text:
\section{Introduction}\label{sec:1}
Due to the emergence of quantum information science, the
characterization and quantification of quantum correlations in a
system have became one of the people's research focuses in the past
few decades \cite{nielsen-qcqi}. Nowadays, when we mention to
quantum correlations, two prominent lines of research on this
problem may immediately appear in our minds. The first one centers
around the entanglement-separability paradigm, under which various
forms of the entanglement measures have been proposed and
extensively studied \cite{rmp-en}. The second one is based on the
noncommutativity of operators in quantum mechanics, and along this
line people also presented a plenty of discord-like quantum
correlation measures \cite{rmp-qd}.

It is well accepted that entanglement is responsible for the
advantage of many quantum communication and computation tasks
\cite{Ekert,DiVincenzo,Pan}. It is also realized recently that
quantum discord \cite{qd}, which reveals quantum correlations from a
different perspective, plays a vital role in quantum protocols such
as the deterministic quantum computation with one qubit \cite{DQC1},
remote state preparation \cite{rsp}, quantum locking of classical
correlations \cite{qlock1,qlock2}, quantum state broadcasting
\cite{qsb}, and quantum state merging \cite{qsm1,qsm2}. Moreover,
the discord consumption has been linked to the quantum advantage for
extracting information via coherent interactions \cite{Gumile}, and
meanwhile, it can also be connected to entanglement via some
measurement processes \cite{qd-en}.

Although both can serve as physical resources for quantum
information processing, quantum discord and entanglement are in fact
fundamentally different in many aspects. The most prominent one
is that quantum discord may be increased by local operations
\cite{qd-in}, while entanglement can only be increased by coherent
operations. Moreover, when considering their evolution for open
quantum system, quantum discord and entanglement also exhibit
distinct singular behaviors. For instance, quantum discord is more
robust against decoherence than entanglement
\cite{robu1,robu2,robu3}, it is immune to certain quantum noises
during certain time intervals \cite{froz1,froz2,froz3,add3,add4},
and therefore exhibits the frozen behavior which is impossible for
entanglement. Quantum discord exhibits sudden changes (SCs) due to
the optimization procedure involved in its definition
\cite{sc1,sc2,sc3,add1,add2,sc4,sc5}, while entanglement undergoes
sudden death \cite{ESD} which does not exist for quantum discord. It
is also worthwhile to note that the sudden death of entanglement is
independent of the chosen entanglement measures, as it occurs
whenever the evolved state becomes separable.

We investigate in this work the singular behaviors of three kinds of
the geometric quantum discords (GQDs) for two non-interacting qubits
subjecting to the independent thermal reservoirs. We will show that
for both the two-sided and the one-sided reservoirs, there are SCs
being observed during the evolution of the GQDs, and the critical
times for their occurrence are strongly dependent on the choice of
the discord measures, which means that this phenomenon is the
combined result of the chosen discord measure and the quantum state
other than a property of the state itself. This statement is further
confirmed by the relativity of different GQDs, i.e., different GQDs
may impose different orderings of quantum states.

We arrange this paper as follows. In Sections \ref{sec:2} and
\ref{sec:3}, we recall the formula for the three GQDs, and present
the analytical solution for the master equation describing the
evolution of the system. Then in Section \ref{sec:4}, we give a
discussion of the SC phenomenon. Finally, Section \ref{sec:5} is
devoted to a summary.

\section{Distanced-based measures of the GQDs}\label{sec:2}
As we mentioned in Section \ref{sec:1}, the quantum discord in a
system can be quantified from different perspectives. We adopt in
this work three forms of the distance-based measures of GQDs
\cite{trace1,trace2,trace-dyn,trace-ana,hellinger,lqu,bures1,bures2,bures-ana}.
They are defined via the minimal distance from $\rho$ to $\Omega_0$,
i.e.,
\begin{eqnarray}\label{eq1}
 D_\alpha(\rho)=\min_{\chi\in \Omega_0} d_\alpha(\rho,\chi)
\end{eqnarray}
where $d_\alpha$ designates the different distance measures one used
in defining the GQDs. They are all well defined and can avoid the
problem for the GQD based on the Frobenius norm \cite{gqd-fro}. We
concentrate in the following on a bipartite system $AB$ described by
the density operator $\rho$, and the three GQDs are defined
respectively via the trace distance
\cite{trace1,trace2,trace-dyn,trace-ana}, the Hellinger distance
\cite{hellinger,lqu}, and the Bures distance
\cite{bures1,bures2,bures-ana}. To facilitate later description, we
call them the TDD, HDD, and BDD for brevity.

We first recall the corresponding measures for the trace distance,
the Hellinger distance, and the Bures distance between $\rho$ and
$\chi$, which are given respectively by
\begin{equation}\label{eq2}
\begin{split}
 & d_{\rm T}(\rho,\chi)=\parallel\rho-\chi\parallel_1,\\
 & d_{\rm H}(\rho,\chi)=\parallel\sqrt{\rho} -\sqrt{\chi}\parallel_2,\\
 & d_{\rm B}(\rho,\chi)=[2(1-\sqrt{F(\rho,\chi)})]^{1/2},
\end{split}
\end{equation}
where $\parallel\cdot\parallel_1$ and $\parallel\cdot\parallel_2$
denote respectively the trace norm and the Frobenius norm, while
$F(\rho,\chi)= [{\rm tr}(\sqrt{\rho}\chi\sqrt{\rho})^{1/2}]^2$
represents the Uhlmann fidelity.

Then, the TDD, HDD, and BDD can be defined respectively via the
minimal $d_{\rm T}(\rho,\chi)$, $d_{\rm H}(\rho,\chi)$, and $d_{\rm
B}(\rho,\chi)$ as
\begin{equation}\label{eq3}
\begin{split}
 &D_{\rm T}(\rho)=\min_{\chi\in\Omega_0}\parallel\rho-\chi\parallel_1, \\
 &D_{\rm H}(\rho)=2\min_{\sqrt{\chi}\in\Omega'_0}\parallel\sqrt{\rho}-\sqrt{\chi}\parallel_2^2, \\
 &D_{\rm B}(\rho)=\sqrt{(2+\sqrt{2})\min_{\chi\in\Omega_0}(1-\sqrt{F(\rho,\chi)})},
\end{split}
\end{equation}
where the set $\Omega_0$ is usually taken to be the
classical-quantum state $\Omega_0=\sum_k (\Pi_k^A\otimes
I_B)\rho(\Pi_k^A\otimes I_B)$, and $\Omega'_0 =\sum_k
(\Pi_k^A\otimes I_B)\sqrt{\rho}(\Pi_k^A\otimes I_B)$, with
$\{\Pi_k^A\}$ being the projection-valued measurements. Moreover,
the constants $2$ and $2+\sqrt{2}$ before min are introduced for the
normalization of $D_{\rm H}(\rho)$ and $D_{\rm B}(\rho)$ for the
two-qubit maximally discordant states.

The calculation of the TDD, HDD, and BDD is a hard task for general
$\rho$, and analytical results exist only for certain special
classes of states. First, for the two-qubit {\it X} state $\rho^X$
which only contains nonzero elements along the main diagonal and
anti-diagonal, the TDD is given by \cite{trace-ana}
\begin{eqnarray}\label{eq4}
 D_{\rm T}(\rho^X)=\sqrt{\frac{\xi_1^2\xi_{\rm max}-\xi_2^2\xi_{\rm min}}
                   {\xi_{\rm max}-\xi_{\rm min}+\xi_1^2-\xi_2^2}},
\end{eqnarray}
where $\xi_{1,2}=2(|\rho_{23}|\pm |\rho_{14}|)$,
$\xi_3=1-2(\rho_{22}+\rho_{33})$, $\xi_{\rm
max}=\max\{\xi_3^2,\xi_2^2+x_{A3}^2\}$, and $\xi_{\rm
min}=\min\{\xi_1^2,\xi_3^2\}$, with
$x_{A3}=2(\rho_{11}+\rho_{22})-1$.

Second, for the $2\times n$ dimensional state $\rho$, and the
decomposed $\sqrt{\rho}=\sum_{ij} c_{ij}X_i\otimes Y_j$, with
$\{X_i:i=0,1,2,3\}$ and $\{Y_j:j=0,1,\ldots,n^2-1\}$ constituting the
orthonormal operator bases for the Hilbert spaces $\mathcal {H}_A$
and $\mathcal {H}_B$, the HDD can be calculated as \cite{hellinger}
\begin{eqnarray}\label{eq5}
 D_{\rm H}(\rho)=2(1-||\bold{r}||_2^2-z_{\max}),
\end{eqnarray}
where $||\bold{r}||_2^2=\sum_j c_{0j}^2$, and $z_{\max}$ represents
the largest eigenvalue of the matrix $ZZ^\dag$, with $Z=(c_{ij})_{
i=1,2,3;j=0,1,\cdots,n^2-1}$.

Finally, although there is no analytic solution, the calculation for
the maximum of $F(\rho,\chi)$ can be simplified as \cite{bures-ana}
\begin{eqnarray}\label{eq6}
 F_{\max}(\rho,\chi)=\frac{1}{2}\max_{||\vec{u}=1||}
 \left(1-{\rm tr}\Lambda+2\sum_{k=1}^{n}\lambda_k(\Lambda)\right),
\end{eqnarray}
where $\lambda_k(\Lambda)$ denote the eigenvalues of
$\Lambda=\sqrt{\rho}(\vec{u}\cdot\vec{\sigma}\otimes
I_B) \sqrt{\rho}$ in non-increasing order, with
$\vec{u}$ being a unit vector in $\mathbb{R}^3$, and
$\vec{\sigma}=(\sigma^x,\sigma^y,\sigma^z)$ is the vector of the
Pauli operators.

From the above equations, one can note that even for the simple
\emph{X} state, there are optimization procedures involved for
obtaining the GQDs, and this may induce SC behaviors of $D_{\rm
\alpha}(\rho)$. We will discuss them explicitly in the following
text.

\section{Solutions of the model}\label{sec:3}
The central system we considered consists of two qubits (labeled as
$S=A,B$) with large enough spatial distance, and the direct
interaction between them can be ignored. We will discuss two
different cases: (i) both qubits $A$ and $B$ are embedded in their
own independent thermal reservoirs, and (ii) only qubit $A$ (or $B$)
is embedded in the thermal reservoir, while the other one is free of
noise.

For the qubit $S$ subject to the thermal reservoir, the evolution of
$\rho^S(t)$ is governed by the master equation \cite{master}
\begin{equation}\label{eq7}
 \frac{d\rho^S }{dt}=\frac{\gamma_S}{2}\sum_{k=1}^2 \left(2\mathcal {L}_k^S
                    \rho^S\mathcal {L}_k^{S\dag}
                   -\{{L}_k^{S\dag}\mathcal {L}_k^S,\rho^S\}\right),
\end{equation}
under the Markovian approximation. Here, $\gamma_S$ is the strength
of the damping rate, $\{\cdot,\cdot\}$ denotes the anticommutator,
while $\mathcal {L}_1^S =\sqrt{\bar{n}+1}\sigma_S^{-}$ and $\mathcal
{L}_2^S =\sqrt{\bar{n}}\sigma_S^{+}$ (with $\sigma_S^{\pm}$ being
the raising and the lowering operators) describe, respectively, the
decay and excitation processes of the qubit $S$, with rates depending on
the temperature which is proportional to the average thermal photons
$\bar{n}$ in the reservoir.

The master equation \eqref{eq7} can be solved analytically. For
convenience of later presentation, we define
\begin{eqnarray}\label{eq8}
  q_1^S=\frac{\bar{n}+(\bar{n}+1)p_S^2}{2\bar{n}+1},
  ~q_2^S=\frac{\bar{n}(1-p_S^2)}{2\bar{n}+1},
\end{eqnarray}
then one can obtain that $\rho^S(t)$ takes the following form
\begin{widetext}
\begin{eqnarray}\label{eq9}
\rho^S(t)=\left(\begin{array}{cc}
      q_1^S\rho^S_{11}(0)+q_2^S\rho^S_{00}(0)   &  p_S\rho^S_{10}(0) \\
      p_S\rho^S_{01}(0)  & 1-q_1^S\rho^S_{11}(0)-q_2^S\rho^S_{00}(0) \\
            \end{array}
\right),
\end{eqnarray}
\end{widetext}
where $\rho_{ij}^S=\langle i|\rho^S|j\rangle$ in the standard basis
$\{|1\rangle,|0\rangle\}$ expanded by the eigenvectors of the Pauli
operator $\sigma_S^z$, and the time-dependent factor
$p_S=e^{-(2\bar{n}+1)\gamma_S t/2}$. Moreover, when $\bar{n}=0$, the
reservoir is at zero temperature, and the solution of Eq.
\eqref{eq9} reduces to that given in Ref. \cite{Bellomo}.

From the analytical expression for the single-qubit reduced density
matrix $\rho^S(t)$, one can obtain the two-qubit density matrix
$\rho(t)$ for arbitrary initial state $\rho(0)$ by using the
procedure presented in Ref. \cite{Bellomo}. The diagonal elements
are given by
\begin{equation}\label{eq10}
\begin{split}
 &\rho_{11}(t)=q_1^A q_1^B \rho_{11}(0)+q_1^A q_2^B \rho_{22}(0)+q_2^A q_1^B \rho_{33}(0)\\
              &\hspace{13mm} +q_2^A q_2^B \rho_{44}(0),\\
 &\rho_{22}(t)=q_1^A [(1-q_1^B) \rho_{11}(0)+(1-q_2^B) \rho_{22}(0)]\\
              &\hspace{13mm} +q_2^A [(1-q_1^B) \rho_{33}(0)+ (1-q_2^B)\rho_{44}(0)],\\
 &\rho_{33}(t)=(1-q_1^A) [q_1^B \rho_{11}(0)+ q_2^B \rho_{22}(0)]\\
              &\hspace{13mm} +(1-q_2^A)[q_1^B \rho_{33}(0)+q_2^B \rho_{44}(0)],\\
 &\rho_{44}(t)=1-\rho_{11}(t)-\rho_{22}(t)-\rho_{33}(t),
\end{split}
\end{equation}
while the nondiagonal elements are given by
\begin{equation}\label{eq11}
\begin{split}
 &\rho_{12}(t)=q_1^A p_B \rho_{12}(0)+q_2^A p_B \rho_{34}(0),\\
 &\rho_{13}(t)=q_1^B p_A \rho_{13}(0)+q_2^B p_A \rho_{24}(0),\\
 &\rho_{14}(t)=p_A p_B \rho_{14}(0),~\rho_{23}(t)=p_A p_B \rho_{23}(0),\\
 &\rho_{24}(t)=p_A (1-q_1^B) \rho_{13}(0)+p_A (1-q_2^B) \rho_{24}(0),\\
 &\rho_{34}(t)=(1-q_1^A)p_B \rho_{12}(0)+(1-q_2^A)p_B\rho_{34}(0),
\end{split}
\end{equation}
and the other nondiagonal elements can be written directly by using
the Hermitian condition $\rho_{ij}(t)=\rho_{ji}^*(t)$.

For the special case of $p_A=p_B=p$, we obtain the two-sided
identical reservoir $\mathcal {E}_{AB}=\mathcal {E}_A\otimes\mathcal
{E}_B$, while for $p_A=p$ and $p_B=1$ ($p_A=1$ and $p_B=p$), it
reduces to the one-sided reservoir $\mathcal {E}_A$ ($\mathcal
{E}_B$).

\section{SC of the GQDs}\label{sec:4}
Based on the solutions in Eqs. \eqref{eq10} and \eqref{eq11} for the
system-reservoir coupling mode presented in Section \ref{sec:3}, we
begin to discuss decay dynamics of the three GQDs. We will show
that they exhibit distinct singular behaviors, which include the
completely different SCs and their relativity on
characterizing quantum correlations. To be explicit, we consider
initial two-qubit state of the following form
\begin{eqnarray}\label{eq12}
 |\Psi\rangle=\alpha|11\rangle + \beta|00\rangle,
\end{eqnarray}
where $\alpha\in[0,1]$, and $\beta=\sqrt{1-\alpha^2}$. The
analytical expressions of $\rho(t)$ for both the two-sided identical
reservoirs $\mathcal {E}_{AB}$ and the one-sided reservoir $\mathcal
{E}_A$ (or $\mathcal {E}_B$) can be written directly from Eqs.
\eqref{eq10} and \eqref{eq11}, which are of the \emph{X} form.

\begin{figure}
\centering
\resizebox{0.46\textwidth}{!}
\includegraphics{figure1.eps}}
\caption{(Color online) $\gamma t$ dependence of $D_{\rm T}(\rho)$
(black), $D_{\rm B}(\rho)$ (red), and $D_{\rm H}(\rho)$ (blue) for
the initial state $|\Psi\rangle$ subject to the two-sided reservoir
$\mathcal {E}_{AB}$ with $\bar{n}=0.6$. The other parameters are
given by $\alpha^2=0.3$ (a), $0.5$ (b), and $0.7$ (c). The hollow
circles denote the SC points.} \label{fig:1}
\end{figure}

We first consider the two-sided identical reservoir $\mathcal
{E}_{AB}$. Fig. \ref{fig:1} is an exemplified plot of the $\gamma t$
dependence of $D_{\rm T}(\rho)$, $D_{\rm B}(\rho)$, and $D_{\rm
H}(\rho)$ for the initial state $|\Psi\rangle$ with $\bar{n}=0.6$
and different values of $\alpha^2$. For this case, as
$\rho_{23}(t)=0$, the TDD can be obtained analytically as
\begin{equation}\label{eq13}
 D_{\rm T}(\rho)=2p^2\alpha\sqrt{1-\alpha^2},
\end{equation}
therefore it is symmetric with respect to $\alpha^2=0.5$, and decays
smoothly and monotonously with increasing $\gamma t$ for any
$\alpha^2$.

The BDD and the HDD are no longer the symmetric functions of
$\alpha^2=0.5$. As displayed in Fig. \ref{fig:1}, while both $D_{\rm
B}(\rho)$ and $D_{\rm H}(\rho)$ still decay monotonously with
increasing $\gamma t$, there are also SCs being observed, i.e., they
are nonsmooth functions of $\gamma t$. In particular, the critical
time $\gamma t_c$ for the SCs and the times of SCs are determined
strongly by the chosen discord measure and the form of the initial state.
For the chosen parameters in Fig. \ref{fig:1}, $D_{\rm B}(\rho)$ decays smoothly
for $\alpha^2=0.3$, while $D_{\rm H}(\rho)$ experiences double SCs at
$\gamma t_c\simeq 0.0115$ and $0.287$, respectively. For $\alpha^2=0.5$, the
single SC of $D_{\rm B}(\rho)$ ($\gamma t_c\simeq 0.478$) occurs
earlier than that of $D_{\rm H}(\rho)$ ($\gamma t_c\simeq 0.5115$),
while for $\alpha^2=0.7$, both $D_{\rm B}(\rho)$ and $D_{\rm
H}(\rho)$ exhibit double SCs, where the first one of $D_{\rm
B}(\rho)$ ($\gamma t_c\simeq 0.0404$) occurs shortly after $D_{\rm
H}(\rho)$ ($\gamma t_c\simeq 0.0066$), and the second one of $D_{\rm
B}(\rho)$ ($\gamma t_c\simeq 0.5985$) turns out to be a little bit
earlier than that of $D_{\rm H}(\rho)$ ($\gamma t_c\simeq 0.6113$).

\begin{figure}
\centering
\resizebox{0.46\textwidth}{!}
\includegraphics{figure2.eps}}
\caption{(Color online) $\gamma t$ dependence of $D_{\rm T}(\rho)$
(black), $D_{\rm B}(\rho)$ (red), and $D_{\rm H}(\rho)$ (blue) for
the initial state $|\Psi\rangle$ subject to the one-sided reservoir
$\mathcal {E}_{A}$ (solid) or $\mathcal {E}_{B}$ (dash-dotted) with
$\bar{n}=0.6$. The other parameters are $\alpha^2=0.3$ (a), $0.5$
(b), and $0.7$ (c). The hollow circles and squares denote the SC
points. Moreover, the lines of $D_{\rm T}(\rho)$, as well as the
lines of $D_{\rm B}(\rho)$ with $\alpha^2=0.3$, are overlapped for
the $\mathcal {E}_{A}$ and $\mathcal {E}_{B}$ cases.} \label{fig:2}
\end{figure}

Next we turn to discuss the cases of the one-sided reservoirs
$\mathcal {E}_{A}$ and $\mathcal {E}_{B}$. We displayed in Fig.
\ref{fig:2} the $\gamma t$ dependence of the GQDs by the solid and
the dash-dotted lines, respectively. As for these two cases, only
one of the two qubits is exposed to the reservoir, the decay of the
GQDs is slower than that for the two-sided reservoir case. First,
for both $\mathcal{E}_{A}$ and $\mathcal {E}_{B}$, the TDD is always
given by
\begin{eqnarray}\label{eq14}
 D_{\rm T}(\rho)=2p\alpha\sqrt{1-\alpha^2},
\end{eqnarray}
which is still a symmetric function about $\alpha^2=0.5$, and decays
smoothly and monotonously with the increasing $\gamma t$.

The BDD and the HDD may do not behave as smooth functions of $\gamma
t$. For the one-sided reservoir $\mathcal {E}_A$, they exhibit
qualitatively the same behaviors as those for the $\mathcal
{E}_{AB}$ case, and the only difference is that the critical times
for the SCs are all obviously delayed. For $\alpha^2=0.3$, the
double SCs for $D_{\rm H}(\rho)$ occur at $\gamma t_c\simeq 0.024$
and $0.536$, respectively. For $\alpha^2=0.5$, the single SC for
$D_{\rm B}(\rho)$ and $D_{\rm H}(\rho)$ occurs respectively at
$\gamma t_c\simeq 0.802$ and $0.816$. Finally, for $\alpha^2=0.7$,
the double SCs for $D_{\rm B}(\rho)$ occur at $\gamma t_c\simeq
0.0611$ and $0.966$, while for $D_{\rm H}(\rho)$ they occur at
$\gamma t_c\simeq 0.01045$ and $0.914$.

For the one-sided reservoir $\mathcal {E}_B$, although the evolved
density matrix differs only in $\rho_{22}(t)$ and $\rho_{33}(t)$
from the $\mathcal {E}_A$ case, the BDD and HDD are not exactly the
same (cf. the solid and the dash-dotted lines in Fig. \ref{fig:2}).
For $\alpha^2=0.3$, $D_{\rm B}(\rho)$ exhibits completely the same
$\gamma t$ dependence compared with that of the $\mathcal {E}_A$
case, while the double SCs of $D_{\rm H}(\rho)$ ($\gamma t_c\simeq
0.022$ and $0.652$) are slightly different. Moreover, the single SC
for both $D_{\rm B}(\rho)$ and $D_{\rm H}(\rho)$ disappears for
$\alpha^2=0.5$. Finally, for $\alpha^2=0.7$, the first SC of $D_{\rm
B}(\rho)$ ($\gamma t_c \simeq 0.1305$) and $D_{\rm H}(\rho)$
($\gamma t_c \simeq 0.027$) occurs later than that of the $\mathcal
{E}_A$ case, while their second SC ($\gamma t_c \simeq 0.822$ and
$0.724$, respectively) occurs earlier than that of the $\mathcal
{E}_A$ case.

The decay rates of the GQDs for $\mathcal {E}_A$ and $\mathcal
{E}_B$ may also be different. As showed in Fig. \ref{fig:2}, for
$\alpha^2 =0.5$, $D_{\rm B}(\rho)$ and $D_{\rm H}(\rho)$ for the
$\mathcal {E}_B$ case decay faster than those for the $\mathcal
{E}_A$ case in the whole $\gamma t$ region. For $\alpha^2 =0.3$ and
$0.7$, however, $\mathcal {E}_A$ and $\mathcal {E}_B$ give different
decay rates of $D_{\rm B}(\rho)$ and $D_{\rm H}(\rho)$ only during
limited $\gamma t$ regions. For $\alpha^2 =0.3$, $D_{\rm H}(\rho)$
for the $\mathcal {E}_B$ case decays slightly faster than that for
the $\mathcal {E}_A$ case when $\gamma t\in[0.022,0.652]$, while for
$\alpha^2 =0.7$, $D_{\rm B}(\rho)$ and $D_{\rm H}(\rho)$ for the
$\mathcal {E}_B$ case decay slower than those for the $\mathcal
{E}_A$ case when $\gamma t\in [0.0611,0.966]$ and $\gamma t \in
[0.01045,0.914]$, respectively.

\begin{figure}
\centering
\resizebox{0.46\textwidth}{!}
\includegraphics{figure3.eps}}
\caption{(Color online) $\gamma t$ dependence of $D_{\rm T}(\rho)$
(black), $D_{\rm B}(\rho)$ (red), and $D_{\rm H}(\rho)$ (blue) for
the initial state $|\Psi\rangle$ subject to the two-sided reservoirs
$\mathcal {E}_{AB}$ (a), one-sided reservoir $\mathcal {E}_{A}$ (b),
and $\mathcal {E}_{B}$ (c), all with the parameters $\bar{n}=0$ and
$\alpha^2=0.5$. The hollow circles denote the SC points.}
\label{fig:3}
\end{figure}

We discussed in the above evolution of the three GQDs, and observed
distinct singular behaviors such as the single and the double SCs
caused exclusively by the reservoir. We discuss in the following two
limiting cases, i.e., the zero temperature ($\bar{n}=0$) and the
infinite temperature ($\bar{n}\rightarrow \infty$) cases.

For the zero temperature case, $\mathcal {L}_2^S =0$, and the only
spontaneous decay term leads to a purely dissipative process. In the
long-time limit, this process drives the corresponding qubit to its
ground state $|1\rangle$, thus the GQDs disappear for arbitrary
initial state. In Fig. \ref{fig:3}, we showed the $\gamma t$
dependence of the three GQDs for $|\Psi\rangle$ with $\alpha^2=0.5$
and $\bar{n}=0$. For $\mathcal{E}_{AB}$, by comparing with Fig.
\ref{fig:1}(b), one can see that they exhibit very similar
behaviors, except that the decay rates are obviously decreased, and
the critical times for the SCs ($\gamma t_c \simeq 0.485$ and
$0.609$, respectively) are also slightly delayed. For
$\mathcal{E}_A$, there is also single SC for both $D_{\rm B}(\rho)$
($\gamma t_c \simeq 0.693$) and $D_{\rm H}(\rho)$ ($\gamma t_c
\simeq0.609$), which occurs earlier than those for the finite
temperature reservoir (cf. Fig. \ref{fig:3}(b) and Fig.
\ref{fig:2}(b)). Moreover, as showed by the blue line in Fig.
\ref{fig:3}(b), the HDD is increased with $\gamma t$ during the
region $\gamma t\in [0.482,0.609]$ (which is a reflection of the
fact that the quantum discord can increase under the local operation
on one party of the system \cite{nonmono}), while the TDD and the
BDD are always decreased. This means that different discord measures
may lead to different orderings of quantum states, and it confirmed
again that what the discord reveals is in fact the combined result
of the chosen discord and the quantum state other than a property of
the state itself. Of course, the increase for HDD is slight and
transient, and after $\gamma t> 0.609$, it decays to zero gradually.
Finally, for $\mathcal{E}_B$, the three GQDs still show
qualitatively the same $\gamma t$ dependence with those for the
finite temperature reservoirs, with however the decay rates are
evidently decreased.

For the infinite temperature case, we have $\mathcal {L}_1^S
=\mathcal {L}_2^S$, the decay and excitation processes occur at
exactly the same rate, and the noise induced by the transitions
between the two levels brings an arbitrary initial state into the
maximally mixed one. For concise of the paper, we do not present the
plots here. But the numerical results showed that for the initial
state $|\Psi\rangle$ with $\alpha^2=0.5$, all the three GQDs decay
smoothly and monotonously with the increasing $\gamma_0 t$ (
$\gamma_0=\bar{n}\gamma$), and there are no SCs being observed for
them.

Finally, as the SCs displayed in Figs. \ref{fig:1}, \ref{fig:2}, and
\ref{fig:3} are obtained via numerical methods, one may wonder
whether they are the real SCs or not, this is because sometimes it
is possible that what one observes as a SC might be the result of a
quick change that is actually not sudden when analyzed for smaller
time intervals \cite{add2}. For the TDD, as its analytical
expressions are given in Eqs. \eqref{eq13} and \eqref{eq14}, it is
evident that it does not experience SC for the model considered
here. For the HDD and BDD, as the square root of the density
operator $\rho$ cannot be derived analytically, analytical solutions
of $D_{\rm H}(\rho)$ and $D_{\rm B}(\rho)$ cannot be obtained. But
the changes observed in the three figures are actually exactly
sudden, and they are caused by the discontinuity of the optimal
angle related to measurement operators $\{\Pi_k^A\}$. In fact, by
writing $\Pi_{1,2}^A=(I_A\pm\vec{u} \cdot\vec{\sigma})/2$, and the
unit vector $\vec{u}=(\sin\theta\cos\phi, \sin\theta\sin\phi,
\cos\theta)$, we found that for the considered state in this paper,
both $D_{\rm H}(\rho)$ and $D_{\rm B}(\rho)$ are independent of the
angle $\phi$, but for the case of the single SC, the optimal
$\theta$ for both $D_{\rm H}(\rho)$ and $D_{\rm B}(\rho)$ is given
by $\pi/4$ before the SC point, and it changes abruptly to $\pi/2$
after the SC point. Moreover, for the case of double SCs, the
optimal $\theta$ changes abruptly from $\pi/2$ to $\pi/4$ and then
back to $\pi/2$ at the SC points. All these correspond to SCs of the
optimal $\{\Pi_k^A\}$, and therefore the changes observed for both
$D_{\rm H}(\rho)$ and $D_{\rm B}(\rho)$ are actually sudden.

\section{Summary}\label{sec:5}
In summary, we have investigated time evolution and the accompanying
singular behaviors of the TDD, BDD, and HDD. To focus exclusively on
the singular behaviors of them as caused solely by the thermal
reservoir, the two qubits of the central system are assumed to be
spatially separated far away from each other, and thus there are no
direct interactions between them, that is, every qubit interacts
with its own independent reservoir.

By solving analytically the master equation describing the evolution
of the two qubits, we analyzed dynamics of the three GQDs, and found
that they are incompatible in characterizing quantum correlations,
although they are all well defined from a geometric perspective. Our
findings are illustrated through two distinct behaviors of the three
GQDs. First, we found that the three GQDs may exhibit completely
different SCs for both the two-sided and the one-sided reservoirs.
The critical times for the SCs and the times of SCs are strongly
dependent on the choice of the GQD measure and the form of the
initial state. Different GQD measures have different SCs, and thus
the SCs are in fact the combined result of the chosen GQD measure
and the quantum state, but not the intrinsic property of a state
itself. This is fundamentally different from the sudden death of
entanglement, which is independent of the entanglement measure.
Moreover, we also revealed the relativity of different GQDs. To be
explicit, we found that the thermal reservoir may lead to a
generation of quantum states manifesting different orderings, and
this implies that different GQDs are incomparable as their behaviors
may not only be quantitatively but also be qualitatively different.

\section*{ACKNOWLEDGMENTS}
This work was supported by NSFC (11205121), and NSF of Shaanxi
Province (2014JM1008).

\newcommand{\PRL}{Phys. Rev. Lett. }
\newcommand{\RMP}{Rev. Mod. Phys. }
\newcommand{\PRA}{Phys. Rev. A }
\newcommand{\PRB}{Phys. Rev. B }
\newcommand{\PRE}{Phys. Rev. E }
\newcommand{\NJP}{New J. Phys. }
\newcommand{\JPA}{J. Phys. A }
\newcommand{\JPB}{J. Phys. B }
\newcommand{\PLA}{Phys. Lett. A }
\newcommand{\NP}{Nat. Phys. }
\newcommand{\NC}{Nat. Commun. }

redpajama_set_name: RedPajamaArXiv


text:
When a loved one dies
May 26, 2017 July 7, 2017 by Jessica Blankenship
Papaw Jerry & Jessica back in time
During the month of April, I took a break from some things and focused on family. In between my full time job and doing my music stuff, I sat in the hospital room and then nursing home room as my grandpa was sick. The sound of the machines beeping, nurses and doctors shuffling in and out to check this and that became a new routine. Relatives would come to visit and spend time sharing stories of our youth growing up in southeast Kentucky, with some moving on to Ohio and Tennessee. Then on the morning of April 26th, time stood still as my grandpa passed away.
In the coming days, we sat together as a family at the funeral home making arrangements, going to the flower shop to pick out the perfect arrangement, shop for the outfit to wear, and preparing for the visitors. My grandma asked me to speak at his funeral and for several days, I wrote, rewrote, and wrote some more trying to figure out the perfect words to say. Needless to say it was the hardest and easiest thing I have ever written in my life. It was an honor to be asked to tell others about my grandpa.
During all of the hustle and bustle, east Kentucky native, Angaleena Presley's new album "Wrangled" was released. I have been meaning to write a full on review of the masterpiece of an album. It has been hard to find the time to dedicate my full attention to it. However, there was one song that has stuck in my head throughout April and now May. "Cheer Up Little Darling" was written by Angaleena Presley and Guy Clark. It would be his last song he wrote before passing away. Musician Shawn Camp plays Guy Clark's No. 10 guitar, which was used to write the song, and Clark's mandola. I personally want to thank them for this song that kept me rolling these last few weeks. If you would like to listen to it, be sure to download it from I-tunes here.
"Cheer Up Little Darling" by Angaleena Presley & Guy Clark
Yeah, it's gonna be real pretty, man
Cheer up little darlin', don't be so sad
There's a time and place for the blues that you've had
Just hold what you've got, babe, and never give in
Seems like a tight spot, but it's just a loose end
There's a time and place for the blues you have had
Hold what you've got, babe, never give in
It feels like a tight spot, but it's just a loose end
You can't fix the world, girl, it's so badly bent
But you can help it along if you save your own skin
The first thing you do, honey, is make you a list
Of the things that you've done and the things that you've missed
Do the best that you can, doll, there's no one to blame
It's just the turn that it took in this crazy old game
I will miss walking up to Mammy and Papaw's house and seeing him sitting in the rocking chair on the front porch. I will miss him saying "how are you doing?" as he sat there smoking a cigarette, drinking his coffee. I will miss his creamed style corn made in the cast iron skillet, complete with chocolate gravy and biscuits. Don't tell Mammy, but Papaw is the only one that could make the best gravy! And most of all, I will miss hearing those words, "Now don't you run off – what's your hurry?" and "I love you" as I had to leave to head home.
In the Bible, we learned that the Lord created the universe and all its beauty in 7 days. I always said that on the 8th day, God made a farmer. On the 9th day, he made a truck driver to haul the farm goods. While he may have parked his truck here on earth, I know that the engine is a roaring as Papaw Jerry drives the streets of Heaven.
Kentucky music women to be featured in new book
Bill Monroe Estate offers rare ownership of prized possessions

redpajama_set_name: RedPajamaCommonCrawl


text:
Konfessionella katolska partiet var ett politiskt parti i Belgien, bildat den 22 oktober 1868 som Förbundet av katolska kretsar och konservativa föreningar (Fédération des Cercles catholiques et des Associations conservatrices eller Verbond van Katholieke Kringen en der Conservatieve Verenigingen).

Bildandet hade föregåtts av tre katolska konferenser i Mechelen under 1860-talet, under vilka liberala katoliker och montanister närmat sig varandra och enats om behovet av en politisk motkraft till de belgiska liberalerna.

Med dessa utkämpades en hård politisk strid om kristendomens ställning i skolan, som gav katolikerna valsegern i valet 1884.

1921 bytte man namn till Katolska unionen av arbetare, borgare, medelinkomsttagare och lantbrukare och 1936 till Katolska blocket, som upplöstes under andra världskriget och efterföljdes av partiet CVP-PSC.

Historiska politiska partier i Belgien
Politiska partier bildade 1868

redpajama_set_name: RedPajamaWikipedia


value, that we were too focused on transformation.
I had too much inventory and quality problems and safety problems.
some concepts were missing in our world of project management.
manpower and machines, and, we never talk about managing space.
And obviously, space is a big, concern in managing any project.
And so we need to bring that into the, discussion, on managing projects.
to begin to think new ways about how we could manage projects.
>> So Lean has been developed in the context of car manufacturing.
methods and tools of Lean are fully applicable in construction.
of a method developed in the framework of construction.
custom products, as opposed to mass produced products.
tradition in terms of design and the actual work itself.
And we have been through the course of the 25 years of Lean construction.
We have been trying to understand how to help people make that change.
So I think the principles absolutely apply.
high quality work, quality number one, safely.
figure out the safest best way to produce that type of quality.
So, that would be just a short answer in terms of trying to connect.
You know, the Lean principles to work.
some of those tools in construction and perhaps get some benefit from them.
because the tools are only as good as the people who are running them at one time.
to be applied at all times to keep it effective and going.
So if you run tools.
People running them lose interest, lose their way, lose company and so on.
value as well, but for my part I find the flow idea a far more powerful one.
on a construction site, or commissioning, or even facility maintenance.
by asking yourself a question.
What are the things that flow?
And what are the barriers to that flow?
invent new tools on new ways of getting results that we want to get.


text:
OSU Head Coach Ryan Day speaks at Big Ten Media Day
CHICAGO (WCMH) — Ohio State Head Coach Ryan Day says his staff is 'recharged' as they head into August and the start of a new football season.
It is Day's first Big Ten Media Day as head coach, following Urban Meyer's departure at the end of the 2018-2019 season.
On following Urban Meyer, Day said, "You don't replace a legend…you try and be yourself." pic.twitter.com/dUBYjTrJMH
— Audrey Hasson (@AudreyNBC4) July 18, 2019
"You don't replace a legend. You don't replace one of the best football coaches in the history of the game," said Day. "What you can do is just be yourself. And I think that's what I'm doing."
Day says he feels great about his players as they prepare for a new season.
"I love our team. I love our players. I love being around the guys," said Day. "These are great kids and they're working really really hard. Like I said, we have a lot to prove. This is a new staff, a new team, and we haven't done anything."
Transfer QB Justin Fields appears to be the front-runner for the starting job, but Day says it's an open competition going into camp.
"I mean, no positions are given, and I think when you look at that whole room, they all just got here, there's been nobody that was on last year's roster. I think it's going to be one of those things where somebody has to to win the job," said Day.
The Buckeyes go to work at fall camp in a couple weeks. The season opener will be August 31 against Florida Atlantic.
More Buckeyes Stories
BOTTOM LINE: No. 21 Ohio State presents a tough challenge for Penn State. Penn State has won two of its three games against ranked opponents this season. Ohio State got past Nebraska by 12 at home on Tuesday.
TEAM LEADERS: Penn State's Lamar Stevens has averaged 16.1 points and 7.1 rebounds while Myreon Jones has put up 13.9 points. For the Buckeyes, Kaleb Wesson has averaged 14.4 points and 9.6 rebounds while D.J. Carton has put up 10.2 points.WONDERFUL WESSON: K. Wesson has connected on 42.9 percent of the 56 3-pointers he's attempted and has made 4 of 10 over the last three games. He's also converted 74.8 percent of his foul shots this season.
No. 21 Buckeyes beat Nebraska, snap 4-game losing streak
COLUMBUS (AP) -- C.J. Walker scored 18 and No. 21 Ohio State snapped a four-game losing streak by beating Nebraska 80-68.
Freshman guard D.J. Carton chipped in 13 for the Buckeyes and Kaleb Wesson had 13 points and 14 rebounds.
Two Buckeyes suspended for Nebraska game
by Justin Holbrock / Jan 14, 2020
COLUMBUS (WCMH) — Ohio State men's basketball head coach Chris Holtmann has suspended sophomore guards Luther Muhammad and Duane Washington Jr. for Tuesday's game against Nebraska.
The news comes from Dan Wallenberg, the associate athletic director for communications at The Ohio State University.


text:
Web Center
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FDA Enforcement Has Started, What Should You Do?
7 steps to take right now. Before FDA regulations took effect, it was enough to prevent an underage tobacco sale. The thinking was, "is this customer old enough to buy tobacco?" ruled the day. That way of thinking falls short now. Retailers are receiving warning letters from the U.S. Food and Drug Administration (FDA) for selling tobacco to a minor AND for not carding someone who is under 27. That's two violations, not one.
FDA Tobacco Enforcement Underway!
Retailers are receiving violation letters Retailers in Mississippi were the first to receive Warning Letters from the U.S. Food and Drug Administration (FDA) after selling tobacco to a minor. Retailers were also cited for failing to check the ID of a tobacco customer under 27 years old.
FDA Needs Your Help...
The FDA says it's looking to retailers to help enforce the sweeping tobacco regulations that took effect June 22. "We understand the retailers really are on the front lines and are the key piece to making this a success," said Kathleen Quinn, the acting director of communications for the Center for Tobacco Products at the FDA. "Our approach is they are our partners and we want to equip them with the tools they need."
Are Age Calculation Tools a Silver Bullet?
Electronic age verification devices take the math out of checking ID. Just swipe the customer's driver's license and the machine does the rest. Sounds simple. We Card calendars take the math out checking ID too. For every day of the month, the calendar provides the birth date required for a legal tobacco sale. In other words, if it's Dec. 19, 2009 and the legal age for buying tobacco is 18, then the customer must have a birth date of Dec. 19, 1991 -- or earlier.
Results From Our Compliance Survey
Top 4 Reasons For Compliance Failure Just about everyone who works behind the counter in a convenience store or a grocery store knows by now that it's against the law to sell tobacco to minors. So why do retailers still fail compliance checks?
We Card: We Laugh, We Learn
Tales From The Trenches Trainers for the We Card program travel the country to train retail employees on how to avoid selling cigarettes to minors. They teach the importance of checking ID's, how to prevent conflict while denying a sale and how to juggle all the tasks of a high-stress job. The stakes are high. In many states an employee can be fined for selling cigarettes to underage customers and the store can lose its license.
What's your policy on Third-Party Sales?
Imagine you're at the register and a customer, who doesn't look much older than your kid brother, asks for a pack of cigarettes. You check his ID and sure enough he's too young, so you refuse the sale. The teenager heads outside and starts talking to a person in a parked car. Next thing you know, the person is standing in front of you asking for the same brand the teenager just asked for. Again, you do what you've been trained to do. "I'm not allowed to sell you those cigarettes because it appears you're buying for a minor," you say.
Click here to see what's required in your state. Alaska keeps its warning against tobacco sales to minors simple, requiring retailers to post these words where customers can easily see them: "The Sale of Tobacco Products to Persons Under Age 19 is Illegal." Delaware requires bold red and black graphics. "Under 18? Don't Even Try to Buy." And in Oklahoma, the state-required sign comes in orange. "It's The Law. We Do Not Sell Tobacco Products to Persons Under 18 Years of Age."
Driver licenses in California are color-coded to identify a minor, so it should be pretty easy for a front line employee to identify a 16-year-old trying to make an illegal purchase of cigarettes. But law enforcement in Yolo County, outside Sacramento, find that frequently the sales associates they cite in enforcement stings have sold to minors who present valid ID's that show they are too young to buy cigarettes.
Are You Carding For e-Cigarettes?
Eleven states prohibit sales to minors... e-Cigarettes are increasing in popularity. States are adding them to the list of restricted products that should not be sold to minors. Today, 11 states consider e-cigarettes as an age-restricted product. In short order, 11 states could change to 12, 15 or even 50 states.
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I would like to wish all readers a Happy New Year. I am sure 2016 will be a wonderful year filled with family happiness, job promotions and of course exciting Science Fiction! To get a glimpse of what you can expect from the new year you can read my two previous posts. The first previews upcoming sci-fi games and the second upcoming movies. The start of 2016 is also very promising. This evening the BBC will air a one-off special episode of Sherlock starring Benedict Cumberbatch and Martin Freeman. I will post a review tomorrow. The day after, on Sunday, I will have finished my write-up on the Star Citizen Alpha 2.0 release. It should give readers an idea of where the game is going.
For me, the editor of this blog, things will get very exciting. After I managed to publish my first novel last year I have no less than two planned for this year. Well, one is a novel and the other is a Computer Science book aimed at novices. Both are almost completed and are scheduled for release sometime on the spring. I originally hoped to release them by the end of January, but editing, proof-reading and creating graphs takes time and should not be rushed. So, what are my new books about? Below there is a small preview of both.
The first book to be released in the new year is Roland, a sci-fi novel set a few decades in the future. Roland is set in a future World War 3 in which democracies have slowly evolved into Fascist governments. The story follows Roland, the titular character, as he deals with the governments oppressive presence, love of friends, death and loneliness. It is very much a coming of age story. At this moment I am also writing a novella set in the same Rolandverse, but it follows different characters. Roland is set for release on February 29th. I have almost concluded proof-reading it and organizing a cover. If everything proceeds smoothly you should be able to pre-order Roland by the end of January. Initially it will only be released for Amazon Kindle.
Compared to Roland this is a very long title, but it truthfully states what it is. As a Computer Science PhD candidate I have had to endure some very bad classes, read a lot of bad computer books and received a lot of poor advice. Over the years I have accumulated a lot of lecture notes and written a number of small tutorials and primers on Linux, programming and penetration testing. I thought it would be a wonderful idea to collect everything in a book so I can share my knowledge. My own experience has shown that each of these topics are interconnected in a way that beginners are best taught each at the same time. This way knowledge gained will have more meaning and hopefully stick with the reader. The book is filled with small sidetracks, examples, anecdotes and even reviews of examples from popular culture. If you desire to get a better understanding of these topics and you have little experience with them, then this book is something for you. I have set a tentative release date of March 31st. Just like Roland it will be available for Kindle with pre-orders possible one month before release!
My new novel – Roland – is now available for pre-order from Amazon!


text:
Trump supporters storm US Capitol, lawmakers evacuated
WASHINGTON — Violent protesters loyal to President Donald Trump stormed the U.S. Capitol on Wednesday and forced lawmakers into hiding, in a stunning attempt to overturn America's presidential election, undercut the nation's democracy and keep Democrat Joe Biden from replacing Trump in the White House.
The National Guard and state and federal police were called in for control, and the mayor of Washington imposed a rare evening curfew. One person was reported to have been shot.
People shelter in the House gallery as protesters try to break into the House Chamber at the U.S. Capitol on Wednesday, Jan. 6, 2021, in Washington. (AP Photo/Andrew Harnik)
The protesters were egged on for weeks by Trump, who since the November presidential election had launched a barrage of false attacks on the integrity of the results. While rallying his supporters outside the White House Wednesday morning, he urged them to march to the Capitol. But later — hours after they fought police and breached the building — he told them in a video that although they were "very special people" and he backed their cause, they should "go home in peace."
Other than a pair of tweets and that minute-long video, Trump was largely disengaged from the occupation of a main seat of the nation's government. It was Vice President Mike Pence, not Trump, who spoke with senior defense leaders about calling up the National Guard.
President-elect Biden, two weeks away from being inaugurated, had declared in Wilmington, Delaware: "I call on President Trump to go on national television now to fulfill his oath and defend the Constitution and demand an end to this siege."
Biden said that democracy was "under unprecedented assault," a sentiment echoed by many in Congress, including some Republicans.
District 4 Congressman Tom McClintock, who represents Lake Tahoe, also was outraged by the violence.
"The attack on the Capitol strikes at the most sacred act of our democracy: the peaceful transfer of power," McClintock said on Twitter. "It is an outrage and a threat to our most fundamental principles as a free people."
Even for a Capitol building that has seen centuries of protests and even violence — including a 1954 shooting involving Puerto Rican nationalists — Wednesday's events were astounding because they appeared to unfold at least initially with the blessing of the president and also because of the underlying goal of overturning the results of a lawful presidential election.
The chaotic protests halted Congress' constitutionally mandated counting of the Electoral College results, in which Biden defeated Trump, 306-232. Republican Senate Majority Leader Mitch McConnell had tried to steer Congress away from Wednesday's formal protest of those results, and he said at the start of proceedings that Trump had clearly lost.
Wednesday's ordinarily mundane procedure of Congress certifying a new president was always going to be extraordinary, with Republican supporters of Trump vowing to protest election results that have been certified by the states. But even the unusual deliberations, which included Vice President Mike Pence and Senate Majority Leader McConnell defying Trump's demands, were quickly overtaken by the chaos.
In a raucous, out-of-control scene, protesters fought past police and breached the building, shouting and waving Trump and American flags as they marched through the halls. One person was reported shot at the Capitol, according to a person familiar with the situation. That person's condition was unknown. At least one explosive device was found but law enforcement said it did not pose a threat.
More than a dozen people were arrested.
As darkness began to set in, law enforcement officials worked their way toward the protesters, using percussion grenades to try to clear the area around the Capitol. Big clouds of tear gas were visible. Police in full riot gear moved down the steps, clashing with demonstrators.
It added up a frightening scene for lawmakers, who were directed to take extraordinary action for their own safety. The protesters abruptly interrupted the congressional proceedings in an eerie scene that featured official warnings directing people to duck under their seats for cover and put on gas masks after tear gas was used in the Capitol Rotunda. Some House lawmakers tweeted they were sheltering in place in their offices.
Rep. Scott Peters, D-Calif., told reporters he was in the House chamber when protesters began storming it. Security officers "made us all get down, you could see that they were fending off some sort of assault, it looked like. They had a piece of furniture up against the door, the door, the entry to the floor from the Rotunda, and they had guns pulled," Peters said.
"And they just told us to take our pins off," he added, referring to lapel pins members wear so Capitol Police can quickly identify them. Then the lawmakers were evacuated.
A clerk helped grabbed the boxes of Electoral College votes as the evacuation took place. Otherwis, said Sen. Jeff Merkley, D-Ore., the ballots likely would have been destroyed by the protesters.
Trump supporters posting on internet forums popular with far-right fringe elements celebrated the chaos. Messages posted on one turned from profane frustration over the content of Trump's speech to glee when supporters stormed the building. At least one leading figure was livestreaming video from inside the Capitol during the siege.
The Pentagon said about 1,100 District of Columbia National Guard members were being mobilized to help support law enforcement at the Capitol.
Pence was closely watched as he stepped onto the dais to preside over the joint session in the House chamber.
Pence had a largely ceremonial role, opening the sealed envelopes from the states after they are carried in mahogany boxes used for the occasion, and reading the results aloud. But he was under growing pressure from Trump to overturn the will of the voters and tip the results in the president's favor, despite having no legal power to affect the outcome.
"Do it Mike, this is a time for extreme courage!" Trump tweeted Wednesday.
But Pence, in a statement shortly before presiding, defied Trump, saying he could not claim "unilateral authority" to reject the electoral votes that make Biden president.
Despite Trump's repeated claims of voter fraud, election officials and his own former attorney general have said there were no problems on a scale that would change the outcome. All the states have certified their results as fair and accurate, by Republican and Democratic officials alike.
Arizona was the first of several states facing objections from the Republicans as Congress took an alphabetical reading of the election results. Then the chaos erupted.


text:
H.R. 1676 (115th)
H.R. 1676 (115th): Palliative Care and Hospice Education and Training Act
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To amend the Public Health Service Act to increase the number of permanent faculty in palliative care at accredited allopathic and osteopathic medical schools, nursing schools, social work schools, and other programs, including physician assistant education programs, to promote education and research in palliative care and hospice, and to support the development of faculty careers in academic palliative medicine.
The bill's titles are written by its sponsor.
Sponsor and status
Eliot Engel
Sponsor. Representative for New York's 16th congressional district. Democrat.
Read Text »
Last Updated: Jul 24, 2018
115th Congress, 2017–2019
Died in a previous Congress
This bill was introduced in a previous session of Congress and was passed by the House on July 23, 2018 but was never passed by the Senate.
What legislators are saying
"Smith Awarded Prestigious National Public Service Award from American Heart Association"
— Rep. Christopher "Chris" Smith [R-NJ4] (Co-sponsor) on Jun 27, 2017
"House E&C subcommittee members advance draft PAHPA bill to combat biological threats"
— Rep. Susan Brooks [R-IN5] (Co-sponsor) on Jun 29, 2018
"Walberg, House Advance Legislation to Lower Health Care Premiums, Increase Patient Choice"
— Rep. Tim Walberg [R-MI7] on Jul 26, 2018
More statements at ProPublica Represent...
What stakeholders are saying
Institute for Spending Reform SpendingTracker.org estimates H.R. 1676 will add $87 million in new spending through 2023.
Minnesota Network of Hospice & Palliative Care
Alzheimer's Impact Movement The Alzheimer's Association and AIM proudly support the Palliative Care and Hospice Education and Training Act(S. 693/H.R. 1676) and its goal of ensuring a high-quality palliative care and hospice workforce.
Your organization's position statement could be on this page! Register your organization's position on this bill »
Earlier Version — Introduced
This activity took place on a related bill, H.R. 6155 (112th).
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H.R. 1676 (115th) was a bill in the United States Congress.
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This bill was introduced in the 115th Congress, which met from Jan 3, 2017 to Jan 3, 2019. Legislation not enacted by the end of a Congress is cleared from the books.
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Institute for Spending Reform
This position statement was submitted by Institute for Spending Reform on Oct 15, 2018.
SpendingTracker.org estimates H.R. 1676 will add $87 million in new spending through 2023.
Minnesota Network of Hospice & Palliative Care supports this bill
This position statement was submitted by Minnesota Network of Hospice & Palliative Care on Jul 23, 2018.
Alzheimer's Impact Movement supports this bill
This position statement was submitted by Alzheimer's Impact Movement on Feb 23, 2018.
The Alzheimer's Association and AIM proudly support the Palliative Care and Hospice Education and Training Act(S. 693/H.R. 1676) and its goal of ensuring a high-quality palliative care and hospice workforce.
Alzheimer's is a progressive, neurodegenerative and fatal disease for which there is currently no treatment or cure. Palliative and hospice care can improve both the quality of care and quality of life for those with advanced dementia. A recent study shows that nursing home residents with dementia who receive palliative care at the end of life , compared with those who do not receive such care, are up to 15 times less likely to die in a hospital, nearly 2.5 times less likely to have a hospitalization in the last 30 days of life, and up to 4.6 times less likely to have and emergency room visit in the last week of life. Individuals with advanced dementia who are enrolled in hospice have a lower rate of dying in the hospital, a lower rate of hospitalization in the last 30 days of life, and better symptom management. Additionally, nearly half of all people with dementia die in hospice care.
The Palliative Care and Hospice Education Training Act would ensure an adequate, well-trained palliative care workforce through workforce training, education and awareness, and enhanced research. in addition, the legislation is consistent with the recommendations made by the Advisory Council on Alzheimer's Care, Research, and Services.


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IRCTC to launch its first premium trains on Lucknow-Delhi & Mumbai-Ahmedabad routes
Ministry of Railways, in order to further professionalize the services, have entrusted IRCTC, to operate two premium passenger trains for the general public using the air-conditioned chair car rakes of Tejas trains. These two trains will ply on Lucknow – Delhi – Lucknow and Ahmedabad – Mumbai- Ahmedabad routes for 6 days in a week and will primarily cater to the business travelers traveling frequently between these cities, besides catering to tourists. Trains would be equipped with all modern onboard facilities for ensuring a high level of comfort to the passengers.
The first train will be operated in Lucknow – New Delhi – Lucknow sector having a total journey time of 6 hours and 15 minutes. This train will have one Executive class air-conditioned chair car having 56 seats and nine air-conditioned chair cars having a capacity of 78 seats each. The total carrying capacity of the train will be 758 passengers. The train will start its journey in the morning at 06:10 hours and will reach New Delhi at 12:25 hours having scheduled commercial halts at Kanpur Central and Ghaziabad.
Another train on Tejas rake will run on Ahmedabad – Mumbai – Ahmedabad route a little later. This train will leave Ahmedabad at 06:40 hours and will reach Mumbai at 13:15 hrs. In the return direction, the train will leave Mumbai at 15:40 hrs and reach Ahmedabad at 22:25 hrs. This train will also run 6 days a week, except Thursday.
According to a press release, the booking will be available exclusively on the IRCTC website and its mobile app "Irctc Rail Connect". There will be no booking at railway reservation counters. However, passengers can get their tickets booked through IRCTC authorised agents. Train will also be available for booking through IRCTC' online travel portal partners like Paytm, Ixigo, PhonePe, MakeMy Trip, Google, Ibibo, Railyatri etc.
To get travel updates directly on your mobile, save and send a message at 9461777617 on Whatsapp to start.
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