt Maria teaches Sarah Marie how to read. then Sarah Marie notices signs in town she hadn't been able to read before, like the one on the bathroom door that says, "White Women" and another that says "Colored Women." Sarah Marie faces a hard realization about the segregated South. but in the fall she reads about events happening in places like Clinton, Tennessee, and Montgomery, Alabama. and by the next summer, when they go back to visit Grandmama, they all sit in the front of the bus.


Vertical Riser pole « Portable Staging NexGen – Manufacturers of Portable Staging for schools, choirs and events.
These stage parts fit into the black plastic node block on each riser set and wraps around the black nylon guide tube.
These can be dislodged and bent from time to time. Please make sure to check occasionally to stop these bending. Opening the riser whilst on the floor makes a big difference, watch our tutorial videos to help stop these stage parts from getting damaged.
I always keep a small selection of stage parts to hand when travelling to sing in various venues. It is real easy to swap out a broken or lost part in seconds, and bring the staging legs back to full working order.
Made of aluminium, these tubes allow the riser to open and close properly. They work in tandem with the nylon guide tubes.
Sliver Aluminium tube. Screw fitted to the location pillar or flat black screw, both in the square nylon block. DIY self fit to the other corresponding parts on the riser leg.


If you need a hazmat evaluation of your property in Rancho Cucamonga, there's no better choice than AQHI, Inc. We thoroughly check residential and commercial properties for contaminants such as lead, asbestos, mold, and fungus. We also test indoor air quality to make sure the air you're breathing each day is free from hazardous materials.
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We check the soil, water, air, and every other molecule of your home—we even have a tool that checks the moisture levels of the air inside your walls. We're Rancho Cucamonga's hazmat experts, so when you need the job done right, go with AQHI, Inc.
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Our hazmat experts will arrive at your home, conduct a thorough investigation for contaminants, and then propose a remediation plan for getting rid of any lead, asbestos, mold, or fungus found in your property. We always follow federal and state laws to the letter, and often exceed standards set by law. Safety is #1 at AQHI, Inc, and we are a fully licensed and insured hazmat company. We work in Rancho Cucamonga and throughout Southern California, and people have enlisted us to take care of all hazmat needs in commercial and residential properties for over 27 years. Jump on the bandwagon by scheduling an appointment by calling 800-481-8080 or emailing Sj@aqhiinc.com today! Find out what makes working with AQHI, Inc such a peaceful, stress-free experience.
AQHI, Inc is a local company that serves Rancho Cucamonga and many other areas of Southern California. When you decide to hire us to conduct your hazmat evaluation and remediation services, you will be treated respect from the moment you call or email to the second our company trucks pull out of your driveway. At AQHI, Inc, you're not just any customer—you're a neighbor, a fellow citizen of our community, and we treat you with the courtesy and compassion you deserve.
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text:
#include <babylon/cameras/arc_rotate_camera.h>
#include <babylon/engines/scene.h>
#include <babylon/helpers/environment_helper.h>
#include <babylon/interfaces/irenderable_scene.h>
#include <babylon/loading/plugins/gltf/gltf_file_loader.h>
#include <babylon/loading/scene_loader.h>
#include <babylon/samples/babylon_register_sample.h>

namespace BABYLON {
namespace Samples {

/**
 * @brief Suzanne Scene (glTF). Suzanne from Blender with smoothed faces.
 * @see https://doc.babylonjs.com/how_to/load_from_any_file_type
 * @see https://github.com/KhronosGroup/glTF-Sample-Models/tree/master/2.0/Suzanne
 */
struct SuzanneScene : public IRenderableScene {

  SuzanneScene(ICanvas* iCanvas) : IRenderableScene(iCanvas)
  {
    GLTF2::GLTFFileLoader::RegisterAsSceneLoaderPlugin();
  }

  ~SuzanneScene() override = default;

  const char* getName() override
  {
    return "Suzanne Scene (glTF)";
  }

  void initializeScene(ICanvas* /*canvas*/, Scene* scene) override
  {
    SceneLoader::ImportMesh({}, "glTF-Sample-Models/2.0/Suzanne/glTF/", "Suzanne.gltf", scene,
                            [scene](const std::vector<AbstractMeshPtr>& /*meshes*/,
                                    const std::vector<IParticleSystemPtr>& /*particleSystems*/,
                                    const std::vector<SkeletonPtr>& /*skeletons*/,
                                    const std::vector<AnimationGroupPtr>& /*animationGroups*/,
                                    const std::vector<TransformNodePtr>& /*transformNodes*/,
                                    const std::vector<GeometryPtr>& /*geometries*/,
                                    const std::vector<LightPtr>& /*lights*/) {
                              scene->createDefaultCameraOrLight(true, true, true);
                              // Set the camera position
                              auto camera
                                = std::static_pointer_cast<ArcRotateCamera>(scene->activeCamera());
                              if (camera) {
                                camera->setTarget(Vector3::Zero());
                                camera->alpha  = 0.9f;
                                camera->beta   = 1.f;
                                camera->radius = Math::PI2 * 0.75f;
                              }
                            });
  }

}; // end of struct SuzanneScene

BABYLON_REGISTER_SAMPLE("Loaders - glTF format", SuzanneScene)

} // end of namespace Samples
} // end of namespace BABYLON

redpajama_set_name: RedPajamaGithub


Dr Shah moved to the Goulburn Valley in 2011 after completing his dental training in the UK. Dr Shah enjoys all forms of general and cosmetic dentistry. His compassionate and practical approach is refreshing and make dental appointments more pleasant.
Sarah completed her qualification in the UK and has been in the Goulburn Valley since 1992. Sarah has a particular focus on preventive treatment and education for a future of healthier teeth and gums!
Tracy brings a wealth of knowledge in Dental Assisting and practice management to the team. She cares deeply for all our patients and ensures that your visits to us are pleasant and comfortable.
Lauren has been part of our team since 2016, she has quickly grown to love her position as a dental assistant.


Hey everybody we're going to go with some traditional CTF this month. The wad we'll be using is zdmegactf-1f, please visit the link below for more information on what maps we'll be playing. Players interested in competing can sign up by visiting the registration link below. If you have any questions please contact Dannyboy in the #zd-tourneys channel. Registration ends on the 7th of May. All team captains are required to post their teams roster in this thread!
DB4D will be attending ladies.
[DH]-Blue: dew, Nevan, ModeneseAdriano, ?
uh uh yo yo! pick me alpha I want to be in that ______ slot !!
Fatal Insolation will be playing as well!
COLORS ARE SO MUCH FUN !
Posted: Tue Apr 30, 2013 10:54 pm Post subject: Te BruTe Force!!!
So Cataclysm will be playing for two teams at a time?
No it's not possible, his team informed me they wouldn't be participating.
Brackets are loaded, feel free to play your games early good luck to all the teams participating and more importantly have fun!
So uhhh Vision, get your team ready, we're usually, always online, just waiting on for you, Te.
Switching out [FI]SiguR for [FI]Avc69 due to Schedule issues.
is the new roster for our team.
FI came in round 1 really strong and beat us pretty good and showed us they really wanted to win.
Our team had to focus in round 2 and 3 to barley give us the win.
These matches gave our team a really good warm up for the next coming games.
FI truly has lots of potential i hope to see them continue to play in pub and in future tournaments was fun games GG Folks!


text: Q: Non-associative algebras and determinant over 3 by 3 matrices I have a non associative algebra $A$ that is not unital. And I have three by three matrices $X$ with coefficients over $A$ with a matrix product $*$. I'd like to define something like the determinant of the matrix. Do you know any definition that goes in that direction?

redpajama_set_name: RedPajamaStackExchange


text:
The Original Yoga
Shyam Ghosh (Author)
Synopsis Very little is known about the author of this book apart from the facts that he is a retired Government of India officer, now in his late nineties, apparently hoary, but healthy. When requested for more bio-data, he wrote back: The real author of the Original Yoga is the Lord Siva. In the mundane world, Patanjali is the prime propagator of yoga. Any other claim to authorship, therefore, cannot but be spurious. It is truth that matters, not the utterer of the truth. No truth ever becomes a bit truer even when repeated by the most distinguished person. Hence all genuine aspirants of yoga should be content to remain unknown and unnoticed by others. His other published work is Hindu Concept of Life and Death.
Notify me when the book will be in stock
Shyam Ghosh
Shyam Ghosh (1904-2000) was a Vedic research scholar. He had studied other Hindu scriptures as well, some of which he had translated and published, including an exhaustive commentary on Patanjali's Yogasutra. Having boycotted his school at the call of Mahatma Gandhi in 1920, Shyam Ghosh joined Freedom Movement and participated in meetings and rallies organized by Pt. Jawaharlal Nehru and other leaders. Later, after a few years of self-education, he worked his way to the editorial staff of three national daily newspapers before being selected as the editor of the AIR journals – The Indian Listener, Awaz and Saran (new defunct). After a few years he was shifted to the Publications Division of the same Ministry as Deputy Director, and then on to the finance Ministry, where he worked as Financial Advisor to the Director-General of Supply and Disposal. He had always been averse to personal publicity, "An author should be judged by his work, and not by his personal qualifications." Was his firm view. Some of this other works are, the Original Yoga (1980, 1999) and Hindu concept of Life and Death (1989, 2002).
Rgveda for the Layman: Satasuktaparidarsanam
The Original Yoga: As Expounded in Sivasamhita, Gherandasamhita, and Patanjala Yogasutra
The Essence of Yoga: Reflections on the Yoga Sutras of Patanjali
Encyclopaedia of Traditional Asanas
Tantra, Mantra, Yoga & Spiritual Bliss
Mantravalli: An Anthology of Sacred Chants
Title	The Original Yoga
Author	Shyam Ghosh
Edition	Reprint
Publisher	Munshiram Manoharlal Publishers Pvt. Ltd.
length	xxiv+262p., Glossary; Index; 23cm.
Subjects Health, Mind & Body > Yoga


text:
est une chanson du groupe anglais Gorillaz, en collaboration avec le groupe de rap américain De La Soul. C'est le premier single tiré de l'album Demon Days sorti en 2005.

Succès commercial
Feel Good Inc a atteint la  dans les classements britanniques tandis qu'aux États-Unis la chanson s'est classée  dans le Billboard Hot 100, la seule du groupe à atteindre le top 20 dans ce classement. Elle s'est classée en tête d'un autre classement établi par Billboard, le Hot Modern Rock Tracks, pendant huit semaines consécutives, une autre première pour le groupe. Elle s'est également inscrite au top 10 de quatorze pays dont l'Espagne où elle a atteint la .

La chanson a été inscrite à la  le Pitchfork Media's Best Songs pour la décennie 2000-2010. On peut considérer que cette chanson est le premier tube planétaire de Gorillaz, ce qui confirma non seulement que Damon Albarn avait encore sa place dans l'univers musical après des années de recherche personnelle dues à la dissolution de son premier et fameux groupe Blur, mais également qu'il faudrait apprendre à compter désormais avec ce groupe parmi les grandes figures de la musique contemporaine. Un EP est également sorti au Japon sous le titre de Feel Good Inc. EP.

Clip vidéo

Le clip commence dans une sorte de tour de concert ou on peut voir Murdoc, Russel et 2D donner concert. Ensuite 2-D s'approche de la fenêtre et aperçoit Noodle sur son île au moulin flottante jouer de la guitare acoustique. À la fin, on voit l'île s'éloigner suivie par deux hélicoptères de chasse.

Liste des titres
 CD
 Feel Good Inc
 Spitting Out The Demons

 DVD
 Feel Good Inc (vidéo)
 Spitting Out The Demons
 Bill Murray

 Édition vinyle collector
 Feel Good Inc
 68 State

 iTunes EP
 Feel Good Inc (animatic)
 Feel Good Inc (live In Harlem)
 Feel Good Inc (Noodle's demo)
 68 State

Personnels

Damon Albarn – écriture, production, mixage
David Jolicoeur – écriture
De La Soul – voix supplémentaire
Simon Tong – guitare additionnelle
Morgan Nicholls – guitare basse
Danger Mouse – production, mixage, programmation boite à rythmes
Jason Cox – production additionnelle, mixage, ingénieur du son
James Dring – production additionnelle, programmation boite à rythmes
Howie Weinberg – mastering
Jamie Hewlett – artwork, design , animation
Pete Candeland – animation
Passion Pictures – production
Rushes – post production
Chu-Li Shewring – sound design
Sebastian Monk – sound design

Classements hebdomadaires

Certifications

Notes et références

Chanson interprétée par Gorillaz
Single musical sorti en 2005
Single certifié double platine au Royaume-Uni
Single certifié or aux États-Unis
Single certifié or en Nouvelle-Zélande
Single certifié platine en Australie
Single certifié platine au Danemark
Single certifié platine en Italie
Single numéro un en Espagne
Single numéro un aux États-Unis

redpajama_set_name: RedPajamaWikipedia


Resolve Optics is an organisation specialising in OEM custom lens assemblies. The company delivers thousands of custom lens assemblies per year – from complex high- speed imaging lenses and material sorting inspection lenses, to radiation resistant lenses for critical medical and nuclear applications.
Over the last 25 years, Resolve Optics have built a world class, quality accredited global supply chain that ensures they can obtain unprecedented quality at competitive prices. In addition, Resolve Optics has invested in an array of in-house testing equipment to insure suppliers meet their demanding requirements.
While some outsourcing companies specialising in custom optics promise low cost and high quality, few can additionally offer the in-depth innovative optical design and technical support that Resolve Optics is internationally renowned for.
Mark Pontin – Managing Director of Resolve Optics commented "We have successfully developed many innovative and custom OEM products for our industry partners over the last 25 years. We welcome the OEM design and manufacture opportunity to develop, design or manufacture special lenses and optical products. Drawing upon our experienced team of optical designers we are able to quickly gain an understanding of the basic physics of the technology associated with each customer's product enabling us to propose novel solutions. Our aim is to provide a fast and flexible optical and mechanical design assessment on all projects. A project engineer is assigned to closely liaise with customers at all points of a development to ensure complete satisfaction with the final product".
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text:
Toggle section links
Medicine & veterinary medicine
Scrapie resistance impacts farming
Life-changing stroke research
Identifying a new bovine disease
Compression stockings for stroke patients
Preventing deaths from pesticides in Asia
Reducing blood transfusions
Removing virus from salmon farms
Enabling women to have children after cancer
Decreasing child pneumonia mortality
Dolly the Sheep
GRACE: assessing heart attack risk
Medicine and veterinary medicine research
Summaries and case studies of the world-leading research carried out in the College of Medicine & Veterinary Medicine.
The College of Medicine & Veterinary Medicine is an internationally leading force in basic-to-clinical translational research. It has a longstanding commitment to 'one health', enabling the translation of expertise from human to animal medicine and vice versa.
College research summary
Pioneering work by Professor Nora Hunter and Dr Wilfred Goldmann has had an international impact on a disease that causes considerable economic hardship to the small ruminant farming industry.
Professor Gillian Mead and Dr David Saunders' life-changing research influenced stroke rehabilitation policy and practice, both in the UK and internationally.
University scientists successfully collaborated to develop effective disease controls and treatments for a new disease, Bovine Neonatal Pancytopenia (BNP), which affects calves.
Compression stocking research improves care of stroke patients
The care of stroke patients has been improved after Edinburgh research showed that the routine use of graduated compression stockings was ineffective in preventing deep vein thrombosis (DVT).
Research focusing on the prevention of deaths from self-poisoning helped to improve medical management and impacted on pesticide policies in Sri Lanka.
Reducing blood transfusions cuts deaths, risks and costs
Edinburgh researchers have discovered benefits from reducing the use of blood transfusions and have been instrumental in the adoption of new practices that have saved money, risks and ultimately patients' lives.
Breeding salmon for resistance to infectious pancreatic necrosis
Researchers have found a new way to identify salmon that are resistant to infectious pancreatic necrosis, benefiting the fish farming industry in Scotland and globally.
Enabling women to have children after cancer treatment
Edinburgh researchers have developed new procedures and protocols for the freezing of ovarian tissue to allow women to have children after cancer therapy.
Paving the way to decreased child pneumonia mortality
Research into the prevalence of childhood pneumonia and the relatively low attention it has historically received has driven new global action and helped produce a dramatic reduction in childhood deaths.
Dolly the Sheep: the first cloned adult animal
The process pioneered at Edinburgh to produce Dolly the Sheep, the first mammal cloned from an adult cell, has been taken up widely, resulting in the cloning of many agricultural and other animals worldwide.
The GRACE risk score: assessing heart attack risk and guiding treatment
A 10-year research programme resulted in the Global Registry of Acute Coronary Events (GRACE) and the GRACE Risk Score, which has saved lives by helping doctors better manage the treatment of heart patients.
This article was published on 30 Mar, 2016


We currently have an ongoing special that if you Pre-Pay for your unit for 1 year then you will get the 12th month Free. We happy accept Cash or Checks for Payment options.
Many of the perks to Bismarck Storage LLC / Property 94 is that we have an onsite manager at the facility.
Having an onsite manager can benefit you in many ways. One being that if you find yourself needing a lock cut or you need to replace one, we are there to help you. Making payments is as easy dropping it off or we have a 24 hour secure drop box just left of our office door.
Are your storage needs different? We can certainly find the right storage solution to fit your needs. Bismarck Storage LLC/ Property 94 has storage in a wide variety of sizes and features to satisfy your unique storage requirements. Basic units and even indoor climate control for sensitive items with onsite management, video, and additional security features.
Location Location Location….. Did we say Location?
We have 3 locations to serve your needs. We have both indoor climate controlled units and outdoor storage options at our Industrial Drive location. For those closer to Lincoln we have a spot for you as well at our Cavalry Ave Storage Units. If your near the Wal-Mart South we have our Morrison Ave location for your needs.
We recommend you contact your Auto Insurance Agent about Rental Storage Policies to keep your goods insured and covered.
Storage Insurance is a Must have!!!


text:
2017	2016	2015	2014	2013	2012	2011	2010	2009	2008	2007	2006	2005	2004	2003	2002	2001	2000	1999	1998	1997	1996	1995	1994	1993	1992	1991	1990	1989	1988	1987	1986	1985	1984	1983	1982	1981	1980	1979	1978	1977	1976	1975	1974	1973	1972	1971	1970	1969	1968	1967	1966	1965	1964	1963	1962	1961	1960	1959	1958	1957	1956	1955	1954	1953
The #1 song in the UK on July 6, 1962
Come Outside - Mike Sarne with Wendy Richard
Problem with Come Outside by Mike Sarne with Wendy Richard
Other #1 songs on July 6, 1962
The Stripper - David Rose
Notable July 6th Birthdays in the 1960s


Now, one evening there was a community supper in the big activity center. These two were at the same table, across from one another. As the meal went on, he made a few admiring glances at her and finally gathered up his courage to ask her, "Will you marry me?"
After about six seconds of 'careful consideration,' she answered. "Yes.
The meal ended and with a few more pleasant exchanges, they went to their respective places. Next morning, he was troubled. Did she say 'yes' or did she say 'no'? He couldn't remember. Try as he would, he just could not recall. Not even a faint memory.
With trepidation, he went to the telephone and called her. First, he explained to her that he didn't remember as well as he used to. Then he reviewed the lovely evening past. As he gained a little more courage, he then inquired of her, "When I asked if you would marry me, did you say 'Yes' or did you say 'No'?"
He was delighted to hear her say, "Why, I said, 'Yes, yes I will' and I meant it with all my heart." Then she continued, "And I am so glad that you called, because I couldn't remember who had asked me."


text:
Q: Reference Request: Fluid dynamics/Elasticity via Lagrangians Would there be a book that does what Landau does in Fluid Mechanics and Theory of Elasticity using Lagrangian's/Action-principles, analogous to the presentation in Landau's mechanics?
I have only found brief mentions of a Lagrangian in Fluid mechanics, e.g. Stone and Lanczos, and nothing useful for elasticity.
Would sincerely appreciate a reference, thanks.
References:

*

*Stone - Mathematics for Physics: A Guided Tour for Graduate Students, p. 25.

*Lanczos - The Variational Principles of Mechanics, p. 360.

A: You should check out Marsden and Hughes's Mathematical Foundations of Elasticity.
It is a book that requires a lot of work (I am going through it myself and it is not easy!), using the very general framework of tensors on manifolds and functional analysis, but it is an invaluable resource.
Two chapters in that book (5. Hamiltonian And Variational Principles and 6. Methods Of Functional Analysis In Elasticity) are particularly relevant to your question.


Traditional Greek breakfast in Mani at Porto Kale guesthouse.
In Mani, breakfast is considered as the most important meal of the day. A good and healthy breakfast provides vitality and energy for a whole day full of activities and leisure. Therefore Porto Kale guesthouse places big value on local and fresh products. All the food served is prepared by the owner and his family for each room individually. We offer seasonal fruits and juices, organic eggs, tasty Greek yoghurt with local thyme honey, home-made jellies and jams with a choice of fresh bread as a standard to all our customers. We know that every guest is different and not everybody enjoys the same daily rhythm. This is why we offer our Greek traditional breakfast at any time desired by our guests, served to their rooms. Enjoying your first meal of the day on your private veranda in the cool shade of an eucalyptus tree while overlooking the sea, will create the perfect start for every day of your holidays in Mani. We take pride in offering this individual service to each of our guests regardless of the duration of their stay.


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text:
CPI(ML): Wrong Turn in West Bengal: A Political Commentary
THE general secretary of the CPI(ML), Dipankar Bhattacharya, has been airing his views in the media for some time, about the tactics that the Left should adopt for the West Bengal assembly election.
He has been criticising the CPI(M) and the Left Front for equating the BJP and the Trinamool Congress and for calling for a fight against both the BJP and the TMC.
He has, in an interview to The Hindu, dated March 8, said that the political line taken by the CPI(M)-led alliance is "narrow, short-sighted, suicidal". He has also said that the Left Front is "underestimating the BJP while targeting Mamata Banerjee and the TMC". As against this "erroneous" line, the CPI(ML) has decided to distance itself from the CPI(M)-led front and go it alone. According to reports, the CPI(ML) will fight in 12 seats, support the Left Front in probably 12 to 15 seats and in the rest, campaign for the defeat of the BJP, thereby indicating that in over 200 seats, their support is for the TMC.
Unlike what the CPI(ML) alleges, the CPI(M) and the Left Front have not equated the BJP and the TMC. There is no underestimation of the threat posed by the BJP in West Bengal. The Hindutva forces are on the offensive in the state and the Lok Sabha election results of 2019 showed the advance and the danger posed by them. However, this does not mean that the Trinamool Congress and its brutal regime can, in any way, be considered to be a partner or ally in the fight against the "fascist offensive" that the CPI(ML) general secretary talks about.
It is necessary to have a proper understanding of the role the Trinamool Congress has played in West Bengal and the character of the Mamata Banerjee government. The TMC was the spearhead for the anti-Communist and anti-Left forces in West Bengal. By fascistic violence and lumpen politics, it targeted the CPI(M) and the Left Front and sought to physically suppress the Left movement. In the past decade, over 220 cadres and supporters of the CPI(M) and the Left have been killed, thousands were driven out of their homes, over a lakh false cases instituted and in many parts, a regime of terror established. The TMC began its political career as an ally of the BJP and as late as 2014, Narendra Modi saw the TMC as a potential ally. It is the establishment of TMC rule and the suppression of the Left that facilitated the rise of the BJP. Unlike what the CPI(ML) believes, `fascism' cannot be fought by an anti-democratic, lumpenised party.
It will be a mistake to consider the TMC as similar to the Rashtriya Janata Dal in Bihar, or, the Samajwadi Party in Uttar Pradesh. In the name of fighting fascism, you cannot sup with the devil. The TMC and the Mamata government have been opposing and unleashing repression on the general strikes of the working class and the struggles of different sections of the working people, even when they are directed against the policies of the Modi government such as the November 26 general strike. A tragic example of the brutal repression was the death of a young man, Maidul who died due to injuries suffered due to severe police beating during a youth-student protest rally.
The CPI(ML) is underestimating the depth of popular discontent against the TMC regime. This is a state assembly election. If the Left Front will not firmly, vigorously and unequivocally fight for the defeat of the TMC government, a large part of this popular discontent will be garnered by the BJP. Even in the Lok Sabha election of 2019, a bulk of people who were suffering under the TMC's autocratic regime, voted for the BJP, as they considered the Left Front too enfeebled to effectively fight the TMC.
The situation in West Bengal today is that even to rouse the people against the BJP's evil designs, it is necessary to furnish your anti-TMC credentials to get their attention.
What electoral tactics the CPI(ML) has spelt out in West Bengal is not going to help the fight against the BJP. Mamata Banerjee has expressed her thanks to various parties for extending support to the TMC and the list includes the general secretary of the CPI(ML). It is unfortunate that the CPI(ML) is perversely considering the TMC as a better political force to fight the BJP than the Left Front in West Bengal.


Now every time I put on a dark lip, it makes me think of Prudence — it is a bit harder to separate the two. A dark lip makes me feel instantly edgy, but I guess I tend to not move towards darker colours as often because of her. Dark reds I like a lot, but purples and blacks remind me of Prudence, so I try to stay away from them.


text:
Home Education Why some Americans refuse to social distance and wear masks
Why some Americans refuse to social distance and wear masks
Illustration by Leigh Wells/Harvard Gazette
By Colleen Walsh
Harvard Staff Writer/The Harvard Gazette
Wearing a mask and social distancing are two important barriers to COVID-19 infection, public health experts and government officials say. But adoption of the seemingly simple precautions has become a cultural battleground in a country where individual rights have long been considered part of the nation's founding ethos. And yet, what do we owe our fellow citizens in the age of a deadly virus with no vaccine that has already taken close to 170,000 lives, and will likely take thousands, and potentially hundreds of thousands, more? In his popular course "Justice," Michael Sandel, Harvard's Anne T. and Robert M. Bass Professor of Government, regularly asks his students to take on such thorny ethical questions. He will do so again this fall with his new class, "Justice: Ethics in an Age of Pandemic and Racial Reckoning." The Gazette recently asked Sandel for his thoughts on the subject.
Michael Sandel
GAZETTE: So many people have chosen not to wear masks or to keep their distance from others, despite the data that those simple measures help save lives. Why is there such resistance?
SANDEL: The wearing of masks has become a flashpoint of partisan disagreement, a new front in the culture wars. On one level, this seems puzzling. Why doesn't everyone simply wear a mask for the sake of public health? For two reasons: First, many Americans consider mask mandates a violation of individual liberty. They don't want the government to force them to wear a mask. Second, many Americans resent governing elites who claim to derive their authority from science. Here, the debate over masks is similar to the partisan disagreement on climate change. Many supporters of President Trump share his resentment of elites and experts. They don't trust experts who tell them they should wear a mask to reduce the spread of the coronavirus any more than they trust experts who say they should pay a carbon tax to alleviate climate change. The resistance to wearing a mask is not about public health; it's about politics.
"The resistance to wearing a mask is not about public health; it's about politics."
Stephanie Mitchell/Harvard file photo
GAZETTE: What are our ethical obligations in the middle of a pandemic?
SANDEL: Our ethical obligations are, first of all, to minimize the possibility that our behavior will expose others to the risk of contracting the virus. This means wearing masks and social distancing. Beyond this, those of us who are fortunate enough to work from the safety of our homes have a responsibility to support those who take risks on our behalf — not only doctors, nurses, and hospital workers, but delivery workers, grocery-store clerks, maintenance workers, child-care workers, home health care workers. This support should take the form of public appreciation for such workers, but also tangible, material support, such as health care, paid sick leave, and wage support.
GAZETTE: There are periods in American history — like World War II — when nearly everyone spent years voluntarily sacrificing in service to a national cause. Why isn't that happening now? What has changed?
SANDEL: The question reminds me of an internet meme early in the pandemic: "Your grandparents were called to war. You are called to sit on your couch. You can do this." Even as the pandemic highlights our mutual dependence, it is striking how little solidarity and shared sacrifice it has called forth. Why do we seem incapable of solidarity at the time we need it most? The answer goes back to the social unraveling that preceded this crisis. The pandemic caught us unprepared — logistically and medically, but also morally unprepared. It arrived at a time of deep polarization and partisan rancor. Four decades of deepening inequality have driven us apart. Resentment of the elites whose policies produced these inequalities led to a populist backlash. The pandemic arrived at just the wrong moment — amid toxic politics, incompetent leadership, and fraying social bonds.
GAZETTE: How might Americans be the same or different from people in other societies in taking these types of individual steps to help society as a whole? Are there examples from COVID that you can point to?
SANDEL: Sadly, the U.S. has handled the COVID crisis worse than most affluent countries. We've had more than 4½ times the number of deaths per capita as Germany, 60 times the number in Japan, and more than 80 times as many as South Korea. Our poor performance is due partly to failed leadership at the federal level and the lack of an adequate public health system. But I think you are right to suggest that social habits and cultural factors come into play. Our ardent individualism is a strength in some settings but not in contending with a pandemic. In Japan, the ready acceptance of face masks seems to have reduced the spread of the virus. In Germany, the electorate supported far greater investment in wage replacement and public health spending than the U.S. government could muster. In South Korea, a single-payer health system made testing easier and more widespread. In addition, a social movement among landlords to reduce or freeze rents helped keep small businesses afloat.
"This moment makes vivid the need for a broader public debate about the inequality this crisis has highlighted and a reconsideration of what we owe one another as citizens."
GAZETTE: What role do leaders have in supporting this social contract?
SANDEL: Trust matters in a pandemic — not only trust in the scientific information and medical advice the government provides, but trust among citizens. Perhaps the single greatest responsibility of leaders in times of crisis is to inspire such trust. Angela Merkel in Germany and Jacinda Ardern in New Zealand are examples of leaders who have led their countries through the crisis effectively, in part by fostering trust. In this country, by contrast, we've seen how evading responsibility and sowing discord undermines the trust and solidarity we need to contend with the pandemic.
GAZETTE: What does this say, when we are all so culturally siloed, about the future of shared American values and what we owe to each other?
SANDEL: From the outset of this crisis, we've heard the slogan, "We are all in this together." We hear it from politicians, advertisers, and celebrities. It evokes our mutual dependence and vulnerability in the face of COVID-19. It points to an inspiring ideal. But it rings hollow, because we know it doesn't describe the facts on the ground. Some of us work from home and hold meetings on Zoom. Others have little choice but to risk their health and lives serving the public and delivering things to our doorsteps, enabling us to avoid risk. The people we now celebrate as "essential workers" are not the best-paid or most-honored workers in our society. This is a moment to ask how to reconfigure the economy to bring the rewards of such workers into better alignment with the importance of the work they do. This moment makes vivid the need for a broader public debate about the inequality this crisis has highlighted and a reconsideration of what we owe one another as citizens.
GAZETTE: What are some of the ethical questions you plan to take up in your fall semester, University-wide course "Justice: Ethics in an Age of Pandemic and Racial Reckoning"?
SANDEL: The course is an updated version of the "Justice" course I've taught over the years. The moral urgency of the pandemic and of this moment of racial reckoning prompted me to offer the course again, updated to include the ethical questions that confront us today. For example: Who should be first in line to get a coronavirus vaccine when one becomes available? Should we use cellphone tracking data to enforce stay-at-home orders? In deciding how to balance public health with reopening the economy, should we place a monetary value on human life? We will also take up current debates about racial injustice, including the question of reparations for slavery and racism. The course invites students to think through the fundamental moral and civic questions the crises of our day make unavoidable — about the meaning of a just society and our obligations to one another.
Interview was lightly edited for clarity and length.
(Reprinted with permission from the Harvard Gazette.)
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