text:
\section{Introduction}

The conversion between bipartite or multipartite quantum states through
local operations is a
central concept of entanglement theory.
For instance, it is the criterion used to classify
entanglement, namely, two states have equivalent entanglement if they can be converted to each
other~\cite{nielsenchuang}.
Moreover, it is usual and natural to consider a state more entangled than other when the first can
{\emph{access}} the former by the allowed transformations \cite{nielsen}.
Two important sets of allowed transformations are the deterministic
{\emph{local operations and classical communication}} (LOCC) and its stochastic
version (SLOCC), where the transformation only needs to succeed with positive
probability \cite{Andreas}.
Although this hierarchisation is relatively simple for bipartite pure
states~\cite{nielsen, horodeckis}, it is
quite involved if mixed or multipartite states are considered~\cite{multipartite1, multipartite2,
EisertWilkens}.

The problem of convertibility of bipartite pure states has been solved by Nielsen\cite{nielsen}, using the concept
of majorisation \cite{nielsen2}.
However, Jonathan and Plenio discovered a surprising effect \cite{daniel}.
They have shown the existence of pairs of states
which are not directly inter-convertible but such that their conversion is possible if another (necessarily
entangled) state is attached to them.
That is, the pair of states
have incomparable entanglement but they may become ordered if an extra system is attached
to the original ones.
Such extra state that makes a transformation possible, without being consumed, is called a {\emph{catalyst}}.
More recently, the problem of convertibility has received experimental attention \cite{exp} and has
also been connected to basic results in thermodynamics \cite{brandao}  and phase transitions
\cite{helena}.

In this paper we explore the following question: can one of the states of a non
inter-convertible pair be used as a catalyst?
We answer this question positively, providing explicit examples.
We also explore how common such processes are for low dimensional systems.
An interesting situation is when a state $\ket{\Psi}$ is not able of self-catalysing a transformation, but a number of copies, $\ket{\Psi}^{\otimes n}$, is.
We exhibit examples where more than one copy is required and study how augmenting the number of copies
can increase the probability of conversion.

In section~\ref{catalisys} we review basic notions of state conversion and catalysis under LOCC.
In section~\ref{selfie} we show explicit examples of self-catalytic processes,
explore, through numerical analysis, how frequent they are under random samples of incomparable
pairs and how it depends on the size of the systems.
Section~\ref{catslocc} review the notions of probabilistic catalysis under SLOCC,
while the natural questions of self-catalysis under SLOCC are discussed in section~\ref{sefslocc}.
We close the main text with final remarks
and further problems in section~\ref{concl}.

\section{Catalisys}\label{catalisys}

We say that a bipartite state $\ket{\alpha}\in H_{A}\otimes H_{B}$ \emph{access} a state
$\ket{\beta}\in H'_{A}\otimes H'_{B}$, if there is some LOCC operation, represented by a completely
positive trace preserving (CPTP) map
$\Lambda$, such that $\Lambda(\ket{\alpha}\bra{\alpha})=\ket{\beta}\bra{\beta}$, where
$H_{A},H'_{A},H_{B}$ and $H'_{B}$ are finite dimensional Hilbert spaces and $\Lambda$ maps density
operators acting on $H_{A}\otimes H_{B}$ to density operators acting on
$H'_{A}\otimes H'_{B}$. In such
case, we write
$\ket{\alpha}\rightarrow \ket{\beta}$.
If there is no LOCC operation able to convert $\ket{\alpha}$ to $\ket{\beta}$, we shall write $\ket{\alpha} \nrightarrow \ket{\beta}$.

For example, for a pair of qubits, the Bell state $\ket{\Phi^{+}}=(\ket{00}+\ket{11})/\sqrt{2}$ can
access any two-qubit pure state.
Indeed, naming our qubits $A$ and $B$, and
writing the target state as $\ket{\beta} = a\ket{00} + b\ket{11}$, we can,
for example, make $A$ unitarily interact
with an auxiliary qubit $A'$, so that $\ket{0_{A}0_{A'}}\mapsto
a\ket{0_{A}0_{A'}}+b\ket{1_{A}1_{A'}}$ and $\ket{1_{A}0_{A'}}\mapsto
a\ket{0_{A}1_{A'}}+b\ket{1_{A}0_{A'}}$. Then,
$[(\ket{0_{A}0_{B}}+\ket{1_{A}1_{B}})/\sqrt{2}]\ket{0_{A'}}\mapsto
(1/\sqrt{2})(a\ket{0_{A}0_{B}}+b\ket{1_{A}1_{B}})\ket{0_{A'}}+(1/\sqrt{2})(a\ket{0_{A}1_{B}}+b\ket{
1_{A}0_{B}} )\ket{ 1_{A'} }$. The lab with qubit $A$ can make a measurement
on the computational basis of the auxiliary $A'$
and send the result to the lab holding qubit $B$. If the result is $0$, they already share the
desired state, while the result being $1$, a NOT operation,
$\ket{0_{B}}\mapsto\ket{1_{B}},\ket{1_{B}}\mapsto\ket{0_{B}}$, can be applied to qubit $B$ to also
leave the system $AB$ in the desired state.
This result generalises for a pair of qu$d$its in the state $\displaystyle{\ket{\Phi^+_d} = \frac{1}{\sqrt{d}}\sum_{i=0}^{d-1} \ket{ii}}$, which justifies $\ket{\Phi^+_d}$ to be called a maximally entangled state.

Nielsen, in the seminal paper~\cite{nielsen}, provided a simple necessary and sufficient condition
for determining whether a general bipartite state $\ket{\alpha}$ can access a state
$\ket{\beta}$ in terms of their corresponding Schmidt vectors~\cite{nielsenchuang}:
\begin{theorem}[Nielsen Criterion]
\label{thm-Nielsen}
If
$\vec{\alpha}=(\alpha_{1},...,\alpha_{n})$ and $\vec{\beta}=(\beta_{1},...,\beta_{m})$ are the
(non-increasing) ordered Schmidt
vectors of $\ket{\alpha}$ and $\ket{\beta}$, respectively, we have $\ket{\alpha}\rightarrow
\ket{\beta}$ if, and only if,
\begin{equation}\label{majoracao}
\sum_{l=1}^{k}\alpha_{l}\leq\sum_{l=1}^{k}\beta_{l}
\end{equation}
for $1\leq k\leq \min\{n,m\}$.
\end{theorem}
By an ordered Schmidt vector
$\vec{\lambda} = (\lambda_{1},\ldots,\lambda_{n})$, we mean
that $\lambda_{1}\geq \lambda_{2}\geq \ldots \geq\lambda_{n}$.
In this work, we shall always assume that
Schmidt vectors are ordered. When vectors $\vec{\alpha}$ and $\vec{\beta}$
satisfy conditions~\eqref{majoracao} we say that $\vec{\alpha}$ is majorised by $\vec{\beta}$ and write
$\vec{\alpha} \preceq \vec{\beta}$.

Revisiting the example above, we have $(1/2,1/2)$ for the ordered Schmidt vector of $\ket{\Phi^{+}}$
and $(|a|^{2},|b|^{2})$ for $a\ket{00}+b\ket{11}$, assuming $|a|\geq |b|$, it is straightforward to
apply the criterion and verify that $\ket{\Phi^{+}}\rightarrow [a\ket{00}+b\ket{11}]$.

A consequence of the criterion is the existence of pair of states $\ket{\alpha}$ and $\ket{\beta}$
such that $\ket{\alpha}\nrightarrow\ket{\beta}$ and $\ket{\beta}\nrightarrow\ket{\alpha}$.
Actually, the only case where such an order is total is for two qubits.
For instance, consider a state of two  four-level systems with Schmidt vector $(0.4,0.4,0.1,0.1)$ and a
two-qutrit state with Schmidt vector $(0.5,0.25,0.25)$.
Indeed,
\begin{subequations}
\begin{align}
0.4=\alpha_{1}&< \beta_{1}=0.5,\\
0.4+0.4=\alpha_{1}+\alpha_{2}&>\beta_{1}+\beta_{2}=0.5+0.25.
\end{align}
\end{subequations}

In Ref.~\cite{daniel}, the authors surprisingly show that it is possible to circumvent
such non-accessibility between these states by making the parts to share an entangled state $\ket{\kappa}$  such that $\ket{\alpha}\otimes \ket{\kappa}\rightarrow \ket{\beta}\otimes \ket{\kappa}$
(see Figure~\ref{cris1}). Since the state $\ket{\kappa}$ allows for a previously
forbidden
conversion, but at the end of the process it remains unaltered, it is called a \emph{catalyst}.
{In this sense, we say that $\ket{\alpha}$ \emph{$\vec{\kappa}$-access} $\ket{\beta}$
when $\ket{\alpha} \nleftrightarrow \ket{\beta}$, but $\ket{\alpha} \otimes \ket{\kappa}
\rightarrow \ket{\beta} \otimes \ket{\kappa}$}.
In this specific example, the two-qubit state $\ket{\kappa}$ with Schmidt vector
$\vec{\kappa}=(0.6,0.4)$ is a catalyst.

\begin{figure}[h]
\includegraphics[scale=0.3]{Figura_self_cat_4}
\caption{Although the conversion $\ket{\alpha}\rightarrow \ket{\beta}$ is not allowed under LOCC,
the state $\ket{\kappa}$ can be used (but not consumed) to make it viable.}\label{cris1}
\end{figure}

Nielsen's criterion ensures that all information about the (possibility of) conversion is
contained in the Schmidt vectors.
Therefore, we can explore our knowledge regarding probability
vectors to provide more examples of catalysts.
If $\vec{\alpha} = (0.5,0.4,0.05,0.05)$, $\vec{\beta} =(0.7,0.15,0.15)$,  $\vec{\kappa_1}=(0.7,0.3)$, and $\vec{\kappa_2}=(0.75, 0.25)$,
we obtain:
\begin{subequations}
\begin{align}
\vec{\alpha} &\nleftrightarrow \vec{\beta} \\
\vec{\alpha}  \otimes \vec{\kappa_1} &\rightarrow \vec{\beta} \otimes \vec{\kappa_1} \\
\vec{\alpha}  \otimes \vec{\kappa_2} &\rightarrow \vec{\beta} \otimes \vec{\kappa_2}.
\end{align}
\end{subequations}
This is a good example of non-unicity of catalysts.
Indeed, for a given forbidden transition, $\ket{\alpha} \nrightarrow
\ket{\beta}$, and fixed local dimensions for the catalysts, the set of
allowed vectors $\vec{\kappa}$ is a polytope~\cite{SFDY05}.

Another interesting geometric fact is that catalysis is possible for every
bipartite scenario, starting from $4\times 3$,
{\textit{i.e.}},  for all effective dimensions $m \geq n$, $m \geq 4$, and $n\geq 3$ it
is possible to choose
$\vec{\alpha}=(\alpha_{1},...,\alpha_{m})\nleftrightarrow(\beta_{1},...,\beta_{n})=\vec{\beta}$ with
catalyst $\vec{\kappa}$.
The essential step, after the previously described examples, is that given
$\vec{\alpha}$, $\vec{\beta}$, and $\vec{\kappa}$, we generically can
increase dimensions by one  and construct
a forbidden transition  $\vec{\alpha'} \nrightarrow \vec{\beta'}$  with
the same catalyst $\vec{\kappa}$, by using
$\vec{\alpha'}=(\alpha_{1},...,\alpha_{n}-\epsilon,\epsilon)$ and
$\vec{\beta'}=(\beta_{1},...,\beta_{n}-\epsilon',\epsilon')$
with small enough $\epsilon>0$ and $\epsilon' \geq 0$.
This allow us to construct examples for all such $m$ and $n$.

\section{Self-Catalysis}\label{selfie}

In this section we address the main question of this paper for the case of LOCC
convertibility: can a bipartite quantum
state be itself the catalyst of a forbidden conversion?
To be more precise, is there  a forbidden
conversion $\ket{\alpha}\nrightarrow \ket{\beta}$, such that
$\ket{\alpha}\otimes
\ket{\alpha}\rightarrow \ket{\beta}\otimes \ket{\alpha}$?
Refining a little bit more, is it possible that we still have
$\ket{\alpha}\otimes
\ket{\alpha}\nrightarrow \ket{\beta}\otimes \ket{\alpha}$, but a larger
number of copies of
$\ket{\alpha}$ would do the job, {\textit{i.e.}},   $\ket{\alpha}\otimes
\ket{\alpha}^{\otimes N}\rightarrow \ket{\beta}\otimes \ket{\alpha}^{\otimes N}$ for some
$N > 1$?

In Table~\ref{Table} we list examples to answer affirmatively these
questions.
The first is an example of self-catalysis from a two-ququart state to a two-qutrit state.
Then, for the same scenario, there is a list of multi-copy self-catalysis, with the corresponding minimal number of copies.
At the last row, we come back to single-copy self-catalysis from a two-ququart state to another two-ququart state.

\textit{Remark}. When $\ket{\alpha}$ has 4 non-zero Schmidt
coefficients and $\ket{\beta}$ has 3 non-zero Schmidt coefficients, the phenomenon of
self-catalysis can be noted, that is, even in the minimal dimensions to occur catalysis
(see \cite{daniel}) the self-catalysis can also carry out.
Geometrically, this mean that even is the smallest dimension (most
restrictive) case, there are cases where the source state $\vec{\alpha}$
belongs to the polytope of catalysts for the reaction $\vec{\alpha}
\nrightarrow \vec{\beta}$.
Moreover, it is possible to construct
examples for any scenario through a similar argument presented at the Section~\ref{catalisys}.

\begin{table}
\begin{tabular}{c| c |c}
\hline
$\vec{\alpha}$ & $\vec{\beta}$ & $\sharp$Copies ($N$) \\
\hline
$({0.900},{0.081},{0.010},{0.009})$ & ({0.950},{0.030},{0.020}) & $N=1$ \\
\hline
$({0.900},{0.088},{0.006},{0.006})$ & & $N=2$ \\
$({0.908},{0.080},{0.006},{0.006})$ & & $N=3$\\
$({0.918},{0.070},{0.006},{0.006})$ &
$({0.950},{0.030},{0.020})$  & $N=4$\\
$({0.925},{0.063},{0.006},{0.006})$ & & $N=5$\\
$({0.928},{0.060},{0.006},{0.006})$  &  & $N=6$ \\
\hline
$({0.900},{0.081},{0.010},{0.009})$ & ({0.950},{0.030},{0.019},{0.001}) & $N=1$ \\
\hline

\end{tabular}
\caption{\label{Table}Number $N$ of copies required to make $\ket{\alpha}^{\otimes N}$ a deterministic
catalyst for the process $\ket{\alpha}\rightarrow \ket{\beta}$.}
\end{table}

\subsection{Stability under small perturbations}\label{stab}

It is important to mention that the phenomenon of self-catalysis, as it happens
with
catalysis, is generically robust against small perturbations of the state vectors involved. That
is, suppose that one is aiming to perform a self-catalytic process $\ket{\alpha}\otimes
\ket{\alpha}\rightarrow \ket{\beta}\otimes \ket{\alpha}$, but it actually implements states
$\ket{\alpha'}$, $\ket{\beta'}$, where $\ket{\alpha'}\approx \ket{\alpha}$ and
$\ket{\beta'}\approx \ket{\beta}$.
Our claim is: \emph{generically, if $\ket{\alpha}$ $\alpha$-access
$\ket{\beta}$, then $\ket{\alpha'}$ $\alpha'$-access $\beta'$}.
It is easy to see
that this will be true, depending only on the inequalities implying that
$\ket{\alpha}\nrightarrow \ket{\beta}$, as well as those assuring that
$\ket{\alpha}\otimes
\ket{\alpha}\rightarrow \ket{\beta}\otimes\ket{\alpha}$,  all be strict (except the last one, which is granted by normalisation).
Denote by $|\vec{\lambda}|$
some norm ({\textit{e.g.}},  Euclidean) of the vector $\vec{\lambda}$ and $n,m$ the sizes
of Schmidt vectors
$\vec{\alpha},\vec{\beta}$.
Now, assuming that we have, for
some $l$, $\sum_{j=1}^{k}\alpha_{j}<\sum_{j=1}^{k}\beta_{j}$ for $1\leq k\leq l$ and
$\sum_{j=1}^{k}\alpha_{j}>\sum_{j=1}^{k}\beta_{j}$ for $l< k\leq n-1$, the same set of inequalities
will hold for the entries of vectors $\vec{\alpha'}$ and $\vec{\beta'}$ if
$|\vec{\alpha}-\vec{\alpha}'|,|\vec{\beta}-\vec{\beta'}|<\epsilon$, as long as
$\epsilon$ is small enough.
This implies that $\ket{\alpha'}$ and $\ket{\beta'}$ are still incomparable. A
similar reasoning can be applied when the incomparability of vectors
$\vec{\alpha}$ and $\vec{\beta}$ is due to two or more changes of signs in the inequalities.
In the same way,
$\sum_{j=1}^{k}(\alpha\otimes\alpha)^{\downarrow}_j<\sum_{j=1}^{k}(\beta\otimes\alpha)^{\downarrow}_j$ for $k<nm-1$,
imply
$\sum_{j=1}^{k}(\alpha'\otimes\alpha')^{\downarrow}_j<\sum_{j=1}^{k}(\beta'\otimes\alpha')^{\downarrow}_j$,
if $|\vec{\alpha}-\vec{\alpha}'|,|\vec{\beta}-\vec{\beta}'|<\epsilon$,
for small enough $\epsilon$, which proves the claim.
Naturally, the argument includes the well-motivated situation when $\ket{\beta'} = \ket{\beta}$ as a special case.

\subsection{Self-Catalysis under LOCC for random Schmidt vectors}\label{tip}

We have numerically investigated how usual the phenomenon of
self-catalysis among pairs of
incomparable bipartite states is.
Fixing the sizes of $\vec{\alpha}$ and $\vec{\beta}$, we
randomly sample pairs of such vectors until we
find incomparable ones.
The sampling of each
vector is done by uniformly sorting unitary vectors in $\mathbb{C}^n\otimes \mathbb{C}^n$, \textit{i.e.} sorting according
to
the \emph{Haar Measure} in the respective state spaces~\cite{Loyd,Zyc}, and
then calculating the correspondent Schmidt vector.
After finding a pair of incomparable states, we test
whether the first of the vectors can be used as a catalyst for the
conversion.
The results show that for this method of sampling and for small dimensional systems, the phenomenon is actually \emph{atypical}.
Moreover, the numerical estimations seem to imply that the phenomenon of
self-catalysis is atypical in any dimension.

\begin{figure}[h]

\includegraphics[scale=0.5]{resultados_LOCC_haar_cores_EDITADO_colorido2}
\caption{Probability of finding a pair of states exhibiting self-catalysis as function of
the dimension of each system, that is
$\mathbb{P}( \ket{\alpha}\otimes\ket{\alpha} \rightarrow
\ket{\beta}\otimes \ket{\alpha} \,\, | \,\,  \ket{\alpha} \nleftrightarrow
\ket{\beta} )$. Each symbol means an average over a distinct
set of random choices. For each size explored there are three different
symbols, which indicates a reasonable stability in this sampling
process.}\label{cris2haar}
\end{figure}

Figure~\ref{cris2haar} shows a numerical estimation for $\mathbb{P}(
\ket{\alpha}\otimes\ket{\alpha} \rightarrow
\ket{\beta}\otimes \ket{\alpha} \,\, \arrowvert \,\,  \ket{\alpha}
\nleftrightarrow
\ket{\beta} )$, that is, the conditional
probability of self-catalysis given a pair of incomparable Schmidt vectors,
 as a function of their sizes.
Note that this probability increases with the size of the Schmidt vectors,
until sizes about $20$ and then starts to slowly decrease, but
in fact for any dimension its order of magnitude
shows that the phenomenon is present, but rare.

We believe the global maximum
present in the Figure~\ref{cris2haar} can be explained by the algebraic
character of the comparison between vectors and the concentration of measure phenomenon. In order to
$\ket{\alpha}\otimes \ket{\alpha}$ access
$\ket{\alpha}\otimes \ket{\beta}$, all the corresponding inequalities of Eq.~\eqref{majoracao} must be satisfied. Then, in one hand, by
increasing the dimension of the vectors, this
comparison become harder to be valid (more inequalities have to be obeyed). On the other hand,
it is known that a measure concentration
phenomenon takes place for increasing dimensionality, in the sense that the Schmidt vectors become
typically closer and closer to the constant vector $(\frac{1}{d},...,\frac{1}{d})$. This phenomenon can be
highlighted by the average normalized entropy of the sorted vectors: if this average is close to $1$, it means that
most vectors are close to the constant vector. This average value is know to be exactly
$\frac{1}{\ln{d}}[\sum_{k=d+1}^{d^{2}}\frac{1}{k}-\frac{d-1}{2d}]$, as conjectured by
Page~\cite{page} and latter
proved by Foong and Kanno~\cite{foong}

If the vectors of the pair are incomparable, that is, some inequalities of Eq. \eqref{majoracao} are not satisfied, but
both are close to the constant vector, it will be easier for the catalyst to make the transition
possible, since the corresponding inequalities for the vectors with the catalyst attached will be
easier to be satisfied.

We can note that the concentration of measure increases very fast at low dimensions (see
Figure~\ref{entropy_haar}), helping the possibility of  ``organising'' the Schmidt
vectors with the catalyst attached, which justifies
the corresponding increasing in the probability for self-catalysis.
However, around dimension $20$ the
concentration happens much slower and presumably is not fast enough to overcome the size effect (which makes majorisation more
difficult), so the probability of finding a self-catalytic pair decreases after this point.
\begin{figure}
\includegraphics[scale=0.5]{entropy_haar_EDITED}
\caption{Mean normalized entropy for Schmidt vectors
sorted by Haar measure. Here we have explored the behavior of the entropy
until effective dimension 100.}\label{entropy_haar}
\end{figure}

Summing up, the probability seems to be a smooth function
of the dimension, reaching its higher value for dimensions of each
Schmidt vector close to $20$ and, apparently, converging to a  value
considerably
smaller than $1$.

\begin{remark} We have restricted the analysis above to events where
$\ket{\alpha}
\nleftrightarrow \ket{\beta}$.
For large dimensions we have checked that this event
\emph{is} typical.
Indeed, in such regime the entries of vectors
$\vec{\alpha}$ and $\vec{\beta}$, before ordering, are essentially
concentrated random variables fluctuating around a fixed value. If we look,
then, to Eq.~\eqref{majoracao} we see that there
is a good chance for the sign of the inequality to change as we vary the index
$k$, implying that the states are incomparable. Therefore, we expect that
$\mathbb{P}(\ket{\alpha} \nleftrightarrow \ket{\beta})
\approx 1$,
for $n \gg 1$.\label{remark4}
\end{remark}

\section{Catalysis under SLOCC}\label{catslocc}

It is possible to generalize the concept of accessibility if we allow for
non-deterministic
processes.
In this case, we can look for the probability
$P_{SLOCC}(\ket{\alpha}\rightarrow
\ket{\beta})$, or $P_{S}(\ket{\alpha}\rightarrow
\ket{\beta})$ for short, of having the state conversion
$\ket{\alpha}\rightarrow\ket{\beta}$ under the best
local strategy, {\textit{i.e.}},  optimizing $P$ under the conditions defining SLOCC.
It is interesting to recall
that the famous result on inconvertibility between W and GHZ states
refers
to such conditions on the multipartite scenario~\cite{GHZW}.

As shown by Vidal~\cite{vidal}, for the bipartite case, the Schmidt vectors
also encode this
maximal probability of conversion, $P_{S}(\ket{\alpha}\rightarrow
\ket{\beta})$ through
\begin{theorem
\label{thm-Vidal}
Let $\vec{\alpha}=(\alpha_{1},...,\alpha_{n})$ and
$\vec{\beta}=(\beta_{1},...,\beta_{n})$
be ordered Schmidt vectors for states $\ket{\alpha}$ and $\ket{\beta}$,
assuming $\alpha_n,\beta_n >0$. Define $\displaystyle{E_k(\lambda) =
1-\sum_{l=1}^{k-1}\lambda_{l}}$. Then, the optimal transformation
probability is given by
\begin{equation}\label{formulavidal}
P_{S}(\ket{\alpha}\rightarrow
\ket{\beta})=\min_{1\leq k\leq n}\left\lbrace
\frac{E_k(\alpha)}{E_k(\beta)} \right\rbrace .
\end{equation}
\end{theorem}

For instance, if
\begin{equation}\label{exampleslocc}
\vec{\alpha}=(0.6,0.2,0.2)\text{ and }
\vec{\beta}=(0.5,0.4,0.1)
\end{equation}
are the Schmidt vectors for the two-qutrit states $\ket{\alpha}$ and
$\ket{\beta}$, respectively, we get:

\begin{subequations}
\begin{align}
P_{S}(\ket{\alpha}\rightarrow \ket{\beta})=0.8, \\
P_{S}(\ket{\beta}\rightarrow \ket{\alpha})=0.5.
\end{align}
\end{subequations}
The following Proposition shows that the optimal probability of conversion
$P_S$ attains $1$ precisely when $\ket{\alpha}$ access $\ket{\beta}$.

\begin{proposition} \label{conexao}
 Let $\vec{\alpha}$ and $\vec{\beta}$ be a pair of random independent
Schmidt vectors with same size $n$, then the event $ \{ \ket{\alpha}
\rightarrow \ket{\beta} \}$ is equal to event $\{P_{S}(\ket{\alpha}
\rightarrow \ket{\beta})=1\}$. In particular $\mathbb{P} (
\ket{\alpha}
\rightarrow \ket{\beta}) = \mathbb{P}(P_{S}(\ket{\alpha}
\rightarrow
\ket{\beta})=1)$.
\end{proposition}

\begin{proof}
Suppose that $\ket{\alpha} \rightarrow \ket{\beta}$, thus:
\begin{equation}
\sum_{i=1}^{k} \alpha_{i} \leq \sum_{i=1}^{k} \beta_{i}, \,\, \forall k
\,\, \in \{1,2,...,n \}.
\end{equation}
Then $E_{k}(\alpha)=1-\sum_{i=1}^{k-1}\alpha_{k} \geq
1-\sum_{i=1}^{k-1}\beta_{k}=E_{k}(\beta), \,\, \forall \,\, k \in
\{1,2,..., n-1 \}$ with the equality holding if $k=1$, therefore
$P_{S}(\ket{\alpha} \rightarrow \ket{\beta})=1.$ The converse is similar.
\end{proof}

Note that from our considerations at the end of section~\ref{tip} we expect
that $\mathbb{P} ( \ket{\alpha} \rightarrow \ket{\beta}) \approx 0$ (using LOCC) for
large $n$, since the event $\{ \ket{\alpha} \rightarrow \ket{\beta} \}$ is
in the complement of $\{ \ket{\alpha} \nleftrightarrow
\ket{\beta}\}$.

We numerically estimate the SLOCC average rate of conversion between random
states.
For this, we generate
incomparable Schmidt vectors following the \emph{Haar measure} and
calculate the probabilities of conversion from the first to the
second and also the maximum conversion probability.
The results shown in Figure~\ref{Fig:ConvRatehaar} clearly shows that,
given a
random pair $\alpha$,
$\beta$, it is very common to have a large probability of conversion from
some of them to the other,
 that is $\mathbb{E}[\text{max}\{P_{S}(\ket{\alpha} \rightarrow
\ket{\beta},P_{S}(\ket{\beta} \rightarrow \ket{\alpha})\}] \gtrsim 0.8$.
Moreover we have a smaller, but still significant, average probability
of
conversion $\mathbb{E}[P_{S}(\ket{\alpha} \rightarrow
\ket{\beta})]$, slightly below $0.6$.

\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{figura_probabilidade_de_conversao_EDITADA2}
\caption{Sampled probability of conversion under SLOCC for randomly chosen
(following the Haar measure)
incomparable states as function
of the dimension of each system. Each symbol represents an average over a distinct set of randomly
chosen pairs. There are two sets of three symbols for each explored
dimension. For each dimension,
the smaller results consider conversion from the first to the second
$\mathbb{E}[P_{S}(\ket{\alpha} \rightarrow
\ket{\beta})]$, and
the larger the maximum
conversion rate $\mathbb{E}[\text{max}\{P_{S}(\ket{\alpha} \rightarrow
\ket{\beta},P_{S}(\ket{\beta} \rightarrow \ket{\alpha})\}]$.}
\label{Fig:ConvRatehaar}
\end{figure}

Also in the probabilistic scenario the presence of an extra state can
improve the probability of conversion between two states \cite{FDY04}. As a chemical catalyst, this extra state
is used, but not consumed, to increase the rate (probability) of a reaction (conversion).  In the
above example, if
$\vec{\kappa}=(0.65,0.35)$, we arrive at:
\begin{subequations}
\begin{align}
\vec{\alpha} \otimes \vec{\kappa}&=(0.39, 0.21, 0.13, 0.13, 0.07, 0.07), \\
\vec{\beta} \otimes \vec{\kappa} &= (0.325, 0.26, 0.175, 0.14, 0.065, 0.035).
\end{align}
\end{subequations}
Therefore $P_{S}(\ket{\alpha} \otimes \ket{\kappa} \rightarrow \ket{\beta}
\otimes \ket{\kappa}) \simeq
0.904$, and $\ket{\kappa}$ can be viewed as a \emph{probabilistic-catalyst}
in the stochastic scenario for the conversion that starts in $\alpha$ and
ends in $\beta$, despite the fact that $P_{S}(\ket{\beta} \otimes
\ket{\kappa} \rightarrow \ket{\alpha} \otimes \ket{\kappa}) =
0.5$, and then $\ket{\kappa}$ does not increases the probability of
conversion for the transformation that starts in $\beta$ and ends in
$\alpha$. In this sense
Jonathan and Plenio pointed out~\cite{daniel} that if $P_{S}(\ket{\alpha}
\rightarrow \ket{\beta})$,
under the best local strategy, is equal to $\alpha_n / \beta_n$, then
this probability can
not be increased by the presence of any catalyst state. Feng
\textit{et.~al.}~\cite{FDY04}
improved
this result, obtaining the following theorem:
\begin{theorem}\label{thm_orientais_prob}
 Let $\vec{\alpha}$ and $\vec{\beta}$ be two $n$--dimensional probability vectors written in non-increasing order.
 There is a probability vector $\vec{\kappa}$ such that $P_{S}(\vec{\alpha}
\otimes \vec{\kappa} \rightarrow \vec{\beta} \otimes \vec{\kappa}) >
P_{S}(\vec{\alpha} \rightarrow
\vec{\beta})$ if, and only if,

\begin{equation}
P_{S}(\vec{\alpha} \rightarrow \vec{\beta}) <
\text{\normalfont{min}}\left\lbrace
\frac{\alpha_n}{\beta_n},1 \right\rbrace .
\end{equation}
\end{theorem}

A deeper connection between LOCC catalysis and its stochastic counterpart
connects the probability of occurrence of the event
$ \{ \alpha \rightarrow \beta \}$ and the maximal probability of stochastic
conversion, $P_{S}(\ket{\alpha}\otimes \ket{\phi}\rightarrow
\ket{\beta} \otimes \ket{\phi})$.

\begin{proposition} \label{boundlemma}
 Let $\vec{\alpha}$ and $\vec{\beta}$ be a pair of random independent
Schmidt vectors with same size $n$. Then
\begin{equation}
\mathbb{P}[\sup_{\phi} \{P_{S}(\ket{\alpha} \otimes
\ket{\phi} \rightarrow
\ket{\beta} \otimes \ket{\phi}) \} > P_{S}(\ket{\alpha} \rightarrow
\ket{\beta}) ] \quad \geq \quad  \frac{1}{2} - \mathbb{P}(\ket{\alpha}
 \rightarrow \ket{\beta}).
\end{equation}
\end{proposition}

\begin{proof}

\begin{subequations}
\begin{align}
 \mathbb{P}[P_{S}(\ket{\alpha} \otimes \ket{\phi} \rightarrow
\ket{\beta} \otimes \ket{\phi}) > P_{S}(\ket{\alpha} \rightarrow
\ket{\beta}) ] & = 1 - \mathbb{P} \{  P_{S}(\ket{\alpha} \rightarrow
\ket{\beta}) = \text{min}(\alpha_{n}/ \beta_{n},1) \} \label{l1} \\
& \geq 1  -
\mathbb{P}\{ P_{S}(\ket{\alpha} \rightarrow \ket{\beta}) = \alpha_n /
\beta_{n} \}
- \mathbb{P} \{ \ket{\alpha} \rightarrow \ket{\beta} \}  \label{l2} \\
& \geq \frac{1}{2} - \mathbb{P} \{ \ket{\alpha} \rightarrow \ket{\beta}
\}. \label{l3}
\end{align}
\end{subequations}
Where~\eqref{l1} comes from Thm~\ref{thm_orientais_prob}, \eqref{l2} from
set theory and Proposition \ref{conexao}, and finally \eqref{l3} from
$\{P_{S}(\ket{\alpha} \rightarrow
\ket{\beta}) = \alpha_n /
\beta_{n} \} \subseteq \{ \alpha_n \leq \beta_{n}\}
$.
\end{proof}
\begin{remark}
   From Remark~\ref{remark4} we know that as $n$ grows, $ \mathbb{P}(\ket{\alpha}
 \rightarrow \ket{\beta})\approx 0$, so we conclude from Proposition~\ref{boundlemma} that $\mathbb{P}[P_{S}(\vec{\alpha} \rightarrow \vec{\beta}) <
\text{\normalfont{min}}\left\lbrace
\frac{\alpha_n}{\beta_n},1 \right\rbrace]\gtrsim 1/2$.
\label{remark2}
\end{remark}

In Ref.~\cite{FDYIEEE} a necessary and sufficient
condition for a state $\ket{\kappa}$ to work as a probabilistic catalyst was provided:
\begin{theorem}\label{theorem_feng}
 Suppose that $\vec{\alpha}$ and $\vec{\beta}$ are two non-increasingly ordered
 $n$--dimensional probability vectors, and $P(\vec{\alpha} \rightarrow \vec{\beta})
 < \min\left\lbrace\frac{\alpha_n}{\beta_n},1\right\rbrace$. Define
 \begin{equation} \label{eq_oriente}
 L= \left\lbrace l; 1<l<n, \,\, \mbox{and} \,\, P(\vec{\alpha} \rightarrow \vec{\beta}) =
\frac{E_{l}(\vec{\alpha})}{E_{l}(\vec{\beta})}\right\rbrace.
 \end{equation}
 %
 Then a non-increasingly ordered $k$-dimensional probability vector $\vec{\kappa}$ serves
 as a probabilistic catalyst for the conversion from $\ket{\alpha}$ to $\ket{\beta}$ if, and
 only if, for all $r_1, r_2, ..., r_k \in L \cup\{n+1\}$ satisfying $r_1 \geq r_2 \geq ...
  \geq r_k \neq n+1$, there exist $i$ and $j$, with $1 \leq j < i \leq k$, such that
  \begin{align}\label{eq_desigualdades}
 \frac{\kappa_i}{\kappa_j} < \frac{\beta_{r_j}}{\beta_{r_{i}-1}} \,\, \mbox{or} \,\,
 \frac{\kappa_i}{\kappa_j} > \frac{\beta_{r_{j}-1}}{\beta_{r_{i}}}.
\end{align}
By definition, whenever  one of the
inequalities~(\ref{eq_desigualdades}) includes an index  $n+1$, it is considered to be
violated, so the other one must necessarily be satisfied.
\end{theorem}
We should stress a couple of facts about the set $L$: first of all, note
that it is a key ingredient for identifying catalysts for a given
conversion, since in a certain sense it determines which indexes are
really important for the comparison between
$\vec{\beta}$ and $\vec{\kappa}$. Secondly observe that typically $L$ has
only
one element $l$, {\textit{i.e.}},  the minimum which determines the
probability of
conversion (see Thm. 2) is non-degenerate.

\section{Self-Catalysis under SLOCC}\label{sefslocc}

The phenomenon of self-catalysis can also take place when considering conversions under SLOCC.
Namely, if a conversion $\ket{\alpha}\rightarrow\ket{\beta}$ takes place with optimal probability
$0<p<1$, it can be the case that the optimal probability for
$\ket{\alpha}\otimes\ket{\alpha}\rightarrow\ket{\beta}\otimes\ket{\alpha}$ be $p' > p$.
Indeed, for the same Schmidt vectors $\vec{\alpha}$ and $\vec{\beta}$ given by Eq.~\eqref{exampleslocc}, there is a gain in the probability of conversion
if we use the state $\ket{\alpha}$ itself as a catalyst. Using Eq.~\eqref{formulavidal}, we have:
\begin{equation}
P_{S}(\ket{\alpha}\otimes\ket{\alpha}\rightarrow
\ket{\beta}\otimes\ket{\alpha})\simeq
0.889>0.800 = P_{S}(\ket{\alpha}\rightarrow \ket{\beta}).
\end{equation}

As it happens in the context of LOCC operations, the conversion between states can depend on the
number of attached copies of $\ket{\alpha}$.
Table~\ref{table2} shows, for the example
we are considering (given by Eq.~\eqref{exampleslocc}), how the probability
of
conversion increases with the number of copies of $\ket{\alpha}$ to be used
as catalyst.

This example may suggest that, by increasing the number of copies of $\ket{\alpha}$, the
probability of conversion approaches one. This is not always the case, however. Note that we
can bound from above the probability in Thm.~\ref{thm-Vidal} for the
pair $\ket{\alpha}\otimes
\ket{\alpha}^{\otimes N}$ and $\ket{\beta}\otimes \ket{\alpha}^{\otimes N}$,
since
$P_{S}(\ket{\alpha}\otimes
\ket{\alpha}^{\otimes
N}\rightarrow \ket{\beta}\otimes \ket{\alpha}^{\otimes N})\leq
\alpha_{n}/\beta_{n},$
for all $N\geq 1$. Therefore, as long as
$\alpha_{n}/\beta_{n}<1$, no matter how
many copies of $\ket{\alpha}$ we have, the probability of conversion will
not exceed $\alpha_{n}/\beta_{n}$. The previous reasoning allows us to
state the following Proposition:
\begin{proposition}
 Let $\vec{\alpha}$ and $\vec{\beta}$ be a pair of ordered Schmidt vectors
with $n$
non-null components. If $\alpha_n / \beta_n < 1$, then
\begin{equation}
 P_{S}(\ket{\alpha} \otimes \ket{\alpha}^{\otimes N} \rightarrow
\ket{\beta} \otimes
\ket{\alpha}^{\otimes N}) \leq \frac{\alpha_n}{\beta_n}, \,\, \forall \,\,
N \geq 0.
\end{equation}
\end{proposition}

For example, given
$\vec{\alpha}=(0.60,0.21,0.10,0.09)$ and
$\vec{\beta}=(0.55,0.25,0.10,0.10)$,
 we have a pair of
incomparable states with
$P_{S}(\ket{\alpha}\rightarrow
\ket{\beta})=0.88$, but since $\alpha_{4} / \beta_{4} = 0.9$,
the probability of conversion under SLOCC using self-catalysis is limited by
$0.9$ and indeed, for this case, $N=1$ is already optimal, since
$P_{S}(\ket{\alpha} \otimes \ket{\alpha} \rightarrow \ket{\beta} \otimes
\ket{\alpha})=0.9$. Analogously,
for the pair $\vec{\alpha^{\prime}}=(0.40,0.34,0.15,0.11)$ and
$\vec{\beta^{\prime}}=(0.50,0.21,0.17,0.12)$, Table~\ref{table3} shows
the behavior of $P_{S}(\ket{\alpha^{\prime}} \otimes
\ket{\alpha^{\prime}}^{\otimes
N} \rightarrow
\ket{\beta^{\prime}} \otimes
\ket{\alpha^{\prime}}^{\otimes N})$ with the number $N$ of copies of
$\ket{\alpha^{\prime}}$, and since $\alpha^{\prime}_{n}/\beta^{\prime}_{n}
= 0.91667 < 1$ the probability of conversion may increase, but can not
reach 1.
Moreover, there is the case (see
Table~\ref{table4}) where, for a given pair $\vec{\alpha}, \vec{\beta}$ of
incomparable Schmidt
vectors, the probability of conversion increases
monotonically
with respect to $N$ and, for an $N _{0}< \infty $,
$P_{S}(\ket{\alpha} \otimes
\ket{\alpha}^{\otimes N_{0}} \rightarrow \ket{\beta} \otimes
\ket{\alpha}^{\otimes N_{0}})=1$ and, by Proposition \ref{conexao}, $\ket{\alpha} \otimes
\ket{\alpha}^{\otimes N_{0}} \rightarrow
\ket{\beta} \otimes \ket{\alpha}^{\otimes N_{0}}$.

\begin{table}
\begin{center}
\begin{tabular}{c | c}
\hline
$\sharp$ Copies (N) & $P_{S}(\ket{\alpha}\otimes\ket{\alpha}^{\otimes N}
\rightarrow \ket{\beta}
\otimes
\ket{\alpha}^{\otimes N})$  \\

\hline
 0&  $\simeq$ 0.800\\
\hline
 1&  $\simeq$ 0.889\\
\hline
  2&  $\simeq$ 0.907 \\
\hline
 3&  $\simeq$ 0.926 \\
\hline
  4&  $\simeq$ 0.932 \\
\hline
 5&  $\simeq$ 0.940\\
\hline
 6&  $\simeq$ 0.944\\
\hline
 7&  $\simeq$ 0.947\\
\hline
 8&  $\simeq$ 0.950\\
\hline
 9&  $\simeq$ 0.952\\
\hline
10&  $\simeq$ 0.955\\
\end{tabular}
\caption{Increase of optimal probability for the conversion
$\ket{\alpha}\rightarrow\ket{\beta}$, where $\vec{\alpha}=(0.6,0.2,0.2)$ and
$\vec{\beta}=(0.5,0.4,0.1)$, with the number of copies $N$ of $\ket{\alpha}$
used as a
catalyst.}\label{table2}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{c | c}
\hline
$\sharp$ Copies (N) & $P_{S}(\ket{\alpha^{\prime}} \otimes
\ket{\alpha^{\prime}}^{\otimes N}
\rightarrow
\ket{\beta^{\prime}} \otimes
\ket{\alpha^{\prime}}^{\otimes N})$  \\

\hline
 0&  $\simeq$ 0.8965\\
\hline
 1&  $\simeq$ 0.9038\\
\hline
  2&  $\simeq$ 0.9072 \\
\hline
 3&  $\simeq$ 0.9092 \\
\hline
  4&  $\simeq$ 0.9105 \\
\hline
 5&  $\simeq$ 0.9109\\
\hline
 6&  $\simeq$ 0.9110\\
\end{tabular}
\caption{Increase of optimal probability for the conversion
$\ket{\alpha^{\prime}}\rightarrow\ket{\beta^{\prime}}$, where
$\vec{\alpha^{\prime}}=(0.40,0.34,0.15,0.11)$ and
$\vec{\beta}=(0.50,0.21,0.17,0.12)$, with the number of copies $N$ of
$\ket{\alpha^{\prime}}$ used as a
catalyst.}\label{table3}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{c | c}
\hline
$\sharp$ Copies (N) & $P_{S}(\ket{\alpha}\otimes\ket{\alpha}^{\otimes N}
\rightarrow \ket{\beta}
\otimes
\ket{\alpha}^{\otimes N})$  \\

\hline
 0&  $\simeq$ 0.600\\
\hline
 1&  $\simeq$ 0.818\\
\hline
  2&  $\simeq$ 0.911 \\
\hline
 3&  $\simeq$ 0.957 \\
\hline
  4&  $\simeq$ 0.981 \\
\hline
 5&  $\simeq$ 0.994\\
\hline
 6&  $=$ 1\\
\hline
\end{tabular}
\caption{Increase of optimal probability for the conversion
$\ket{\alpha}\rightarrow\ket{\beta}$, where
$\vec{\alpha}=(0.928,0.060,0.006,0.006)$ and
$\vec{\beta}=(0.950,0.030,0.0195,0.0005)$, with the number of copies $N$ of
$\ket{\alpha}$ used as a
catalyst.}\label{table4}
\end{center}
\end{table}

A particular case of Theorem~\ref{theorem_feng} allows us to obtain a
necessary and sufficient
condition for having probabilistic self-catalysis for a single copy:

 \noindent
 \textbf{Criterion}\emph{
  Let $\vec{\alpha}$ and $\vec{\beta}$ be two $n$--dimensional Schmidt vectors with
$P_{S}(\vec{\alpha} \rightarrow \vec{\beta})
 < \min\left\lbrace \frac{\alpha_n}{\beta_n},1 \right\rbrace$ and
 \begin{equation}
 L= \left\lbrace l; 1<l<n, \,\, \mbox{and} \,\, P_{S}(\vec{\alpha}
\rightarrow \vec{\beta}) =
\frac{E_{l}(\vec{\alpha})}{E_{l}(\vec{\beta})}\right\rbrace .
 \end{equation}
 %
The vector $\vec{\alpha}$ serves as a probabilistic self-catalyst for the transformation from
$\ket{\alpha}$ to $\ket{\beta}$ if, and  only if, for all $r_1, r_2, ..., r_n \in L \cup\{n+1\}$
satisfying $r_1 \geq r_2 \geq ... \geq r_n \neq n+1$, there exist $i$ and $j$, with $1 \leq j <  i
\leq n$, such that
\begin{align}
 \frac{\alpha_{i}}{\alpha_{j}} &< \frac{\beta_{r_j}}{\beta_{r_{i}-1}}
\label{ineq5} \\
&\mbox{\text{or}}  \nonumber \\
 \frac{\alpha_{i}}{\alpha_{j}} &> \frac{\beta_{r_{j}-1}}{\beta_{r_{i}}}.\label{ineq4}
\end{align}
Whenever  one of the inequalities~\eqref{ineq5} or~\eqref{ineq4}
has an index $n+1$, it is considered to be
violated, so the other one must necessarily be satisfied.

\begin{remark}
 Again, we observe that typically $L$ has only one element $l$, since the minimum which
determines the probability of conversion is non-degenerate with probability $1$.
\label{remark3}
\end{remark}

\subsection{{Self-Catalysis under SLOCC for random Schmidt vectors}}\label{tip2}

The criterion above, together with Proposition~\ref{boundlemma} and the
behaviour of $L$, have interesting consequences for the probability of
occurrence of self-catalysis.

From Remark~\ref{remark2}, we know that the event
$[P_{S}(\ket{\alpha}\rightarrow\ket{\beta})<\text{min}\{\frac{\alpha_{n}}{\beta{n}},1\}]$ has
probability $\gtrsim 1/2$. Conditioning on this event we can then analyze the validity of
Ineqs.~\eqref{ineq5} and \eqref{ineq4}. With probability $1$ we must have
$L=\{l\}$, for some $1<l<n$. Therefore, we can choose the indexes
$r_{i}$ in only two ways: either $r_{1}=r_{2}=...=r_{n}=l$ or $r_{1}=r_{2}=...=r_{j}=n+1$ and $r_{j+1}=...=r_{n}=l$, for some $j$.
For the second case, one can always satisfy one of the inequalities using the index $n+1$. For the
first case, the r.h.s of the inequalities always have index $l$ and we can lower bound the
probability for at least one of Inequalities \eqref{ineq4} to be valid:
\begin{align}
\mathbb{P}[\max_{1\leq j<i\leq n}\{\frac{\alpha_{i}}{\alpha_{j}}\}>\frac{\beta_{l-1}}{\beta_{l}}]\geq
\mathbb{P}[\max_{1<j\leq n}\{\frac{\alpha_{j-1}}{\alpha_{j}}\}>\max_{1<j\leq n}\{\frac{\beta_{j-1}}{\beta_{j}}\}]=1/2,
\end{align}
using that $\max_{1\leq j<i\leq n}\{\frac{\alpha_{i}}{\alpha_{j}}\}=\max_{1< j\leq
n}\{\frac{\alpha_{j-1}}{\alpha_{j}}\}$, since $\vec{\alpha}$ is ordered, and the fact
that the two random variables on the second term are independent and identically distributed. Putting these together
we get that the probability for having SLOCC self-catalysis is $\gtrsim \frac{1}{4}.$

\begin{figure}[htbf]
\centering
\includegraphics[scale=0.5]{figura_SLOCC_haar_EDITADA_colorida2}
\caption{Probability of finding a pair of states exhibiting self-catalysis
under SLOCC, $\mathbb{P}[P_{S}(\ket{\alpha} \otimes \ket{\alpha}
\rightarrow
\ket{\beta} \otimes \ket{\alpha}) > \mathbb{P}(\ket{\alpha} \rightarrow
\ket{\beta}) ]$, as function
of the dimension of each system. Each symbol represents an average over a
new set of randomly (Haar) chosen pairs. There are three symbols for each
explored
dimension.}
\label{cris3haar}
\end{figure}

We have also numerically investigated the typicality of probabilistic self-catalysis by
1) sorting a pair of incomparable Schmidt vectors
of same fixed dimension; 2) counting how many
of them show the effect; and 3) computing the average gain in probability. To
be more specific, we
consider as a success case the situation where the pairs are such that
$p_{1}=P_{S}(\ket{\alpha}\rightarrow\ket{\beta})<P_{S}(\ket{\alpha}\otimes
\ket{\alpha}\rightarrow\ket{\beta}\otimes \ket{\alpha})=p_{2}$ and we compute the average value of
$p_{2}-p_{1}$. In order to avoid counting cases where $p_{1}<p_{2}$ due to numerical
fluctuations, we only consider as valid those vectors where $p_{2}>\left(1+10^{-5}\right)p_{1}$.
The results are shown in Figures~\ref{cris3haar} and \ref{cris4haar}.

\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{figura_SLOCC_media_haar_EDITADA_colorida2}
\caption{Average gain of probability,  $\mathbb{E}[P_{S}(\ket{\alpha}
\otimes \ket{\alpha} \rightarrow \ket{\beta} \otimes
\ket{\alpha})-P_{S}(\ket{\alpha} \rightarrow \ket{\beta})]$, as a function
of the size of Schmidt vectors (sampled following Haar measure), considered
only those pairs where
self-catalysis occurs, {\textit{i.e.}},  $P_{S}(\ket{\alpha}
\otimes \ket{\alpha} \rightarrow \ket{\beta} \otimes
\ket{\alpha}) > P_{S}(\ket{\alpha} \rightarrow \ket{\beta})$.}
\label{cris4haar}
\end{figure}
First note that they are consisted with the lower bound of $1/4$ estimated before. Comparing with the
deterministic case, probabilistic self-catalysis is much more frequent, as expected. Even more, here we \emph{do not} have
the same qualitative behaviour: the probability of having a pair of incomparable states exhibiting
self-catalysis increases monotonically with the size of the Schmidt
vectors
and seems to be
converging to a  value around $0.6$.
Meanwhile, this is not the behaviour of the average probability gain, shown in
Figure~\ref{cris4haar}.
The average gain in probability is
relatively small for all system sizes, and decreases even more for larger
sizes. But note that we consider only one copy of $\ket{\alpha}$
attached, and while for many copies the average gain must be
larger. It is important to recall Fig.~\ref{Fig:ConvRatehaar}, however, which tell
us that a pair $\left( \vec{\alpha}, \vec{\beta}\right)$ has, on average, a probability
of direct conversion close to $0.6$, which naturally bounds the catalytic gain to about $0.4$.

Finally,  Figure~\ref{Fig:Dpagainstphaar} represents, for
randomly chosen pairs of
incomparable Schmidt vectors with size $n=45$, the self-catalytic
probability gain \emph{versus} direct conversion rates.
The diagonal straight line just represents saturation of probability.
Some concentration close to the horizontal axis is natural, representing the cases where self-catalysis does not happen.
However it is not clear why there is the bold concentrated
cloud in red, where the majority of the pairs fit.
It empirically means that the most typical situation for a pair of Schmidt vectors of the same size is to have a large probability of conversion and to have a considerable (but not maximal) self-catalytical gain.

\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{colorido_tam45_editado2}
\caption{Self-catalytic probability gain and direct conversion rates for
randomly (Haar) chosen pairs of
incomparable Schmidt vectors with size $n=45$. The colour represents the
number of pairs per pixel.}
\label{Fig:Dpagainstphaar}
\end{figure}

\section{Conclusions}\label{concl}

In this paper we have shown the possibility of self-catalytic entanglement
conversion for
both LOCC and SLOCC scenarios by providing explicit examples of them.
We have explored numerically and by arguments of typicality  how frequent they are by showing that it is much more
common in SLOCC case
than the deterministic one, but despite the fact that self-catalysis
under SLOCC become more common as the systems sizes increase, the direct
self-catalysis decreases with the dimension.
Moreover, we also investigated how the phenomenon may depend on the number of copies used as catalyst, finding examples of different behaviours, running from cases where there is no gain in considering multiple copies, to cases where the conversion becomes deterministic for a finite number  of copies.
Since we rest on numerical techniques, we could not guarantee the existence of a transition where the probability of conversion asymptotically goes to $1$.
Finally, we computed the average gain in probability in the SLOCC case, showing that
this gain, as in the LOCC case, has a global maximum, and also decreases
with larger systems sizes.

{In answering the question about existence of self-catalysis, we obtained
many results, not
only on deterministic and probabilistic self-catalysis, but also on ordinary
catalysis.
About the typicality of self-catalysis, our data support two conjectures:
under LOCC, the
probability of finding a self-catalytic reaction increases monotonically
attaining a global maximum for a dimension about $20$.
We believe that the origin of this
overall non-monotonic behaviour is due the competition between the
\emph{sizes} of each Schmidt vectors and the measure \emph{concentration}
phenomenon}. On the other hand, the data suggests that under SLOCC the probability of finding a self-catalytic reaction increases
monotonically with the dimension.

In a sense, we estimated numerically the volumes of the sets of pairs of
Schmidt vectors where the
phenomena take place, but it was not possible to characterize completely the asymptotic behavior
with the vectors sizes.
{In fact, some of our numerical results have a reasonable dependence on the way  we sort
the random Schmidt vectors}.
That is something to be explored elsewhere.
Perhaps a better understanding of
the geometry of the sets involved could
help in that analysis.

\begin{acknowledgments}
We would like to thank Tha\'is Matos Ac\'acio, Jos\'e Am\^ancio,  Luiz Fernando Faria, Eduardo Mascarenhas,  Fernando de Melo, Roberto Oliveira,  and the anonymous referee for useful comments and stimulating discussions, as well as CNPq, CAPES and FAPEMIG for financial support.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


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Apr 30 Our antibiotics are no match for superbugs, and it's a 'global crisis,' UN report says
Originally published on CNN
Common diseases such as tuberculosis are becoming untreatable while lifesaving medical procedures like surgeries carry increasing risks because of "alarming levels" of resistance to antimicrobial drugs, according to a new United Nations committee report. Resistance to antibiotics, antivirals, antifungals and antiprotozoals, which has been seen in countries of all income levels, is now a "global crisis," the authors wrote.
The Ad hoc Interagency Coordination Group on Antimicrobial Resistance, made up of scientific experts and representatives of major UN scientific agencies, was convened in March 2017 and tasked with providing practical guidelines for global action to address antimicrobial resistance.
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Focus on One Health
Two million Americans develop antibiotic resistance infections each year, and 23,000 die from those infections, according to the US Centers for Disease Control and Prevention.
The new report emphasizes that misuse and overuse of existing antimicrobial agents in humans as well as animals and plants accelerate the development and spread of resistance. Factors contributing to the emergence and spread of drug-resistant pathogens include inadequate access to clean water, sanitation and hygiene; poor infection and disease prevention; lack of equitable access to affordable antimicrobials, vaccines and diagnostics; and weak health, food and feed production, food safety and waste management systems.
Ultimately, antimicrobial resistance threatens global progress toward improved health, food security, clean water and sanitation, and sustainable consumption and production, according to the Interagency Coordination Group.
Melinda Pettigrew, a professor of epidemiology in the Department of Epidemiology of Microbial Diseases at the Yale School of Public Health, wrote in an email that the report is especially notable for its emphasis on One Health, a concept that recognizes how the health of people is connected to the health of animals and the environment and that seeks to achieve optimal health outcomes by recognizing this interconnection, according to the CDC.
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Three urgent threats in the United States
The problem of antimicrobial resistance includes "more resistant" infections that lead to extensive, expensive treatment, if not death, experts say. Dramatically increased health care expenditures due to uncontrolled antimicrobial resistance, plus the impact on food production and livelihoods, could result in economic damage comparable to that of the 2008-09 global financial crisis, according to the authors of the new report.
In the United States, the CDC describes three "urgent" antimicrobial resistant threats: C. difficile or C. diff, which causes life-threatening diarrhea and colitis (an inflammation of the colon) in, mostly, patients who have recently been given antibiotics, infects 500,000 people annually and leads to 15,000 deaths. Carbapenem-resistant Enterobacteriaceae (CRE), which is known as the "nightmare bacteria" and is usually acquired in health care facilities, causes 9,000 infections and results in about 600 deaths annually. Finally, N. gonorrhoeae, which causes the sexually transmitted disease gonorrhea and has developed resistance to antibiotics, infects 246,000 people each year.
Meanwhile, the World Health Organization reports that the number of drugs in development is simply not enough. Although dozens of antibiotics and biologics -- medical products made from natural sources -- may be in the pipeline, new drugs have just a 14% chance of gaining approval for use in humans.
The report concludes with a summary of five recommendations: Accelerate progress (including implementation of One Health National Antimicrobial Resistance Action Plans); innovate to secure the future (including development of new antimicrobials); collaborate for more effective action; invest for sustainable response; and strengthen accountability and global governance.
What can people do? Pettigrew advises, "Work to reduce the use of antibiotics in livestock and fish farming -- especially for growth promotion. And make sure family and community members and children have their vaccines!"
Pettigrew said that the ongoing measles outbreak illustrates this point, although all antiviral vaccines, including the flu shot, can help in the fight against antimicrobial resistance.
Vaccines "prevent secondary bacterial infections that might require antibiotics," and they "prevent the use of broad spectrum antibiotic, empiric, and inappropriate prescribing," she explained. "Fewer antibiotics will be used if we prevent people from getting sick in the first place."
Apr 30 Deadly armpit superfungus from Japan has killed eight people in UK hospitals
Apr 26 Research finds some bacteria travel an alternate path to antibiotic resistance
Oct 16 Drug-resistant bugs could cause more deaths than cancer by 2050
Jan 20 Anti-vaxxers, air pollution, superbugs: These are 10 big global health threats, WHO report says
May 30 Tourists are spreading superbugs across international borders, report warns

redpajama_set_name: RedPajamaCommonCrawl


text:
Marc Prensky (; * 15. März 1946 in New York City) ist ein US-amerikanischer Autor, Lehrer und Manager.
Er prägte den Begriff Digital Native und schuf Arbeiten zu digitalen Bildungsformen, u. a. zu Digital Game-based Learning.

Leben
Prensky hat Abschlüsse am Oberlin College (1966), an der Yale University (1968) und an der Harvard Business School (1980) gemacht. Zu Beginn seiner beruflichen Tätigkeit arbeitete er als Lehrer und Unternehmer. Derzeit ist er als Autor und Berater tätig.

Leistungen
Prensky beschäftigt sich mit der Frage, wie im digitalen Zeitalter moderne Bildungsformen aussehen können. Er prägte 2001 die Begriffe Digital Native und Digital Immigrant durch seinen Aufsatz "Digital Natives, Digital Immigrants".

Werke

Aufsätze
 Marc Prensky: Digital Natives, Digital Immigrants. (PDF; 135 kB) In: On The Horizon, MCB University Press, Vol. 9, No. 5, Oktober 2001, .
 Marc Prensky: Digital Natives, Digital Immigrants, Part II: Do They Really Think Differently? (PDF; 252 kB) In: On The Horizon, MCB University Press, Vol. 9, No. 6, Dezember 2001, .
 Viele weitere seiner Aufsätze sind auf seiner Webseite frei zugänglich.

Bücher
 Digital Game-Based Learning (2001)
 Don't Bother Me Mom – I'm Learning (2006)
 Teaching Digital Natives – Partnering for Real Learning (2010)
 From Digital Natives to Digital Wisdom: Hopeful Essays for 21st Century Learning (2012)
 Brain Gain: Technology and the Quest for Digital Wisdom (2012)

Weblinks
 http://www.marcprensky.com/
 Video interview with Marc Prensky about Game Based Learning from 1999

Einzelnachweise

Autor
US-Amerikaner
Geboren 1946
Mann

redpajama_set_name: RedPajamaWikipedia


text:
Le Vaux de Vicomte
The Château de Vaux-le-Vicomte is a baroque chateau located 55 km southeast of Paris. It was built between 1658 to 1661 for Nicolas Fouquet, Marquis de Belle Ile, Viscount of Melun and Vaux, and Superintendent of the finances of Louis XIV.
The château was one of the most influential architectural works built in Europe in the mid-17th century and the most elaborate house built in France after the Chateau de Maisons-Lafitte, designed by Francois Mansart from 1630 to 1651 (below).
Once a small château located between the royal residences of Vicennes and Fontainebleau, the estate of Vaux-le-Vicomte was purchased by Nicolas Fouquet in 1641. At that time he was an ambitious twenty-six year-old member of the Parliament of Paris. Fouquet was an avid patron of the arts and attracted many artists with the gifts and encouragements he poured on them.
When Fouquet became King Louis XIV's superintendent of finances in 1657, he commissioned Le Vau, Le Brun and Le Nôtre to renovate his estate and garden to match his grand ambition. Fouquet's artistic and cultivated personality subsequently brought out the best in the three.
At Vaux-le-Vicomte the architect Louis Le Vau, the gardener Andre le Notre, and the painter-decorator Charles Le Brun, worked together on a large-scale project for the first time. Their collaboration marked the beginning of the Louis XIV 'en suite' style of architecture, interior design, works of art, and garden landscaping. The garden's use of a baroque axis that extends to infinity is an example of this style.
To secure the necessary grounds for the elaborate plans for Vaux-le-Vicomte's garden and castle, Fouquet purchased and demolished three villages. The displaced villagers were then employed in the upkeep and maintenance of the gardens. It was said to have employed eighteen thousand workers and cost as much as sixteen million livres.
The château and its patron became for a short time a focus for fine feasts, literature and arts. The poet La Fontaine and the playwright Moliere were among the artists close to Fouquet. At the inauguration of Vaux-le-Vicomte, a Molière play was performed, along with a dinner event organized by Francois Vatel and an impressive firework show.
The château was lavish, refined, and dazzling to behold, but these characteristics proved tragic for its owner: the king had Fouquet arrested shortly after a famous fête that took place on 17 August 1661 where Molière's play 'Les Fâcheux' debuted. The celebration had been too impressive and the superintendent's home too luxurious. Fouquet's intentions were to flatter the King: part of Vaux-le-Vicomte was actually constructed specifically for the king, but Fouquet's plan backfired. Jean-Baptiste Colbert led the king to believe that his minister's magnificence was funded by the misappropriation of public funds. Colbert, who then replaced Fouquet as superintendent of finances, arrested him.
After Fouquet was arrested and imprisoned for life, and his wife exiled, Vaux-le-Vicomte was placed under sequestration. The king seized, confiscated or purchased 120 tapestries, the statues, and all the orange trees from Vaux-le-Vicomte. He then sent the team of artists (Le Vau, Le Nôtre and Le Brun) to design what would be a much larger project than Vaux-le-Vicomte, the palace and gardens of Versailles.
Madame Fouquet recovered her property ten years later and retired there with her eldest son. In 1705, after the death of her husband and son, she decided to put Vaux-le-Vicomte up for sale.
The Marechal de Villars became the new owner although he had never even set eyes on the place. In 1764, the Maréchal's son sold the estate to the Duc de Praslin, whose descendants would maintain the property for over a century. In 1875, after thirty years of neglect, the estate was sold to Alfred Sommier at a public auction. The château was empty, some of the outbuildings had fallen into ruin. The huge task of restoration and refurbishment began under the direction of the renowned architect Gabriel-Hippolyte Destailleur. When Sommier died in 1908, the château and the gardens had recovered their original appearance. His son and daughter-in-law completed the task. It remains in the hands of his family today, owned by the Comte de Vogue.
Above: The Muse's Salon
Below: The Grand Salon
Above: The King's Room
Below: The Hercules Room
Below: Mrs Fouquet's Room
Above: The crown fountain
Below: Night shot of the Chateau with thousands of candles
Below: A small video of views of the chateau
Labels: Baroque Castle, David Lord Cowell, Gardens, Le Brun, Le Notre, Le Vau, Le Vaux de Vicomte, Lord Cowell
Opera Unleashed...
Opera Unleashed is an opera company recently formed by a school friend of mine, Evelyne Bourton. Evelyne graduated from the University of Waikato with a BMus(1st class Hons) and a post-graduate diploma in linguistics as a Sir Edmund Hillary Scholar.
A regular on the concert platform, she has sung in concert with Dame Malvina Major and renowned New Zealand Bass Grant Dickson who she is currently studying under.
She has sung as a soloist with Hamilton Cantando Choir, University of Waikato Choir and was the winner of the 2008 Hamilton Civic Choir solo artist scholarship. As a member of the Chapman Tripp New Zealand Opera Chorus she has sung in operas including Lucia di Lammermoor, Turandot and the Mahler 3rd Symphony.
She sang the role of The Dragonfly and the French Lady's Chair in The Opera Factory's production of 'The Bewitiched Child' by Maurice Ravel and has been a soloist in three of David Griffths operas. She has been a multiple prize winner at the N.Z Aria competition and has sung with the APO.
This year her opera company is performing La Finta Giardiniera.
The most important event of La finta giardiniera takes place a year before the action begins on stage. Count Belfiore, in a fit of passion, stabs his lover, the Marchioness Violante Onesti. He then flees, believing he has killed her. But Violante is not dead. The wound is grievous, but it heals. Her heart, on the other hand, does not. She is still very much in love with the Count. She disguises herself as a simple gardener's girl and, accompanied by her loyal servant, sets out in search of Belfiore.
A garden with a wide staircase leading to the Mayor's mansion.
The Mayor, Cavalier Ramiro and Serpetta descend the staircase as Sandrina and Nardo work in the garden. Together they praise the lovely day. But their happiness is feigned: Sandrina is wretched because Don Anchise is in love with her; Nardo is frustrated by Serpetta, who teases him but refuses to respond to his affections; Ramiro is bitter about being tossed aside by Arminda; and, because she has set her own cap at the Mayor, Serpetta is angry at Sandrina.
The Mayor is the only happy person in the group. Today is his niece's wedding day, and her suitor is due to arrive at any moment. He also is giddy over his plan to propose to Sandrina, which he does at the first opportunity. Sandrina demurs and, when Serpetta rudely interrupts, makes her escape.
Arminda's betrothed, none other than Count Belfiore, arrives and is swept off his feet by her great beauty. But Arminda is quick to let him know that she is someone to be reckoned with: Woe to you if I catch you being unfaithful, she warns. I will box your ears. The Count then boasts of his deeds and ancestry to the Mayor. His family tree, he says proudly, can be traced to Scipio, Cato and Marcus Aurelius. Don Anchise responds with a mixture of awe and skepticism, as though he doesn't care what sort of buffoon this fellow is - as long as he marries his niece.
In the garden, Arminda finds Sandrina and casually mentions that she is to marry Count Belfiore. Stunned by the news, Sandrina faints. When the Count arrives, Arminda leaves him to watch over Sandrina while she rushes off to fetch her smelling salts. He is shocked to find that this simply dressed gardener's girl is none other than Violante.
As is so appropriate for an opera buffa finale, everything gets turned on its head. Arminda returns and immediately runs into the last person she expects to encounter, her former lover Ramiro, who is approaching from the opposite direction. Sandrina awakens and finds herself looking directly into the eyes of Belfiore. What are they to do?
The Mayor enters and demands an explanation. But no one knows quite what to say. Sandrina wavers, unable to make up her mind about revealing her true identity, and nearly driving Belfiore out of his mind in the process. Arminda suspects that she's being deceived, but she isn't quite sure. The Mayor blames everything on Serpetta; Serpetta in turn blames Sandrina; and Ramiro, on the periphery, is certain only of the fact that Arminda still does not love him.
A hall in the Mayor's palace.
Ramiro discovers Arminda and insists that she hear him out. He upbraids her for her inconstancy. When she refuses to listen, he departs, but not before promising revenge upon his rival. Belfiore enters in some distress, muttering: I have no peace since I found Sandrina. Arminda, overhearing this, confronts him angrily before exiting in the grand manner of a spurned seria heroine.
Sandrina is in the worst kind of dilemma. She has finally found her true love, but she is about to lose him forever to another woman. For reasons of her own, she has refused to reveal her identity. Yet when she encounters Belfiore, the question comes gushing out: Why did you stab me and desert me? The Count, overjoyed, responds: Then you are Violante! But Sandrina quickly reassumes her disguise. No, she says, that is what the poor girl said as she died. No matter, Belfiore says, you have the face of my Violante. He begins to serenade her but, partway through, the Mayor enters. Belfiore takes the Mayor's hand, believing it belongs to Sandrina -- then retreats in embarrassment when he discovers his mistake.
Alone with Sandrina, the Mayor again attempts to woo her. But once again he is interrupted, this time by Ramiro, who arrives with the news from Milan that Count Belfiore is wanted for murder. Don Anchise immediately summons Belfiore for questioning. The Count, thoroughly baffled, implicates himself. In order to save him, Sandrina reveals herself as Violante, and the proceedings break up in some confusion. The Count approaches Sandrina, but she pushes him away. I am not your Violante, she says, I only pretended to be in order to save you.
Moments later, Serpetta arrives to tell the Mayor, Nardo and Ramiro that Sandrina has run away. In reality, Arminda and Serpetta have conspired to abduct her, and she has been carried off and abandoned in the wilderness. The Mayor immediately organizes a search party.
A deserted, mountainous spot.
Sandrina is nearly frightened out of her wits. But, in small groups, her rescuers soon begin to arrive: the Count and Nardo, Arminda, Serpetta and the Mayor. Mistaken identities multiply in the darkness: The Mayor mistakes Arminda for Serpetta, and she him for the Count; the Count believes Serpetta is Sandrina, while she believes him to be the Mayor. Nardo alone manages to find his mistress by following her voice. Ramiro, the gallant cavalier, arrives with footmen carrying torches.
All this confusion is too much for poor Belfiore and Sandrina. While the others bicker, they begin to lose their minds. I am the terrible Medusa! cries Sandrina. I am the fearless Alcides! responds the Count. Everyone looks on in astonishment as they begin to dance.
The Count and Sandrina are certifiably insane, as Nardo discovers. Still believing that they are gods from classical Greece, they pursue him until he distracts them by pointing at the sky. Look at difference between the sun and the moon! he cries. Observe all the lovesick stars! They are entranced and Nardo is able to make his escape.
Events are taking their toll on the Mayor's judgment, too. Arminda begs for permission to marry the Count, and Ramiro demands that he order her to marry him. But he becomes confused and gives in to them both: Do what you want, he says, just do not trouble me any more.
A garden.
The Count and Sandrina gradually awaken after sleeping, at a discreet distance from one another, in the garden. Their madness has passed. Belfiore makes one final appeal, and Sandrina admits that she is, indeed Violante. However, she says, she loves him no more. Sadly, the Count agrees that they should go their separate ways.
But (this is an opera buffa, after all) their feet begin to drag, and they turn back. The mutual attraction of their love is too strong: They fall into each other's arms and then immediately run off to get married.
The Mayor and Arminda are dumbfounded when they hear the news. After they recover from their initial shock, they, along with everyone else, take it all in stride. Arminda decides to marry Ramiro, and Serpetta even decides that Nardo isn't such a bad choice, after all. Only the Mayor is left out, and he accepts his fate philosophically. Perhaps, he says, he will someday meet another gardener's girl.
Peter and I, patrons of the arts that we are, hope to have regular Glyndebourne styled afternoons at Willowbrook Park with Evelyne's help.
Until then, you can enjoy the work of her company at the Hamilton Gardens Festival...
Labels: David Lord Cowell, Evelyne Bourton, Glyndebourne, Opera Unleashed
Favourite Designers Part II: Jacques Garcia...
Jacques Garcia, (born September 23, 1947) is a French architect and interior designer. He has designed many private residences, including his own, the revamped Chateau du Champ de Bataille. He is also noted for his hospitality design, designing many hotels including in Hotel Costes in Paris, Hotel Danieli in Venice, and Hotel Metropole in Monte Carlo.
Above: Foyer of the Hotel Danieli, Venice
Below: Hotel Costes, Paris
Chateau du Champ de Bataille
The Château du Champ-de-Bataille, located in the region Haute Normandie, between the commune of Neubourg and Sainte Opportune du Bosc, in the Department Eure, is a baroque castle built in 17th century. It is situated in the Campagne du Neubourg between the rivers Risle and Iton.
The history of the castle goes back to the tenth century, went it was the site of a definitive battle, hence the name Champ-de-Bataille - Battlefield. The two families who reigned over the region under the French feudal system battled for power. They were led by Guillaume Longue Épée, and Robert le Danois. William (Guillaume) won, and with his victory, Normandy gained its independence.
Much later, in 1651, there was a seminal event: Marquis Alexandre de Créqui-Bernieulle (1628-1703), a friend of the Prince of Conde, was exiled to the region by Cardinal Mazarin, who governed France during the minority of Louis XIV. Crequi then decided to build a magnificent palace which recall the splendor of the Court that he would never know. He built the Château du Champ-de-Bataille between 1653 and 1665. Unfortuantely, Crequi died backrupt, but bequeathed the castle to his nephew, Anne-François d'Harcourt, Duc de Beuvron and governor of Normandy. At that time the house was very dilapidated. D'Harcourt then undertook considerable work to restore the glories of yesteryear. But the Revolution interrupted this gigantic task, which remained unfinished for generations.
It languished in the hands of several owners in a ruinous state until Jacques Garcia purchased it in 1999, and transformed it to the glorious house and gardens it is today.
Initially there were no gardens on the property, but Garcia designed the gardens from period plans.
Above: The French Parterres
Below: The Temple of Leda
Which leads to an elegant segue: the story of Leda and the Swan. This tale comes from Greek mythology. The legend recalls that Zeus came to Leda in the form of a swan and raped or seduced her on the same night she slept with her husband, King Tyndareus of Sparta. According to later Greek mythology, Leda bore Helen and Polydeuces by Zeus whilst bearing Castor and Clytemnestra Tyndareus. In some versions, she laid two eggs from which the children hatched.
Above: Reproduction of a lost Michaelangelo Painting of Leda and the swan
Below: The most famous depiction of Leda and the swan, by Leonardo da Vinci.
Below: A more controversial depiction (does one censor art?)
(If this offends anyone's sensibilities please let me know and I shall remove it directly)
"Leda and the Swan" is a poem by William Butler Yeats describing the swan's seduction of Leda.
A sudden blow: the great wings beating still
Above the staggering girl, her thighs caressed
By the dark webs, her nape caught in his bill,
He holds her helpless breast upon his breast.
How can those terrified vague fingers push
The feathered glory from her loosening thighs?
And how can body, laid in that white rush,
But feel the strange heart beating where it lies?
A shudder in the loins engenders there
The broken wall, the burning roof and tower
And Agamemnon dead.
Being so caught up,
So mastered by the brute blood of the air,
Did she put on his knowledge with his power
Before the indifferent beak could let her drop?
The Official website for his chateau is here
Labels: Chateau du champ de bataille, David Lord Cowell, Favourite Designers, Jacques Garcia, Leda and the swan, Lord Cowell
Favourite Designers Part I: Juan Pablo Molyneux......
Kreativ Blogger Award...
My 30th Birthday!
The Music Room...
The Carriage House / Stables...
The Study....
Wainscoating and Boiserie...
The Gentleman's Gentleman....


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Ardic dinner was in the Ataköy Marina restaurant. We are proud to say that activities will continue during the year.
We wish success to all workers .


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For decades Wakulla has been a mystery, but with each exploratory dive a few of the cave's secrets are beginning to surface.
Until last month, divers thought the cave stretched out like a gargantuan throat from its underwater mouth -- a long, watery tunnel. After several weeks of penetrating a half-mile beyond the cave's mouth, divers have found a vast limestone labyrinth that forks off in at least three directions. Each prong of the fork has offshoots heading into even more uncharted territory.


I just recently graduated from the University of New Hampshire with a degree in Zoology. My strong points include subjects pertaining to life sciences such as biology,chemistry, biochemistry, physiology, ornithology, genetics, evolution & ecology, wildlife ecology, animal behavior, and human behavior & public health.
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Learn more about how our consultant pharmacists work with you and residents to recommend ideal prescription solutions.


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Top 10 Edgar Broughton Band Songs
Eric Kamau 1 month ago No Comments
Feature Photo: Michael Warwick / Shutterstock
Our top 10 Edgar Broughton Band songs list presents the music of an English rock outfit that started its musical pursuits as a blues rock-oriented act. Founded in 1968, the band rose into the late '60s underground blues scene under the moniker The Edgar Broughton Blues Band. Led by the Broughton brothers, drummer Steve and guitarist/vocalist Edgar Broughton, the band flourished in its hometown Warwick, before relocating to London, UK.
The move was necessitated by the need to sign a recording contract to reach a wider audience within and beyond the UK. While the band has been cited as a blues rock outfit, its members were not afraid of venturing into psychedelic rock, acid rock, and protopunk sounds. The move to blend several sub-genres into its blues rock sound saw the band change its name, adopting its well-known moniker, The Edgar Broughton Band.
Unfortunately, the move to blend several musical sub-genres into its signature blues rock sound was at the expense of its guitarist Victor Unitt, who would leave the band before returning to the lineup a while later. Guitarist Victor Unitt was a brief member of The Pretty Things, an English ensemble whose name was inspired by the 1955 Willie Dixon-penned hit "Pretty Things."
Edgar Broughton Band's Career Beginnings
After relocating to London, The Edgar Broughton Band was signed to Blackhill Enterprises, a rock music management establishment founded by Andrew King, Peter Jenner, and members of Pink Floyd. Blackhill Enterprises helped the band secure a recording contract with Harvest Records. Other acts/artists who were signed by Harvest Records in the late '60s and early '70s include Deep Purple, Babe Ruth, David Gilmour, Kevin Ayers, Kate Bush, Roy Harper, and Be-Bop Deluxe. Under Harvest Records, the Edgar Broughton Band issued five studio albums, two of which made it to the UK Albums Chart.
Edgar Broughton Band's Album Releases over the Years
In 1969, the Edgar Broughton Band issued its debut studio album, Wasa Wasa. At the time Wasa Wasa was being recorded, the Edgar Broughton Band was a trio composed of the Broughton brothers and bassist Arthur Grant. The album saw the band blend its signature blues rock sound with psychedelic rock and acid rock inflections. Handling the production of the album was Peter Jenner who has produced for The Clash, Denzil, Ian Dury, T. Rex, and Pink Floyd. "Evil," "Death of An Electric Citizen," "American Boy Soldier," and "Love in the Rain" are the best Edgar Broughton Band songs from the album.
Sing Brother Sing, issued in 1970, marked the band's sophomore studio album. The album finds the band giving us a taste of its psychedelic rock and experimental rock explorations. Notable personnel featured in the technical part of the album was producer Peter Jenner and engineer Peter Mew, who is best known for his work with former Soft Machine founding member Kevin Ayers. The album features musical gems including "Aphrodite," "Psychopath," "Officer Dan," and "There's No Vibrations, But Wait!" Sing Brother Sing made it to number eighteen on the UK Albums Chart.
The Edgar Broughton Band returned in 1971 with its momentous eponymous third studio album. Often regarded as "The Meat Album" by the band's fans, Edgar Broughton Band is by far the most successful album by the band, featuring multiple classics. The album finds the Edgar Broughton Band reuniting with guitarist Victor Unitt who had just left The Pretty Things after recording album tracks for the band's fifth LP, Parachute.
Edgar Broughton Band also featured several other additional musicians making it the band's most musically invested album. Among the featured additional personnel were Roy Harper, Mike Oldfield, David Bedford, violinist Johnny van Derek, P. Harold Fatt, and The Ladybirds. "Evening Over Rooftops," "House of Turnabout," "Thinking of You," and "The Birth" are the best Edgar Broughton Band songs from the album. Edgar Broughton Band peaked at number twenty-eight on the UK Albums Chart.
In 1972, the Edgar Broughton Band returned with its fourth studio album, In Side Out. The album saw the band take on a blend of psychedelic rock and hard rock. Members of the Edgar Broughton Band handled the production of this album. "It's Not You," "Gone Blues," and "Chilly Morning Mama" are the standout tracks off In Side Out.
The band went ahead to issue its fifth studio album Oora in 1973. Songs on this album continued to showcase the Edgar Broughton Band's psychedelic rock influences tinged with hard rock sounds. Working with the Edgar Broughton Band in Oora as part of the additional personnel are David Bedford and Madeline Bell, a former member of the impressive British pop group Blue Mink. Some of the best Edgar Broughton Band songs from Oora include "Eviction," "Things on My Mind," and "Oh You Crazy Boy!"
Three years after parting with Harvest Records, the Edgar Broughton Band issued its sixth studio album, Bandages. Bandages was issued through NEMS Records with songs on the album branding it a primarily psychedelic rock record. "Love Gang" and "Speak Down the Wires" are some of the notable musical gems from Bandages. The band was without the Victor Unitt, with his position filled by guitarist/vocalist John Thomas.
In 1979, the band issued its seventh studio album Parlez-Vous English, as The Broughtons. This punk-oriented album release came after the band's short hiatus. Parlez-Vous English was issued through Infinity Records, a label that was home to Hot Chocolate, New England, Solomon Burke, TKO, and the British pop group Racey. Among the additional personnel on this album is instrumentalist Peter Hope-Evans, a former member of the blues rock act Medicine Head. "Anthem," "Little One," "Meglamaster," and "April in England" are some of the best Edgar Broughton Band songs from the album. The Edgar Broughton Band's eighth album Superchip is a 1982 record issued through Sheet Records. The album is the final record by the band with some of its musical gems including "Innocent Bystanders," "Subway Information," and "The Last Electioneer."
Edgar Broughton Band's Legacy
The Edgar Broughton Band was among the most alluring underground blues acts of the late '60s and early '70s. While blues rock influences mark the band's groundbreaking work, the band has made significant psychedelic rock-tinged records throughout its career. The Edgar Broughton Band was cited by some critics as contemporaries of Hawkwind, Groundhogs, and The Pink Fairies
However, the Edgar Broughton Band cut its own path with its politically conscious approach in some of its songs. The Edgar Broughton Band's career in the early '70s saw the band compared to the American acts/artists MC5 and Captain Beefheart owing to Edgar's similar gritty vocal style. Free concerts made the band quite popular in the UK. Here we present the ten best Edgar Broughton Band songs of all time.
#10- Death of An Electric Citizen
Ushering us to the ten best Edgar Broughton Band songs list is the grand hit "Death of An Electric Citizen." The song was among the first songs the band recorded after signing a recording deal with Harvest Records in June 1969. "Death of An Electric Citizen" went on to be featured as the lead track off the band's debut studio album Wasa Wasa. The song is delivered with Captain Beefheart-inspired vocals.
"Death of An Electric Citizen" feels more like an acid rock hit but the heavy blues tinges echo the Edgar Broughton Band's blues origins. The song features unrelenting basslines and stunningly intense guitar riffs. Members of Electric Citizen, a hard rock/stoner rock band formed in Cincinnati cited the song "Death of An Electric Citizen" to have been the inspiration for the band's name.
#9- Love Gang
Without a doubt, the best place to hunt for the best Edgar Broughton Band songs is the band's albums issued through Harvest Records. The perfect place to find a classic from the Edgar Broughton Band beyond the band's tenure with Harvest Records is the 1976 album Bandages. "Love Gang" proves to be the one breathtaking song from the album worth listing on our ten best Edgar Broughton Band songs list.
The song is the band's fairly successful attempt at issuing a proper boogie rock ballad. However, the band barely managed to avoid its psychedelic rock influences on this track. Nevertheless, it is the lyrical distinction of this hit that makes it quite a solid pick from the album, robbing the hits "Signal Injector," "John Wayne," and "Speak Down the Wires" a chance to feature on our list.
#8- Love in the Rain
Coming in at number eight on our top ten Edgar Broughton Band songs list is the tremendous hit "Love in the Rain." While "Love in The Rain" brings the inner Captain Beefheart self of Edgar's vocals, something more portentous is happening in this song. Like the hit "Evil," which misses out on the list, "Love in the Rain" is an outstanding track that brings out the band's slight drift to some heavy guitar work.
"Love in the Rain" and "Evil"' are among the songs that bring out the pure hard rock influences in the band's acid rock/psychedelic rock sounds of Wasa Wasa. The song's heavy guitar riffs might not sound alluring to blues-rock purists. However, this "heavy rock-oriented" guitar riffs in the two songs open a discussion as to why the Edgar Broughton Band was featured in the Nigel Planer-narrated BBC Four Heavy Metal Britannia documentary as one of the bands that "unknowingly" gave life to heavy metal.
#7- Thinking of You
"Thinking of You" is one of the most sought-after hits by the Edgar Broughton Band from the band's eponymous third studio album. This swinging hit has its arrangement evoking memories of the exquisite 1970 hit "Working Class Hero" by John Lennon, the former co-lead vocalist of The Beatles. Edgar Broughton's vocals in this song are far from his usual Captain Beefheart-Esque grunts.
In this song, Edgar unleashes a vocal delivery that falls somewhere between Pink Floyd's Syd Barrett and Uriah Heep's Ken Hensley. "Thinking of You" features singer Mike Oldfield on the mandolin. The band's drummer Steve Broughton repaid the favor by playing drums in Mike's debut studio album Tubular Bells, issued in May 1973.
#6- Apache Dropout
Number six on our ten best Edgar Broughton Band songs list is the superb hit "Apache Dropout." The song was issued as a single on November 20, 1970, with "Freedom" serving as its B-Side track. "Apache Dropout," a psychedelic rock-infused track, sampled two songs including the 1960 hit "Apache" by The Shadows and "Dropout Boogie" by Captain Beefheart and His Magic Band.
Issued under the production of Peter Jenner, "Apache Dropout" rose to become one of the most popular songs by the Edgar Broughton Band. The song was a mainstream success, rising to a peak position of number thirty-three on the UK Singles Chart. "Apache Dropout" is the Edgar Broughton Band's final song to ever make it to the UK Singles Chart.
#5- Aphrodite
Introducing us to the musical finesse of the Edgar Broughton Band's sophomore studio album is the impressive hit "Aphrodite." "Aphrodite" is highly revered by the band's fans thanks to its beautiful laid-back groove. The vocal delivery on this track feels original with some alluring backing vocals from bassist Arthur Grant. "Aphrodite" is the most popular hit off Sing Brother Sing, forming part of the reason the album made it to the UK mainstream charts.
#4- House of Turnabouts
"House of Turnabouts" is yet another impressive cut off the Edgar Broughton Band's eponymous third studio album. The clear vocals from Edgar Broughton are supplemented by background vocals from the band's bassist Arthur Brown. Away from the nervous-sounding yet anthemic vocals, the first thing we notice with the song's instrumentation is the impressive psychedelic rock influences. Additional hard rock influences in this song echo loud especially after the first verse, thanks to the heavy funky guitar riffs.
#3- Out Demons Out
The third song on our ten best Edgar Broughton Band songs list is the raucous hit "Out Demons Out." As we mentioned earlier, the Edgar Broughton Band had its music a tad bit politically charged. "Out Demons Out" is the one big hit that backs this claim up. The song seemed to have drawn inspiration from The Fugs. Earlier on October 21, 1967, The Fugs arrived at the Pentagon with magician and filmmaker Kenneth Anger as part of the anti-war demonstration.
The massively attended protest found The Fugs chanting repeatedly "Out Demon Out!" Later in 1970, the Edgar Broughton Band dropped their own "rendition" of the ritual chant, the hit "Out Demons Out." The song was a success on the UK Singles Chart, rising to the thirty-ninth spot. "Out Demons Out" stayed on the UK Singles Chart for five weeks. During live concerts back in the '70s, this song used to get the masses wild! The song reminds us of yet another hit "Up Yours," which can easily be described as a goofy political satire.
#2- Hotel Room
"Hotel Room" is one of the most sought-after singles of 1970 by the band. This mellow track feels quite anthemic, giving us a taste of the band's solid musicianship. "Hotel Room" feels quite harmonious with its guitar riffs evoking the pure rock ecstasy of the '70s. This haunting tune strikes a balance between blues rock and folk-rock influences.
The dreamy quality is a result of a beautiful arrangement and impressive lyrical content. While "Hotel Room" never made it to the charts, the song grew to become one of the fan-favorite tracks by the band. "Hotel Room" was covered by The Blue Aeroplanes, featuring on the band's 2007 album Harvester.
#1- Evening Over Rooftops
Number one on our ten best Edgar Broughton Band songs list is the alluring hit "Evening Over Rooftops." The song is the most sought-after hit by the band off its eponymous third studio album. "Evening Over Rooftops" is a bold psychedelic rock ballad tinged with folk and country rock influences. The poetic lyrical content and masterpiece delivery guided "Evening Over Rooftops" to become the band's signature hit.
Victor Unitt had just returned to the band's lineup after a short stint with The Pretty Things when he joined Edgar Broughton in writing the incredible lyrics to this song. Joining the band in this track on the backing vocals is The Ladybirds, a British female trio whose origins are traced to the vocal group The Vernon Girls.
Top 10 Edgar Broughton Band Songs article published on Classic RockHistory.com© 2022
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