Description: Just found out the Boom Bot was part of a series, so I thought I would treat all of you by uploading it to SpudMud. Now with fifty more explosive levels of Boombot! The gameplay is the same as the original Boombot, but there are some new items for an added challenge.


Cox technique is a non-force chiropractic adjustment and the safest alternative to back surgery. The safety and effectiveness of this research-based approach is supported and proved by over 40 years of studies. It is the only chiropractic technique where research and studies have been federally funded 3 times. To receive Federal funding for research studies supports its efficacy and shows that the government considers this technique a valuable tool for saving money on medical costs.
The difference between the Cox Technic and other types of spinal traction is that with the Cox Technic every step is monitored by a doctor and tested on patient's tolerance, where other types of spinal traction are monitored by a computer. Flexion/distraction manipulation is strictly customized for each patient according to his/her conditions and symptoms. For this reason I strongly feel that the Cox Technic is far superior in safety and effectiveness than any other spinal traction approach.
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text:
7 Things About 570s gt4 You'll Kick Yourself for Not Knowing
McLaren Racing Limited is a British motor racing group based at the McLaren Innovation Centre, Woking, Surrey, England. McLaren is best referred to as a Formula One constructor and likewise has a history of contending in American open wheel racing as both an entrant and a chassis contractor, and has won the Canadian-American Obstacle Cup (Can-Am) cars racing championship. The group is the 2nd oldest active, and second most effective Formula One team after Ferrari, having actually won 182 races, Chauffeurs' Championships and 8 Manufacturers' Champions. The team is a wholly owned subsidiary of the McLaren Group. Founded in 1963 by New Zealander Bruce McLaren, the group won its first Grand Prix at the 1968 Belgian Grand Prix, but their biggest initial success was in Can-Am, which they controlled from 1967 to 1971. Additional American triumph followed, with Indianapolis 500 wins in McLaren cars and trucks for Mark Donohue in 1972 and Johnny Rutherford in 1974 and 1976. After Bruce McLaren passed away in a testing mishap in 1970, Teddy Mayer took control of and led the team to their very first Solution One Constructors' Championship in 1974, with Emerson Fittipaldi and James Hunt winning the Drivers' Champion in 1974 and 1976 respectively. The year 1974 likewise marked the start of a long-standing sponsorship by Phillip Morris' Marlboro cigarette brand.In 1981, McLaren combined with Ron Dennis' Job Four Racing; Dennis took control of as team principal and shortly after arranged a buyout of the original McLaren shareholders to take complete control of the team. This started the group's most successful period: with Porsche and Honda engines, Niki Lauda, Alain Prost, Click for info and Ayrton Senna took in between them seven Motorists' Championships and the team took 6 Contractors' Championships. The mix of Prost and Senna was particularly dominant-- together they won all but one race in 1988-- but later on their rivalry soured and Prost left for Ferrari. Fellow English group Williams provided the most consistent obstacle during this duration, the two winning every fitters' title between 1984 and 1994. Nevertheless, by the mid-1990s, Honda had actually withdrawn from Formula One, Senna had actually transferred to Williams, and the group went 3 seasons without a win. With Mercedes-Benz engines, West sponsorship, and former Williams designer Adrian Newey, more championships can be found in 1998 and 1999 with chauffeur Mika Häkkinen, and during the 2000s the group were consistent front-runners, motorist Lewis Hamilton taking their newest title in 2008. Ron Dennis retired as McLaren team principal in 2009, turning over to long time McLaren staff member Martin Whitmarsh. However, at the end of 2013, after the team's worst season considering that 2004, Whitmarsh was ousted. McLaren announced in 2013 that they would be utilizing Honda engines from 2015 onwards, replacing Mercedes-Benz.
The group raced as McLaren Honda for the very first time since 1992 at the 2015 Australian Grand Prix. In September 2017, McLaren revealed they had settled on an engine supply with Renault from 2018 to 2020. McLaren will return to utilize Mercedes-Benz engines from the 2021 season up until a minimum of 2024. After initially returning to the Indianapolis 500 in 2017 as a partner to Andretti Autosport to run Fernando Alonso, McLaren revealed in August 2019 that they would run in conjunction with Arrow Schmidt Peterson Motorsports beginning in 2020 to run the full IndyCar Series, the combined entry being called Arrow McLaren SP. Bruce McLaren Motor Racing was founded in 1963 by New Zealander Bruce McLaren. Bruce was a works driver for the British Solution One team Cooper with whom he had actually won three Grands Prix and come 2nd in the 1960 World Championship. Wishing to contend in the Australasian Tasman Series, Bruce approached his employers, however when team owner Charles Cooper demanded using 1.5-litre Formula One-specification engines instead of the 2.5-litre motors allowed by the Tasman rules, Bruce chose to set up his own group to run him and his prospective Solution One teammate Timmy Mayer with customized Cooper vehicles.

redpajama_set_name: RedPajamaCommonCrawl


The flaw in their logic is that when it comes to higher unemployment benefits or any other stimulus spending, the resources given to the unemployed have to be taken from someone else….While the unemployed may spend more as a result of higher unemployment benefits, those people from whom the resources are taken will spend less. In an economy, the income effects from a transfer payment always sum to zero. Quite simply, there is no stimulus from higher unemployment benefits.
Unlike the private sector, government is unable to foster economic growth since it does not have any wealth of its own. Nancy Pelosi who claimed that "we could slip back and have another recession" if Congress didn't pass an unpaid $34 billion unemployment insurance bill, fails to acknowledge the danger of excessive unemployment benefits.
Currently, unemployed American workers can generally collect unemployment benefits for up to 99 weeks—about 2 years. But how much further is Nancy Pelosi willing to extend these unemployment benefits? At what point is enough, enough?
Unemployment insurance also extends the time a person stays off the job. Clark and I estimated that the existence of unemployment insurance almost doubles the number of unemployment spells lasting more than three months.
Take for example Denmark whose citizens use to collect unemployment benefits for up to four years. Of course, being laid off from a job is terrible news. Although, it sure lessens the blow to know that the nanny state will provide you generous unemployment benefits for the next four years of your life.
It shows that between 2005-7, the number of people who got jobs during their four years of benefits — the green line – rose at the beginning before dropping sharply, then spiked as benefits were about to run out, only to plummet after. The red line shows similar behavior in 1998, when Denmark's benefit period was five years.
it shows that people are not seeking all the jobs they could get, but just the jobs they would like to have.
As expected, Speaker of the House Nancy Pelosi irrationally believes that paying people not to work somehow creates jobs. As shown in Denmark, excessive unemployment benefits give people less incentives to actively search for a job. Be on the lookout for Nancy Pelosi to pass yet another deficit increasing unemployment insurance extension during a Lame Duck Session that will add to billions to the national debt while not creating net new jobs.


WASHINGTON, D.C. – Aug. 1, 2018 – Cause of Action Institute ("CoA Institute") today sued the U.S. Department of Justice (DOJ) for failing to respond to three FOIA requests pertaining to the use of personal email by former FBI Director James Comey, former FBI Chief of Staff James Rybicki, and DOJ's Director of Public of Affairs Sarah Isgur Flores.
The recent Office of Inspector General (OIG) report on the Hillary Clinton email scandal disclosed that Comey had used his personal email to conduct official business, but that OIG was, "never given access to all the work-related emails." Comey claimed he either forwarded emails from his personal account to his official account or to Chief of Staff Rybicki. In an unrelated incident last year, Flores was cited as using her Gmail account to issue a statement on behalf of the Attorney General in response to a Washington Post article. CoA Institute filed three FOIAs relating to these matters, and in each case, DOJ failed to respond within the statutory timeframe.
June 14, 2018 – Office of Inspector General FOIA request for "all emails sent or received by former FBI Director James Comey or former FBI Chief of Staff James Rybicki on a personal email account … conducting official government business, that were acquired or reviewed by" OIG.


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Things To Consider For Renting a Roll Off Dumpster in Sunny Isles, Florida.
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Weight restrictions are often not considered, but are something to be considered when renting a dumpster since it can impact your costs. A dumpster may look large, but once you start using it you'll find it can fill up very rapidly. Even if you have extra room in your dumpster, be aware that it could still be at capacity for its maximum weight. You'll also want to be aware of any local regulations in Sunny Isles that may add to your deposal costs depending on the materials you're using. Also, be mindful of the weight in your dumpster and where you're putting it, as you could easily damage your property if it becomes too heavy. You should also inform the rental business of the materials you intend to dispose of in the dumpster, that way they can advise you ahead of time of the type you'll need.
Considering the manner of items you'll be throwing in the dumpster is also an important step to take before you make a choice on which one to rent. If you're only trying to get rid of a handful of appliances or furniture, such as couches, then you could always consider a junk removal service that might be more effective. Sometimes you might just need to get rid of junk or debris in your garage, and in that case a 10 cubic yard dumpster could be ideal for that job. You'll want to make sure you inquire as to what kinds of trash might be prohibited by your dumpster rental company before making a choice. Items like big appliances, oil cans, antifreeze batters and grease waste are often prohibited by dumpster rental companies in Sunny Isles. It's very important to ask these kinds questions when speaking with your rental company.
Estimating your dumpster rental cost will involve several different factors. Fortunately, most rental businesses will give you an estimate at no cost, but you should still know how it is determined. Larger containers, regardless of how full they are, will always cost more. The reason for this is because bigger dumpsters require a higher dumping fee at the landfill, higher transportation costs and more effort to manage.
When selecting a dumpster rental company it's usually best to go with one in the local Sunny Isles area. Local businesses will already be aware of the regulations, recycling and disposal processes associated with local laws, which can save you some effort in the long run. They'll also have a heads up as to where the ideal locations to dispose of waste materials are. Try to provide your rental company with as many details as you can, this will enable them to give you the best price estimate.


text:
\section{Introduction}
Any (generalised) measurement performed on  a physical system is described
by a probability operator-valued measure (POVM) acting on the Hilbert space
of the system. Naimark theorem  \cite{nmk40,akh63,hel73,hel76,hol01} ensures
that any POVM may be implemented as a projective measurement in a larger
Hilbert space, which is usually referred to as the {\em Naimark extension}
of the POVM. As a matter of fact, there are infinite Naimark extensions
and the theorem also ensures that a {\em canonical extension} exists,
i.e. an implementation as an indirect measurement, where the system
under investigation is coupled to an independently prepared probe
system  \cite{per90} and then only the probe is subject to a (projective) measurement
\cite{bin07,ber10,mtqm}.
\par

The problem of finding the Naimark extensions of a POVM is indeed a
central one in quantum technology. On the one hand, it provides a
concrete model to realize the measurement\cite{ban97,Zmeas}, and thus to
assess entanglement cost\cite{ecos03} and/or implementations
on different platforms \cite{tam1,ben07,ben10,lev89,spar14,fey16}. On  the other hand,
it permits to evaluate the post-measurement state and thus to
investigate the tradeoff between information gain and measurement
disturbance \cite{igmd1,igmd2,igmd3,igmd4,igmd5,igmd6,wil13},
as well as any procedure aimed at quantum control \cite{iqc}.
\par
Let us consider a set of operators $\{\Pi_m\}$ that constitute a POVM
for the physical system $S$ described by the Hilbert space
$\mathcal{H}_S$, i.e.
\begin{equation}
\sum_{m=1}^M \Pi_m = I_S,\quad \Pi_m = \Pi_m^\dagger,
\quad \Pi_m \geq 0\,.
\end{equation}
The elements of the set are not necessarily projectors,
$\Pi_n \Pi_m \neq \Pi_n \delta_{n,m}$.
The Naimark theorem states that it is possible to {\em extend}
each POVM elements to a larger (product) Hilbert space
$\mathcal{H}_A \otimes \mathcal{H}_S$ (see \ref{a:kronecker})
such that the extended measurement operators are projectors in
the product space.
In particular, it is possible to define the auxiliary Hilbert
space $\mathcal{H}_A$ such that the system
Hilbert space $\mathcal{H}_S$ is isomorphic to a subspace in $\mathcal{H}_A \otimes \mathcal{H}_S$,
where the density operator $\rho$ defined on $\mathcal{H}_S$ corresponds to the
density operator $\pure{e_1} \otimes \rho$ defined on $\mathcal{H}_A \otimes \mathcal{H}_S$.
The state $\ket{e_1}$ may be chosen as the state corresponding
to the first vector of the canonical basis of $\mathcal{H}_A$.
Naimark theorem states that we can find projectors $\{E_m\}$
\begin{equation}
E_m E_n = E_m \delta_{m,n}, \quad E_m = E_m^\dagger, \quad E_m \geq 0,
\label{vincoliEm}
\end{equation}
each of them corresponding to a POVM element $\Pi_m$ in the following
sense. The distributions of the $m$-th outcome, as obtained from
$\{E_m\}$ and $\{\Pi_m\}$ on the states $\pure{e_1} \otimes
\rho$ and $\rho$ respectively, are the same, i.e.
\begin{equation}
\hbox{Tr}_A\big[\Pi_m\,\rho\big] = \hbox{Tr}_{AS}\big[
\left(\pure{e_1} \otimes \rho\right)\, E_m\big]\,.
\label{BornRule}
\end{equation}
At the operatorial level, this is expressed by the following
the set of relations
\begin{equation}
\Pi_m = \hbox{Tr}_A \left[ \left(\pure{e_1} \otimes
{\mathbb I}_S\right)\, E_m\right]
\end{equation}
which, solved for the $E_m$ given the $\Pi_m$, provide the
desired Naimark extension of the POVM.
\par
As it was originally suggested by Helstrom\cite{hel76} the projectors
$\{E_m\}$ may be built by placing a copy of $\Pi_m$ in the upper-left
block position of the matrix representation of $\{E_m\}$ (corresponding
to the element 1 in the matrix $e_1 \cdot e_1^T$). At the same time, no
explicit recipes had been provided on how to find the remaining blocks.
The aim of this  paper is to describe an  iterative method for effectively
building those blocks upon exploiting the sole requirements of orthogonality
and positivity.
\par
The problem has been addressed before \cite{per95,pre98}, and constructive
methods to find the projective measurement have been suggested. In short,
these methods amount to set up and solve a linear problem which gives the
coefficients of the projectors in the canonical basis of the enlarged Hilbert
space $\mathcal{H}_A \otimes \mathcal{H}_S$.
However, the focus has been on solving the problem for rank-1 POVM elements.
Our iterative method, also based on solving a linear problem,
shows two main advantages compared to existing techniques.
On the one hand, it is more efficient in terms of computational steps
and, on the other hand, it may applied also to POVMs containing
elements with rank greater than one.
\par
The paper is structured as follows. In the next Section we introduce
the iterative method, first illustrating the basic idea and then, in
Sections \ref{costruzioneProiettore} and
\ref{costruzioneOrtogonale}, describing in details its two building
blocks, i.e. the constrained building of an idempotent matrix and the
constrained building of a matrix orthogonal to a given one.
In Section \ref{ss:algo} we put everything together and illustrate
the overall algorithm
to build the Naimark extension of a generic rank-n POVM. In Section
\ref{s:exa}, we illustrate few examples of application, whereas Section
\ref{s:outro} closes the paper with some concluding remarks.
\section{An iterative method to build the Naimark extension of rank-n POVMs}
In the following, we will write  projectors as matrices of suitable
sizes composed by blocks.
The first step in building the projectors $E_m$  is analogue to the
original Helstrom recipe,
i.e. we define the upper-left block in the matrix of $E_m$ equal
to $\Pi_m$.
The algorithm then builds the projectors one at a time, upon defining
their blocks iteratively.
As we will see soon, initially the blocks of the first projector are
mostly zero, and the building
of the following projectors populates other blocks. In this sense,
the amount of non-zero rows
and columns grows during the building of the projectors, and the
size of the necessary auxiliary
Hilbert space $\mathcal{H}_A$ is obtained only at the end of the
procedure.
\par
The algorithm initially builds the blocks of $E_1$ in order
to satisfy the constraints \eqref{vincoliEm} on
itself, i.e.
\begin{equation}
E_1 \cdot E_1 = E_1, \quad E_1 = E_1^\dagger, \quad E_1 \geq 0
\end{equation}
Then, we build \emph{some blocks} of $E_2$ in order to satisfy
the orthogonality with $E_1$,
\begin{equation}
E_1 \cdot E_2 = 0
\end{equation}
and then imposing the other constraints
\begin{equation}
E_2 \cdot E_2 = E_2, \quad  E_2 = E_2^\dagger,\quad  E_2 \geq 0
\end{equation}
we define the remaining  blocks. As we will see, this second step do not modify the previously
defined blocks of $E_2$.
\par
Analogously, the algorithm builds $E_3$ (if any) imposing its orthogonality with $E_1,\ E_2$,  and then imposing
that $E_3 \cdot E_3 = E_3$. The generalisation is straightforward, the element $E_m$ is built in order to satisfy
at first the ortogonality with the previously built projectors, and then imposing the condition $E_m\cdot E_m = E_m$.
The algorithm is thus an iterative one, since it employs the projectors already found, until all the elements are built.
\par
The algorithm requires basically two steps repeated several times: building a matrix with some assigned blocks
such that it is orthogonal to another matrix, and the completion of the matrix in order to make it idempotent, that is,
satisfying
$$
E_m \cdot E_m = E_m.
$$
These steps are analysed in some details in the following two Sections, whereas the overall algorithm
is summarised in Section \ref{ss:algo}.
\subsection{Building an idempotent matrix}
\label{costruzioneProiettore}
At first, let us consider the problem of building an idempotent matrix when some of its blocks are assigned.
This is the case of the evaluation of $E_1$, which has the block $\Pi_1$ in the upper-left position. If $\Pi_1$ is already
idempotent, we can just put $\Pi_1$ in the corner and set the remaining blocks to zero. If this is not the case, we can
define the blocks around $\Pi_1$ such that $E_1 \cdot E_1 = E_1$, possibly employing the minimum amounts of
blocks, and setting the others to zero. In what follows, we ignore the subscripts that refers to the $m$-th element.
The general problem becomes to find the adjacent blocks of the upper-left corner in order to make the matrix
$E$ idempotent.
\par
As we will see in the following, it is enough to assume the following matrix structure for $E$
\begin{equation}
E = \left(\begin{matrix}
\Pi & A & 0 & \dots \\
A^\dagger & B & 0 & \dots \\
0 & 0 & 0 & \dots \\
\vdots & \vdots & \vdots & \ddots \\
\end{matrix}\right)
\end{equation}
with $\Pi$ a given block, while $A,\ B$ are blocks to find ($A^\dagger$ and $B\geq 0$ have been used so that $E = E^\dagger$).
In the case $\Pi^2 = \Pi$ we can omit $A,\ B$ since the matrix is already idempotent. Otherwise, we have to add the blocks $A,\ A^\dagger,\ B$ and the matrix
$E$ grows in sizes. The constraint
\begin{equation}
E \cdot E = \left(\begin{matrix}
\Pi & A \\
A^\dagger & B  \\
\end{matrix}\right)
\cdot
\left(\begin{matrix}
\Pi & A \\
A^\dagger & B  \\
\end{matrix}\right)
=\left(\begin{matrix}
\Pi & A \\
A^\dagger & B  \\
\end{matrix}\right)
=E
\end{equation}
gives the following equations:
\begin{align}
\Pi^2 + A A^\dagger = \Pi \label{eq1} \\
\Pi A + A B = A \label{eq2} \\
A^\dagger A + B^2 = B \label{eq3}
\end{align}
Equation \eqref{eq1} can be solved exploiting the singular value decomposition (SVD) for
$\Pi = V \Lambda V^\dagger$ and $A=USW^\dagger$.
Setting $U=V,\ W=I,\ S=\sqrt{\Lambda (I-\Lambda)}$ leads to  $A=V\sqrt{\Lambda (I-\Lambda)}$. Assuming for the moment a full rank matrix $\Pi$, with eigenvalues strictly included in the range $(0,1)$, equation \eqref{eq2} allows us
to find $B = I- \Lambda$.

Finally, equation \eqref{eq3} is verified with the above solutions,
and the blocks of $E$ can be built as
\begin{equation}
E = \left(\begin{matrix}
\Pi & A & 0 & \dots \\
A^\dagger & B & 0 & \dots \\
0 & 0 & 0 & \dots \\
\vdots & \vdots & \vdots & \ddots \\
\end{matrix}\right)
= \left(\begin{matrix}
V \Lambda V^\dagger & V \sqrt{\Lambda (I-\Lambda)} & 0 & \dots \\
\sqrt{\Lambda (I-\Lambda)}V^\dagger & I-\Lambda & 0 & \dots \\
0 & 0 & 0 & \dots \\
\vdots & \vdots & \vdots & \ddots \\
\end{matrix}\right)
\end{equation}
\par
A different route may be also employed upon exploiting positivity of the elements of the POVM. Indeed, for positive semi-definite
$\Pi$ we have the decomposition $\Pi~=~Y Y^\dagger$ (with $Y$ having no particular properties), which may be used instead
of SVD, which is generally demanding in terms of computational time. Notice that if $\Pi$ is not full rank, the decomposition
is still available, with $Y$ being a rectangular matrix with the same rank.

With this decomposition, equation \eqref{eq1} is solved by
\begin{equation}
A=Y\sqrt{I - Y^\dagger Y}, \ A^\dagger = \sqrt{I - Y^\dagger Y} \  Y^\dagger. \label{solutionY}
\end{equation}
and $B = I- Y^\dagger Y$ follows. Finally, equation \eqref{eq3} is verified by re-writing $A^\dagger A = \sqrt{B} (I-B) \sqrt{B}$.

For rectangular $Y$, equation \eqref{solutionY} still holds upon defining $Y^{-1}$ as the Penrose inverse of Y, that is,
the rectangular matrix satisfying $Y^{-1} Y = I$ on the support of $Y$.
In addition, the decomposition $E=Z Z^{\dagger}$ is also readily available from $Y$,
\begin{align}
E =&  \begin{pmatrix}
YY^\dagger &  Y\sqrt{I - Y^\dagger Y} & 0 & \dots \\
\sqrt{I - Y^\dagger Y} \  Y^\dagger & I - Y^\dagger Y & 0 & \dots \\
0 & 0 & 0 & \dots \\
\vdots & \vdots & \vdots & \ddots \\
\end{pmatrix}
\label{soluzioneCostruzioneProiettore}\\
& =
\begin{pmatrix}
Y \\
\sqrt{I-Y^\dagger Y} \\
0 \\
\vdots
\end{pmatrix}
\cdot
\begin{pmatrix}
Y^\dagger & \sqrt{I-Y^\dagger Y} & 0 & \ldots
\end{pmatrix}
= Z Z^{\dagger}
\label{scomposizioneIdempotente}
\end{align}

\subsection{Building a matrix orthogonal to a given one}
\label{costruzioneOrtogonale}
In this section we consider the problem of building a matrix (with some assigned blocks)
such that it is orthogonal to a given one. This occurs in building, e.g., $E_2$, which has the
upper-left block equal to $\Pi_2$ and must verify $E_1 \cdot E_2 = 0$. If we have $\Pi_1 \cdot \Pi_2 = 0$,
it is enough to set the blocks adjacent to $\Pi_2$ equal to zero. In the most general case, this does not hold,
and to satisfy the orthogonality condition we have to explicitly determine the blocks around
$\Pi_2$. The expression
\begin{equation}
E_1 \cdot E_2 = \left(\begin{matrix}
Y_1Y_1^\dagger & Y_1 \sqrt{I-Y_1^\dagger Y_1} \\
\sqrt{I-Y_1^\dagger Y_1}\ Y_1^\dagger & I-Y_1^\dagger Y_1 \\
\end{matrix}\right)
\cdot
\left(\begin{matrix}
\Pi_2 & A \\
A^\dagger & B  \\
\end{matrix}\right)
=0\,,
\label{equazioneOrtogonale}
\end{equation}
where $\Pi_1 = Y_1 Y_1^\dagger$, provides the constraints
\begin{align}
Y_1Y_1^\dagger \Pi_2 + Y_1 \sqrt{I-Y_1^\dagger Y_1} A^\dagger=0 \label{eq4} \\
Y_1Y_1^\dagger A + Y_1 \sqrt{I-Y_1^\dagger Y_1} B=0 \label{eq5} \\
\sqrt{I-Y_1^\dagger Y_1} Y_1^\dagger A + (I-Y_1^\dagger Y_1) B = 0 \label{eq6}\,.
\end{align}
Equation \eqref{eq4} allows us to find $A = -\Pi_2 Y_1 \sqrt{I-Y_1^\dagger Y_1}$,
whereas equation \eqref{eq5} provides the expression  $B=\left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1} Y_1^\dagger \Pi_2 Y_1 \sqrt{I-Y_1^\dagger Y_1} $.
The third equation, \eqref{eq6}, is indeed verified by these solutions.

At this stage, upon
imposing the orthogonality with $E_1$, we found that $E_2$ has the structure
\begin{equation}
E_2 = \left(\begin{matrix}
\Pi_2 & -\Pi_2 Y_1 \sqrt{I-Y_1^\dagger Y_1} & \quad \ast & \dots \\
- \left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1} Y_1^\dagger \Pi_2 &  \quad \left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1} Y_1^\dagger \Pi_2 Y_1 \sqrt{I-Y_1^\dagger Y_1} & \quad\ast & \dots \\
\ast & \ast & \quad\ast & \dots \\
\vdots & \vdots & \quad \vdots & \ddots \\
\end{matrix}\right),
\label{soluzioneCostruzioneOrtogonale}
\end{equation}
where the blocks indicated by $\ast$ are left unused and may be exploited to impose other conditions on $E_2$.
If a decomposition $\Pi_2=XX^{\dagger}$ is available, the big block just defined in \eqref{soluzioneCostruzioneOrtogonale} has a simple decomposition,
\begin{align}
&\begin{pmatrix}
X X^\dagger & -X X^\dagger Y_1 \sqrt{I-Y_1^\dagger Y_1}  \\
- \left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1} Y_1^\dagger X X^\dagger &  \quad \left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1} Y_1^\dagger X X^\dagger Y_1 \sqrt{I-Y_1^\dagger Y_1}
\end{pmatrix} \notag \\
& =\begin{pmatrix}
X \\
-\left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1} Y_1^\dagger X
\end{pmatrix}
\cdot
\begin{pmatrix}
X^\dagger &  \quad - X^\dagger Y_1 \sqrt{I-Y_1^\dagger Y_1}
\end{pmatrix}
=Y_2 \cdot Y_2^\dagger
\label{scomposizioneOrtogonale}
\end{align}

Notice that the blocks just defined depend on $\Pi_1$ (via its decomposition) and upon $\Pi_2$. If we have
to impose the orthogonality of matrix $E_m$ with $E_1$, only the non-zero blocks in $E_1$ would be involved. Thus, the solution would be the same substituting $\Pi_m = X_m X_m^\dagger$ in place of $\Pi_2=X_2 X_2^\dagger$.
\subsection{The algorithm}
\label{ss:algo}

The algorithm builds the projectors one at a time, using the previously built projectors. For each projector $E_m$ two steps are performed: first
the {\em orthogonal construction} of Section \ref{costruzioneOrtogonale}, which defines some blocks of
$E_m$ such that the projector is orthogonal to all the projectors previously evaluated. In the second step, leveraging the
{\em idempotent construction} illustrated in Section \ref{costruzioneProiettore}, some other blocks are defined
so that $E_m^2 = E_m$. Before applying the orthogonal or idempotent construction, it is checked whether $E_m$ is already ortogonal to the other projectors or idempotent. If this is the case, the step is simply skipped
\par
The algorithm starts building $E_1$ with the idempotent construction, as the orthogonal one is not necessary.
$\Pi_1$ is copied in the upper-left block of $E_1$ and
the solution \eqref{soluzioneCostruzioneProiettore} is evaluated with $Y = Y_1,\ Y_1 Y_1^\dagger=\Pi_1$, where $Y_1$ has been obtained for instance from the singular value decomposition of $\Pi_1 = V_1 \Lambda_1 V_1^{\dagger}$, giving $ Y_1=V_1 \sqrt{\Lambda_1}$. If $\Pi_1$ is full rank, the
projector $E_1$ has a nonzero $2\times2$ blocks in the upper-left corner. The remaining blocks are zero.
\begin{align*}
E_1 &= \left(\begin{array}{cc|cc}
      Y_1 Y_1^\dagger &  \quad Y_1 \sqrt{I-Y_1^\dagger Y_1} &       &  \\
      \ast &      I-Y_1^\dagger Y_1 & \multicolumn{2}{c}{\smash{\raisebox{.5\normalbaselineskip}{\quad 0}}} \\
			\hline \\[-\normalbaselineskip]
        &         &      &       \\
      \multicolumn{2}{c|}{\smash{\raisebox{.5\normalbaselineskip}{ 0}}}
			& \multicolumn{2}{c}{\smash{\raisebox{.5\normalbaselineskip}{\quad 0}}}
    \end{array}\right)
\end{align*}
The projector $E_2$ is then built using the two steps. The block $\Pi_2$ is copied in the upper-left corner, a
decomposition $\Pi_2=X_2^{(1)} X_2^{(1)\dagger}$ is evaluated (by SVD if needed) and the three blocks around are
defined as in \eqref{soluzioneCostruzioneOrtogonale} leveraging on the decompositon $\Pi_1 = Y_1Y_1^\dagger$ previously evaluated.
At this point, the big block just defined has a decomposition $Y_2 Y_2^{\dagger}$ as in \eqref{scomposizioneOrtogonale}, and if not idempotent, the adjacent blocks need to be evaluated accordingly employing the idempotent construction of equation \eqref{soluzioneCostruzioneProiettore}.
\begin{align*}
E_2
		& = \left( \begin{array}{c|c}
		\overbrace{
		\begin{matrix}
		\Pi_2 &  -\Pi_2 Y_1 \left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1}\\
		\ast &    \left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1} Y_1^\dagger  \Pi_2 Y_1 \left(\sqrt{I-Y_1^\dagger Y_1}\right)^{-1}
		\end{matrix}
		}^{Y_2 Y_2^\dagger}
		& \qquad Y_2 \sqrt{I-Y_2^\dagger Y_2}  \qquad \\
		\hline
	  \ast & \qquad  I-Y_2^\dagger Y_2  \qquad
		\end{array} \right)
\end{align*}
Notice that in this case in the original matrix  $E_2$ the $4 \times 4$ blocks in the upper left corner are defined,
for a total of  $4D$ rows and $4D$ columns (if  $\Pi_1,\ \Pi_2$ are full rank), with $D$ being the dimension of the system Hilbert space.
In the evaluation of $E_3$ the same orthogonal and idempotent construction are repeated, with the difference that the
first must be repeated twice to get the orthogonality with $E_1$ ed $E_2$. As usual, first the block
$\Pi_3 = X_3^{(1)} X_3^{(1)\dagger}$ is copied in the upper left corner. The first blocks around $\Pi_3$ are evaluated with \eqref{soluzioneCostruzioneOrtogonale}.

The newly defined big block has decomposition $X_3^{(2)} X_3^{(2)\dagger}$ obtained from \eqref{scomposizioneOrtogonale} where $X=X_3^{(1)},\ Y_2 = X_3^{(2)}$.
The orthogonal construction  \eqref{soluzioneCostruzioneOrtogonale} is repeated to get the orthogonality with $E_2$, employing the block $X_3^{(2)} X_3^{(2)\dagger}$ and the term $Y_2$ previously defined in the idempotent contruction of $E_2$. A new bigger block is
obtained with decomposition $Y_3 Y_3^\dagger$ as in \eqref{scomposizioneOrtogonale}.

The idempotent construction is then used employing $Y = Y_3$ as in equation \eqref{soluzioneCostruzioneProiettore}.
Finally, we get the matrix (note that not all the blocks have the same size)
\begin{align*}
E_3
		= \left( \begin{array}{c|c}
		\overbrace{
		\begin{array}{c|c}
			\overbrace{
			\begin{matrix}
			\Pi_3 &  -\Pi_3 \ Y_1 R_1^{-1}\\
			\ast &    R_1^{-1} Y_1^\dagger \ \Pi_3 \ Y_1 R_1^{-1}
			\end{matrix}
			}^{X_3^{(2)} X_3^{(2)\dagger}}
			& \qquad -X_3^{(2)} X_3^{(2)\dagger} \ Y_2 R_2^{-1} \\
			\hline
			\ast & \displaystyle R_2^{-1} Y_2^\dagger \ X_3^{(2)} X_3^{(2)\dagger} \ Y_2 R_2^{-1}
		\end{array}
		}^{Y_3 Y_3^\dagger} & Y_3 R_3 \\
		\hline
		\ast & I-Y_3^\dagger Y_3
		\end{array} \right)
\end{align*}
with $R_1 = \sqrt{I-Y_1^\dagger Y_1},\ R_2 = \sqrt{I-Y_2^\dagger Y_2},\ R_3 = \sqrt{I-Y_3^\dagger Y_3} $.
Notice that the expression of $E_3$ depends upon the decompositions
$Y_1Y_1^\dagger,\ Y_2Y_2^\dagger$ of the upper-left blocks of the
preceding projectors. This holds for each projector $E_m$.
\par
The method used to evaluate  $E_3$ may be iterated for any subsequent projector. First, the block $\Pi_m = X_m^{(1)} X_m^{(1)\dagger}$ is
copied in the upper left corner. The adjacent blocks are defined imposing the orthogonality with $E_1$,
 following the orthogonal construction. The just defined block has decomposition $X_m^{(2)} X_m^{(2)\dagger}$, and the
 orthogonal construction is repeated using $Y_n$, which is the term used in the idempotent construction of
 $E_n,\ n<m$. At the end of each orthogonal constructions, the newly defined big block is decomposed as $X_m^{(i)} X_m^{(i)\dagger},\ i<m$, and the orthogonal construction is repeated until $i$ reach $m$.
The term $X_m^{(m)} = Y_m$ is then used in the idempotent construction to get the final block structure of $E_m$.
\par
At this stage, upon following the procedure leading to $E_m$, a recursive
construction may be also obtained for its decomposition $E_m = Z_m Z_m^\dagger$.
For further details, see \ref{a:cholesky}.
If all the $\Pi_m$ are full rank and with eigenvalues in the range $[0,1]$,
then the size of the projectors grows exponentially.  In fact, the projector
$E_1$ has in this case $2\times2$ non-zero blocks, for a total of $2D$ rows
and $2D$ columns; the projector  $E_2$ populates $4\times 4$ blocks, the
projector  $E_3$ has $8 \times 8$ non-zero blocks, and so on.
An exception occurs if some of the blocks already satisfy the ortogonality
conditions. For instance,  if $\Pi_2$ is already orthogonal to $\Pi_1$,
there is no need to used the adjacent blocks to obtain
its orthogonality. This is also the case if the block is idempotent, since
the adjacent blocks may left unused.
\section{Examples}\label{s:exa}
Here we apply our procedure to obtain the Naimark extension of POVMs
already presented in the literature. In this way, we are able to show
the main features of the algorithm, and its advantages compared to
existing ones.
\subsection{Three elements POVM}
Helstrom considered the example a three-elements POVM
$\{\Pi_1, \Pi_2, \Pi_3\}$,
$\Pi_1+\Pi_2+\Pi_3 = {\mathbb I_S}$,
defined by $\Pi_k = \frac23 \pure{\psi_k},\
k=1,2,3$, where \cite{hel76}
\begin{equation}
\ket{\psi_1}=\frac1{\sqrt{2}}
\left(\begin{array}{c}
1 \\ 1\end{array}\right)\,, \quad
\ket{\psi_2}=\frac1{\sqrt{2}}
\left(\begin{array}{c}
e^{-i \pi/3} \\
e^{i \pi/3}
\end{array}\right)\,,\quad
\ket{\psi_3}=-\frac1{\sqrt{2}}
\left(\begin{array}{c}
e^{i \pi/3} \\
e^{-i \pi/3}
\end{array}\right)\,.
\end{equation}
i.e.
\begin{align}
\Pi_1=& \frac13 \left(
\begin{array}{cc}
1 & 1 \\
1 & 1 \\
\end{array}
\right), \quad
\Pi_2 = \frac13 \left(
\begin{array}{cc}
1& e^{- 2i \pi/3} \\
e^{2i \pi/3} & 1 \\
\end{array}
\right)\,, \quad
\Pi_3 = \frac13 \left(
\begin{array}{cc}
1& e^{2i \pi/3} \\
e^{-2i \pi/3} & 1 \\
\end{array}
\right)
\label{example3povm}
\end{align}
The extension originally obtained by Helstrom was based on a
two-dimensional auxiliary Hilbert space with basis
$\ket{v_1}=(1,0)^T$, $\ket{v_2}=(0,1)^T$,  and it is given by
$E_k^H=\pure{\xi_k}$, $k=1,..,4$, where
\begin{align}
& \ket{\xi_1} = \sqrt{2/3}\ket{v_1}\ket{\psi_1}
+ \sqrt{1/3}\ket{v_2}\ket{\psi_3}, \\
& \ket{\xi_2} = \sqrt{2/3}\ket{v_1}\ket{\psi_2}
- \sqrt{1/3}\ket{v_2}\ket{\psi_3}, \\
& \ket{\xi_3} = \sqrt{2/3}\ket{v_1}\ket{\psi_3}
+ \sqrt{1/3}\ket{v_2}\ket{\psi_3}, \\
& \ket{\xi_4} = \ket{v_2}\ket{\psi_3'}, \\
& \ket{\psi_3'}=\frac1{\sqrt{2}}
\left(\begin{array}{c}
-e^{i \pi/3} \\
e^{-i \pi/3}
\end{array}\right)\,.
\end{align}
The iterative algorithm in this case is particularly efficient since the orthogonality construction gives also
idempotent matrices. Overall, a two-dimensional auxiliary Hilbert space is still required, but only the upper
left 3-by-3 corner has non-zero coefficients.
\begin{align} \label{pvm}
& E_1 =  \frac{1}{3} \left(
\begin{array}{cccc}
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
1 & 1 & 1 & 0 \\
0 & 0 & 0 & 0
\end{array}
\right), \quad
E_2 = \frac{1}{3} \left(
\begin{array}{cccc}
 1 & e^{-\frac{2 i \pi }{3}} & e^{\frac{2 i \pi }{3}} & 0 \\
 e^{\frac{2 i \pi }{3}} & 1 & e^{-\frac{2 i \pi }{3}} & 0 \\
 e^{-\frac{2 i \pi }{3}} & e^{\frac{2 i \pi }{3}} & 1 & 0 \\
 0 & 0 & 0 & 0
\end{array}
\right), \\
& E_3 = \frac{1}{3} \left(
\begin{array}{cccc}
 1 & e^{\frac{2 i \pi }{3}} & e^{-\frac{2 i \pi }{3}} & 0 \\
 e^{-\frac{2 i \pi }{3}} & 1 & e^{\frac{2 i \pi }{3}} & 0 \\
 e^{\frac{2 i \pi }{3}} & e^{-\frac{2 i \pi }{3}} & 1 & 0 \\
  0 & 0 & 0 & 0
\end{array}
\right)\,.
\end{align}
The correctness of both solutions is verified by checking the
properties of orthogonality, idempotence, and the upper left
corner equal to the original POVM.
\par
The extension proposed by Helstrom gives 4-by-4 matrices with no zero coefficients,
and therefore differs for the block adjacent the left upper corner.
Here we report the matrix expression of $E_1^H$ for
comparison with $E_1$ in Eq. (\ref{pvm})
\begin{align}
& E_1^H= \frac{1}{3} \left(
\begin{array}{cccc}
 1 & 1 & \frac{e^{\frac{2 i \pi }{3}}}{\sqrt{2}} & \frac{e^{-\frac{2 i \pi }{3}}}{\sqrt{2}} \\
 1 & 1 & \frac{e^{\frac{2 i \pi }{3}}}{\sqrt{2}} & \frac{e^{-\frac{2 i \pi }{3}}}{\sqrt{2}} \\
 \frac{e^{-\frac{2 i \pi }{3}}}{\sqrt{2}} & \frac{e^{-\frac{2 i \pi }{3}}}{\sqrt{2}} & \frac{1}{2} &
   -\frac{1}{2} e^{-\frac{i \pi }{3}} \\
 \frac{e^{\frac{2 i \pi }{3}}}{\sqrt{2}} & \frac{e^{\frac{2 i \pi }{3}}}{\sqrt{2}} & -\frac{1}{2}
   e^{\frac{i \pi }{3}} & \frac{1}{2} \\
\end{array}
\right)\,.
\end{align}
\subsection{Four elements POVM}
Helstrom also considered a four-elements POVM
$\{\Pi_1,\ \Pi_2,\ \Pi_3,\ \Pi_4\}$, with\cite{hel76}
$$
\Pi_k = \frac{1}{2}\pure{\psi_k}, \quad
  \ket{\psi_k} = \frac{1}{\sqrt{2}}
\left(\begin{array}{c}
e^{-i(k-1)\frac{\pi}{4}} \\
e^{i(k-1)\frac{\pi}{4}}
\end{array} \right), \quad k= 1,2,3,4 \, ,
$$
i.e.
\begin{equation}
\Pi_1= \frac{1}{4}\left(
\begin{array}{cc}
 1 & 1 \\
 1 & 1 \\
\end{array}
\right),
\Pi_2=\frac{1}{4}\left(
\begin{array}{cc}
 1 & -i \\
 i & 1 \\
\end{array}
\right),
\Pi_3=\frac{1}{4}\left(
\begin{array}{cc}
 1 & -1 \\
 -1 & 1 \\
\end{array}
\right),
\Pi_4=\frac{1}{4}\left(
\begin{array}{cc}
 1 & i \\
 -i & 1 \\
\end{array}
\right).
\end{equation}
Again, the iterative algorithm easily finds the extension since the
orthogonal construction directly gives idempotent matrices, without
the need of the idempotent construction.
\begin{align}
&E_1=\frac{1}{4}\left(
\begin{array}{cccc}
 1 & 1 & \sqrt{2} & 0 \\
 1 & 1 & \sqrt{2} & 0 \\
 \sqrt{2} & \sqrt{2} & 2 & 0 \\
 0 & 0 & 0 & 0 \\
\end{array}
\right)
&E_2=\frac{1}{4}\left(
\begin{array}{cccc}
 1 & -i & -e^{-\frac{i \pi }{4}} & e^{-\frac{i \pi }{4}} \\
 i & 1 & -e^{\frac{i \pi }{4}} & e^{\frac{i \pi }{4}} \\
 -e^{\frac{i \pi }{4}} & -e^{-\frac{i \pi }{4}} & 1 & -1 \\
 e^{\frac{i \pi }{4}} & e^{-\frac{i \pi }{4}} & -1 & 1 \\
\end{array}
\right) \\
&E_3=\frac{1}{4}\left(
\begin{array}{cccc}
 1 & -1 & 0 & i \sqrt{2} \\
 -1 & 1 & 0 & -i \sqrt{2} \\
 0 & 0 & 0 & 0 \\
 -i \sqrt{2} & i \sqrt{2} & 0 & 2 \\
\end{array}
\right)
&E_4=\frac{1}{4}\left(
\begin{array}{cccc}
 1 & i & -e^{\frac{i \pi }{4}} & -e^{\frac{i \pi }{4}} \\
 -i & 1 & -e^{-\frac{i \pi }{4}} & e^{\frac{3 i \pi }{4}} \\
 -e^{-\frac{i \pi }{4}} & -e^{\frac{i \pi }{4}} & 1 & 1 \\
 e^{\frac{3 i \pi }{4}} & -e^{\frac{i \pi }{4}} & 1 & 1 \\
\end{array}
\right)
\end{align}

\subsection{Rank-2 POVMs}
In a more recent paper, rank-2 POVM elements have been
introduced to describe generalized measurements involving
sets of Pauli quantum observables chosen at random, the
so-called {\em quantum roulettes} \cite{spar13}. More precisely,
quantum roulettes are generalized measurements obtained by selecting
the observable $\sigma_k$  with a probability $\{z_k\}$ in the set of
nondegenerate and isospectral observables $\{\sigma_k\}$. The
POVM elements are defined as linear combination of the projectors
associated with the observables outcomes.
\par
In Ref. \cite{spar13}, the \emph{canonical} Naimark extension is sought,
i.e. the implementation of the generalized measurement in a larger
Hilbert space using a projective indirect measurement on the ancillary
system after its coupling with the system. In this scenario,
Eq. \eqref{BornRule} is rewritten as
$$
\hbox{Tr}_A\big[\Pi_m\,\rho\big] = \hbox{Tr}_{AS}\big[ \left(\pure{\omega_A} \otimes \rho\right)\, U^\dagger (P_m \otimes {\mathbb I_S}) U\big]\,,
\label{BornRuleCanonical}
$$
where $\ket{\omega_A}$ is the ancillary state, $U$ describes the coupled
evolution between the systems, and ${P_m}$ is the projective measurement
in the ancillary system. A first example of POVM is that of a roulette
obtained from the Pauli operators $\{\sigma_1, \sigma_3\}$ with
probabilities $\{z, 1-z\},\ z\in(0,1)$, giving the elements
$$
\Pi_1 = \frac{1}{2} \left(
\begin{array}{cc}
 2-z & z \\
 z & z \\
\end{array}
\right), \quad
\Pi_{-1} = \frac{1}{2} \left(
\begin{array}{cc}
 z & -z \\
 -z & 2-z \\
\end{array}
\right).
$$
The solution proposed uses the ancillary state $\ket{\omega_A} = \frac{1}{\sqrt{2}}\left(\ket{0} + e^{i\phi} \ket{1}\right)$, the projectors
$$
P_1=\frac{1}{2} \left(
\begin{array}{cc}
 2-z & \sqrt{z(2-z)} \\
 \sqrt{z(2-z)} & z \\
\end{array}
\right), \quad
P_{-1}={\mathbb I} - P_1,
$$
and the unitary
$$
U=\left(
\begin{array}{cccc}
f & 0 & 0 & 0\\
0 & 0 & if^\ast & 0 \\
0 & if^\ast & 0 & 0 \\
0 & 0 & 0 & f
\end{array}
\right), \quad f=\sqrt{\sqrt{\frac{2-2z}{2-z}} + i\sqrt{\frac{z}{2-z}}} \, .
$$
On the other hand, upon
applying the iterative algorithm gives these solutions straightaway,
$$
E_1=\left(
\begin{array}{cccc}
 1-\frac{z}{2} & \frac{z}{2} & \frac{\sqrt{(1-z) z}}{\sqrt{2}} &
   0 \\
 \frac{z}{2} & \frac{z}{2} & 0 & \frac{\sqrt{(1-z) z}}{\sqrt{2}}
   \\
 \frac{\sqrt{(1-z) z}}{\sqrt{2}} & 0 & \frac{z}{2} & -\frac{z}{2}
   \\
 0 & \frac{\sqrt{(1-z) z}}{\sqrt{2}} & -\frac{z}{2} &
   1-\frac{z}{2} \\
\end{array}
\right), \quad
E_{-1} = {\mathbb I_{AS}} - E_1,
$$
which is equivalent to the canonical one up to a rotation in
the ancillary state.

The paper presents also another example with rank-2 diagonal POVM elements,
$$
\Pi_1 = \left(
\begin{array}{cc}
\frac{1}{2}+ f & 0 \\
0 & \frac{1}{2}-f
\end{array}
\right), \quad
\Pi_{-1} = {\mathbb I} - \Pi_1.
$$
The proposed extension employs the ancillary state $\ket{\omega_A}=
\ket{e_1}$, the projectors of the observable $\sigma_3$,
i.e. $P_1=\pure{e_1},\ P_{-1}= \pure{e_2}$, and the unitary
$$
U=\left(
\begin{array}{cccc}
 \sqrt{\frac{1}{2}+f} & 0 & 0 & i\sqrt{\frac{1}{2}-f} \\
 0 & \sqrt{\frac{1}{2}-f} & i \sqrt{\frac{1}{2}+f} & 0 \\
 0 & i \sqrt{\frac{1}{2}+f} & \sqrt{\frac{1}{2}-f} & 0 \\
 i\sqrt{\frac{1}{2}-f} & 0 & 0 &  \sqrt{\frac{1}{2}+f} \\
\end{array}
\right),
$$
which gives
\begin{equation}
U^\dagger \ (P_1 \otimes {\mathbb I_S}) U = \left(
\begin{array}{cccc}
 \frac{1}{2}+f & 0 & 0 & \frac{1}{2} i \sqrt{1-4 f^2} \\
 0 & \frac{1}{2}-f & \frac{1}{2} i \sqrt{1-4 f^2} & 0 \\
 0 & -\frac{1}{2} i \sqrt{1-4 f^2} &  \frac{1}{2}+f & 0 \\
 -\frac{1}{2} i \sqrt{1-4 f^2} & 0 & 0 & \frac{1}{2}-f \\
\end{array}
\right).
\label{Can2}
\end{equation}
In this case the iterative algorithm is particularly easy to
apply since we have diagonal POVM elements, and it gives the
solution
$$
E_1 = \left(
\begin{array}{cccc}
 \frac{1}{2}+f & 0 & \frac{1}{2} \sqrt{1-4 f^2} & 0 \\
 0 & \frac{1}{2}-f & 0 & \frac{1}{2} \sqrt{1-4 f^2} \\
 \frac{1}{2} \sqrt{1-4 f^2} & 0 & \frac{1}{2}-f & 0 \\
 0 & \frac{1}{2} \sqrt{1-4 f^2} & 0 &  \frac{1}{2}+f \\
\end{array}
\right)
$$
which is equivalent to \eqref{Can2} since in both cases
we can see $\Pi_1$ in the upper left bock.
\section{Conclusions}
\label{s:outro}
In this paper we have addressed the problem of finding the Naimark extension of a probability operator-valued
measure, i.e. its implementation as a projective measurement in a larger Hilbert space. As a matter of fact,
the extension of a POVM is not unique and we have exploited this degree of freedom to introduce an iterative
method to build the projective measurement from the sole requirements of orthogonality and positivity.
Our method improves existing ones, as  it is more effective in terms of computational steps needed to determine
the POVM extension. Even more importantly, our method may be employed also to extend POVMs containing
elements with rank larger than one.
\par
Since a Naimark extension provides a concrete model to realize the generalized measurement, we foresee applications
of our method to assess technological solutions on different platforms and  to investigate the tradeoff between information
gain and measurement disturbance in generalized measurements.
\section*{Acknowledgments}
This work has been supported by EU through the
Collaborative Project QuProCS (Grant Agreement 641277) and by UniMI
through the H2020 Transition Grant 15-6-3008000-625.

redpajama_set_name: RedPajamaArXiv


As Texas residents know, business disputes come in all shapes and sizes. In a recent case, a lawsuit between two companies comes in the form of a square.
Hudson + Broad Inc., a visual design firm, has filed a suit against J.C. Penney Co. alleging breach of contract and misappropriation of trade secrets. According to the suit, J.C. Penney backed out of an agreement for the design company to produce lighted square frames on display at the entrances of J.C. Penney's stores.
The company alleges that the design of the large square Plexiglas LED light displays is proprietary. Furthermore, the suit claims that J.C. Penney agreed to buy 1,800 of them to be installed at department stores across the country after they were shown during the store's rollout of its new "Fair and Square" marketing plan. The frames were designed to change colors to allow for a unique color scheme every month.
However, the visual design company says that instead, the department store informed the firm on April 3 that it was contracting production of the LED displays to another outside company.
While there are no further details regarding the companies' specific agreement, the department store stands to lose a lot if the design firm is successful in this suit. In this case, the business seeks $20 million in compensatory damages and $20 million in punitive damages.
Sometimes, business agreements naturally fall through. However, if an arrangement breaks due to the fault or breach of an involved party, this may lead to contract liability.


Litsea Cubeba Oil is obtained from the small pepper-like fruits of the tree. This is a refreshing, uplifting and stimulating essential oil with antiseptic, carminative and healing properties. This oil is often used to help dispel flatulence and indigestion, and to treat skin disorders such as acne, dermatits, greasy skin and spots. It is also considered to be useful in cases of nervous stress, high blood pressure and arrhythmia. Used in the bath or shower litsea cubeba may aid tissue toning and reduction of cellulite and help to cleanse the system.
This is a mixture of Terpene hydrocarbons and has fresh, citrus, terpenic odor. It is a colorless to pale yellow clear liquid. Used in household products, colognes, shampoos, soaps, and detergents.
Lovage Root oil is obtained by steam distillation from the fresh roots. An amber or olive-brown liquid with a rich, spicy-warm, root-like odor. It is used as a fragrance component in soaps, cosmetics and perfumes. The oils and extracts are used as savoury flavoring agents and in liqueurs and tobacco.
Mace oil is extracted from the arils of the mace fruit. The Mace tree is a member of the Myristicaceae plant family. It is a pale yellow clear liquid. Mace oil is used in soaps, lotions, air fresheners, aromatherapy products, incense, spa products, medicines.
This oil is cold pressed from the unripened fruit of the mandarin tree and is expressed to produce a green oil. The scent is described as tangy citrus, with a fresh, sweet, green note. It is a digestive stimulant and mild antispasmodic. It is a tonic for the stomach and is used to calm the intestines. It is also stimulating to the liver and can help to regulate metabolic processes and may also aid the breakdown of fats.
This oil is cold pressed from the unripened fruit of the mandarin tree. Yellow Mandarin is particularly floral and sweet and beautifully complements heady white floral notes such as Jasmine Sambac, Tuberose Absolute and Orange Blossom Absolute. It has a beautiful, floral-citrusy scent.
Mandarin Yellow Italia Oil has a beautiful, yet slightly bitter aroma as compared to other citrus oils. This is a unique aroma which will display interesting results in natural perfumery. his scent calms the mind, helpful for anxiety, nervous tension, insomnia, restlessness. It is soothing, calming, refreshing, uplifting.
Spain Oil is derived from the plant. The entire plant is extremely aromatic. This oil is widely used in perfumery and spice applications. It has a fresh sweet spicy eucalyptus camphor like odor. It is a pale yellow to orange amber clear liquid.
Mentha oil is derived from a plant named Mentha arvensis or common mint. Mentha oil is extremely useful in a wide variety of industries namely food industry, pharmaceutical industry and also in perfumery and flavoring industries. Also, the constituents and derivatives of mentha oil like mint and menthol are used widely. Menthol is the major derivative product of mentha oil.
Mentha Piperita India Oil is an essential oil which is being used in various applications like perfumes, cosmetics, analgesic balms, ointments, lotions, creams, tooth paste shampoos and conditioners. This oil is derived from the leaves of Mentha Piperita plant.
Mentha Piperita USA Oil is anti-fungal, anti-viral, and anti-bacterial, as well as anti-inflammatory. It is a strong digestive and is helpful for nausea. It is used for many respiratory problems including flu, bronchitis, asthma, and pneumonia. It is a fever reducer, but also stimulating. A must have for headaches and migraines, and will also relive a stuffed up head. Peppermint is a liver tonic and re generator.
This oil is Steam distilled from the flowering tops and leaves. It has fresh, bright, penetrating minty top notes and sharp green camphorous undertones. It blends well with lavender, rosemary, eucalyptus and lemon. This oil is used in food products and medicines.
The essential oil of Myrtle is a natural astringent with balancing and harmonizing actions to the skin making it useful in cases of acne and oily skin in particular.It is a mildly sedative oil, can be helpful for those with difficulty sleeping especially those with insomnia. It may be helpful in the cessation of smoking as the green color is associated with the heart charka and helps to release blocks associated with the emotions.


Ranok has just added our magnificent No Problem! series to its language course offer. Customers will be able to learn Polish, English and German languages in Ukrainian or Russian. Courses are already available online and in bookstores around Ukraine.


text:
Nvidia heralds PC as the "best gaming experience" in E3 press conference
By Ian Birnbaum 2013-06-14T20:05:00.164Z
Nvidia did some well-deserved strutting at E3 yesterday, showing off the superiority of the PC as a gaming platform. With charts and graphs and maybe just a teeny tiny hint of bitterness that AMD processors are powering both the PS4 and the Xbox One, Nvidia's Tony Tamasi told the room that "the PC is the most powerful gaming platform out there."
You'd never hear this kind of honesty in the previous console generation, where AMD and Nvidia both had hardware inside the Xbox 360 and PS3. On finding themselves out of the console game for the foreseeable future, though, this is a sign of Nvidia becoming a more open, vocal proponent of PC gaming this generation.
"It's becoming more and more obvious that PC gaming is the best gaming experience right now, and that won't change even with next generation consoles launching," Tamasi's presentation stated.
It's been a great E3 for PC , but there have been some frustrations. Games that are almost certainly coming to PC had their PC versions go unannounced during the console press conferences, and the Sony versus Microsoft title fight has absorbed a lot of the news coverage of the event. It's nice to see that Nvidia, now that it's free to speak its mind, chose to do some legwork to pump up the PC gamers in the crowd. It might not have been the press conference that PC games really deserved , but it was definitely a start.
Via PCGamesN
For the latest from E3, check out our complete coverage .


text:
Review analysis
This entry is our analysis of a review or synthesis of research findings added to the Effectiveness Bank. The original review was not published by Findings; click Title to order a copy. Free reprints may be available from the authors – click prepared e-mail. The summary conveys the findings and views expressed in the review. Below is a commentary from Drug and Alcohol Findings.
Implementing managed alcohol programs in hospital settings: A review of academic and grey literature
https://findings.org.uk/PHP/dl.php?file=Brooks_HL_1.cab&s=dy&sf=sfnos
Implementing managed alcohol programs in hospital settings: A review of academic and grey literature.
Brooks H.L., Kassam S., Salvalaggio G. et al.
Drug and Alcohol Review: 2018, 37(1), p. S145–S155.
Unable to obtain a copy by clicking title? Try asking the author for a reprint by adapting this prepared e-mail or by writing to Dr Hyshka at ehyshka@ualberta.ca. You could also try this alternative source.
Is it feasible (and desirable) to give regular doses of alcohol to hospital inpatients when supervised withdrawal or short-term abstinence from drinking is not a realistic goal?
Summary Abstinence is frequently a necessary condition for receiving inpatient care. However, supervised/medication-assisted withdrawal and short-term abstinence are not always realistic goals, especially among homeless and otherwise socioeconomically disadvantaged groups who experience significant social and structural barriers to accessing adequate care, including care intended to ameliorate withdrawal symptoms (1 2 3 4 5). The featured review aimed to establish the feasibility of implementing an alternative to inpatient care premised on abstinence – 'managed alcohol programmes', which provide a harm reduction framework for dispensing regular doses of beverage alcohol to people who have been drinking heavily for a long time prior to hospital admission, and for whom abstinence is not realistic (eg, because they have not responded well to pharmacological management with benzodiazepines).
From summary and commentary
Abstinence is frequently a necessary condition for receiving inpatient care, but may not always be a realistic goal – particularly among people who are homeless or otherwise socioeconomically disadvantaged.
An alternative is to implement managed alcohol programmes in hospitals, whereby patients can be given tailored doses of alcohol to address their cravings and prevent withdrawal.
Studies in hospitals and the wider community indicate that this is a feasible approach, though further work is needed to develop and evaluate protocols for the administration of alcohol in acute care settings.
While no research has explicitly described the implementation of formal managed alcohol programmes in hospitals, reviewers found that 28 studies have examined the administration of alcohol in hospitals in order to prevent alcohol withdrawal syndrome, and 14 studies have examined managed alcohol programmes in the wider community, which were typically part of a broader package of health and social care services, including permanent supportive housing or shelter-based accommodation.
Main findings
Participants in community-based managed alcohol programmes tend to be chronically homeless or lacking stable housing, with severe alcohol use disorders, and unable or unwilling to participate in abstinence-based housing or treatment.
The integration of social support with patient care is an essential component of community-based managed alcohol programmes, which have reported positive social outcomes among their participants, including fewer encounters with the police, improved personal hygiene, and uptake of other health and social support.
There are considerable differences in alcohol-dispensing practices in hospitals and community-based managed alcohol programmes. Community studies reported that 148–207 g of alcohol (usually in the form of beer or wine) is typically dispensed every 1–1.5 hours during regular waking hours, whereas doses of alcohol dispensed in hospitals ranged from 30–100 g in the form of spirits, or larger amounts of beer, every 1–4 hours, generally until withdrawal symptoms subsided. Hospital-based studies also frequently described the administration of intravenous alcohol.
In order for a hospital-managed alcohol programme to best replicate the successes of community-managed alcohol programmes, patients should be provided an in-hospital consultation with an addiction medicine specialist and direct referral or access to community support, such as counselling and addiction treatment, primary health care, social services, community-based managed alcohol programmes (when available), and harm reduction-oriented housing options. They may also need to reconsider the use of intravenous alcohol, which could deter patients, who would probably prefer to drink alcohol, and undermine the objective of stabilising alcohol consumption.
Several hospital studies noted concerns about sending conflicting messages about the detrimental effects of drinking to patients. To mitigate this concern, beverage alcohol could be dispensed in a medicine bottle and the patient monitored while consuming the alcohol. Procedures in community settings that discourage drinking outside of the programme could also be incorporated into a hospital-managed alcohol programme. Participant agreements can ensure hospital staff and patients are aware of their responsibilities and obligations within the managed alcohol programme. Agreements should be flexible enough to accommodate minor infringements, particularly while patients are being stabilised, though flexibility may not always accord with highly regimented and standardised acute care environments.
The authors' conclusions
Managed alcohol programmes in the wider community have been well-described. Findings from these, along with studies of in-hospital administration of alcohol, indicate that hospital-based managed alcohol programmes are feasible and could potentially prevent uncomfortable alcohol withdrawal symptoms, stabilise patients' drinking, discourage non-beverage alcohol consumption, encourage patients to stay in hospital and complete treatment, and connect them to other health and social support. However, further work is needed to develop clear eligibility criteria to help clinicians identify patients appropriate for participation, for example so that patients most at risk of leaving against medical advice and not completing treatment can be engaged.
commentary Managed alcohol programmes have been widely tested in community settings where they fulfil a range of functions, including reducing harm, giving marginalised (and often homeless) people an alternative to street drinking, and connecting people with health and social services they might otherwise be excluded from. The featured review did not identify any studies of formal managed alcohol programmes in hospital settings, but did find examples of alcohol being dispensed to inpatients in order to prevent alcohol withdrawal syndrome. On the whole, what distinguished these practices from a managed alcohol programme was the lack of access to wraparound social care, and the different approach to or motivation for administering alcohol. The authors concluded that hospital-based managed alcohol programmes would be feasible and appropriate for patients who would otherwise be vulnerable to unmanaged or undermanaged symptoms of alcohol withdrawal, resorting to non-beverage alcohol consumption such as rubbing alcohol or hand sanitisers, and ultimately leaving hospital without completing their treatment.
Evidence regarding the implementation and effectiveness of managed alcohol programmes predominantly comes from Canada, where they emerged "out of a need for a more compassionate approach to care for people vulnerable to the harms of severe alcohol dependence and homelessness".
"Toronto's Seaton House, one of the first [managed alcohol programmes] in Canada, was started following an inquiry into the tragic freezing deaths of three men on the streets of Toronto in 1996. The recommendations from that inquiry were to develop a [24-hour] shelter program for men with severe alcohol dependence. In the early days of that program they began by storing personal alcohol for men so that they would stay inside overnight instead of ending up outside in the snow during freezing temperatures. This gradually evolved into inviting men to stay for breakfast and providing them with a glass of wine to 'settle their shakes' while encouraging them to eat. Daily alcohol administration started with one man being offered regular doses of alcohol throughout the day to prevent him from being picked up by the police for public intoxication."
Though not all managed alcohol programmes are the same, their commonalities tend to include:
• supporting the goal of preventing and reducing the harms of drinking, in particular those associated with 'binge drinking', drinking non-beverage alcohol, and consuming alcohol in unsafe settings;
• catering to a cohort of people experiencing chronic homelessness, frequent public intoxication, repeated (but unsuccessful) attempts at abstinence-based treatment, and high levels of engagement with police or emergency department services;
• integration with housing (eg, shelter, transitional or permanent supportive housing);
• provision of food;
• increasing access to primary care;
• encouragement to access recreational and social activities both inside and outside of the programmes.
The pursuit of abstinence for all homeless people with a history of substance use problems may not be realistic, and furthermore, may overlook the reasons why homeless people drink or take drugs in the first place. While the default assumption might be that drugs and alcohol have a significant role in precipitating or perpetuating homelessness, substance use can also be functional – it can be one of the ways that people survive being homeless (eg, keeping warm and passing time), as well as a way to derive some pleasure while being homeless. This may remain the case even if people drinking and taking drugs are doing so at harmful levels or through risky means, and even among people whose health is already compromised as a result of their homelessness. Within a harm reduction approach, substance use is accepted "for better or for worse", enabling services to directly "minimize its harmful effects rather than simply ignore or condemn them."
Evidence that managed alcohol programmes can improve the health and wellbeing of homeless dependent drinkers was acknowledged in Scotland's 2018 "Rights, Respect and Recovery" drugs and alcohol strategy, which compared to UK government policy was more willing to centre harm reduction-based interventions.
When abstinence remains a requirement for inpatient care and the person has been drinking heavily for a long time prior to hospital admission, NICE guidelines describe three ways of administering medication to assist with alcohol withdrawal:
• Fixed-dose regimens start with a standard dose which is then reduced over several days.
• Symptom-triggered regimens tailor treatment to the severity of withdrawal signs and symptoms which are regularly assessed and monitored. Medication is provided if the patient needs it, and treatment is withheld if there are no symptoms of withdrawal.
• Front-loaded regimens provide a large dose of long-acting medication at the start and then 'as required'.
NICE found insufficient evidence on front-loading. Compared to fixed-dose regimens, symptom-triggered dosing reportedly involved significantly lower doses of benzodiazepines over a shorter period without an increase in the incidence of seizures or delirium tremens or in the severity of withdrawal symptoms. However, most of the studies were based on patient samples consisting mainly of men who had been admitted to specialist addiction services. Symptom-triggered dosing requires patients to be closely monitored and health care workers with the specialist clinical knowledge needed to identify signs and symptoms that imply a change in severity of withdrawal. In the experience of the expert group that informed the development of the NICE guidelines, acquiring the required skills was not a major task.
The Pennine Acute Hospital Trust in England evaluated a locally-implemented, symptom-triggered approach to alcohol withdrawal management, which, it was hypothesised, would improve care by providing an individualised treatment plan for patients. The change from a fixed-dose regimen was associated with an average reduction of almost 60% in the length of hospital stay and a 66% reduction in the amount of chlordiazepoxide used in detoxification, as well as highlighting that 10% of the sample group did not display any signs of withdrawal and did not require any medication. Even with the reductions in medical treatment, no patient developed any severe signs of withdrawal post-intervention such as seizures or delirium tremens. However, the scope of this study did not include the effectiveness of a symptom-triggered approach among homeless people, for whom medication-assisted withdrawal and short-term abstinence may not be realistic or desirable, particularly when not accompanied by support to materially change their living conditions on leaving hospital.
Last revised 11 July 2019. First uploaded 01 June 2019
STUDY 2005 'Real-world' studies show that medications do suppress heavy drinking
REVIEW 2011 Medical treatment of alcohol dependence: a systematic review
DOCUMENT 2017 Drug misuse and dependence: UK guidelines on clinical management
STUDY 2014 Alcohol treatment in England 2013–14
DOCUMENT 2013 Alcohol treatment in England 2012–13
MATRIX CELL 2018 Drug Treatment Matrix cell D3: Organisational functioning; Medical treatment
STUDY 2014 Primary care-based buprenorphine taper vs maintenance therapy for prescription opioid dependence: a randomized clinical trial
DOCUMENT 2018 Medications for opioid use disorder: for healthcare and addiction professionals, policymakers, patients, and families: Treatment Improvement Protocol: TIP 63
STUDY 2015 Risk of mortality on and off methadone substitution treatment in primary care: a national cohort study


Generally speaking, it shouldn't matter. The sensor that drives the cruise control is also used by the EFI, and all the cruise wiring is in the dash harness or the CC harness. So at least in theory, you won't be changing anything related to the CC system.
I would if you could grab the cruise control box, and the vacuum ball, wiring harness. The cruise on these cars is a separate wiring harness. There is a module in the passenger dash I would also grab. However I'm like 90% sure your 2.8 cruise will carry over... but if you have access to the donor car why not grab the parts to be safe.
Thanks guys.I cant get anything from "donor"vehicle because the guy only wants to sell me the rebuilt engine,wire harness and ecm.All this is from an 87 Camaro and he is going to install L.S 3 engine.
Does the stock computer control cruise control at all?
Need help installing Cruise in a Non-cruise car!


Introduction Evidence from individual longitudinal studies suggests that anxiety and depression may impact quality of life. However, systematic reviews synthesising current evidence have mainly focused on specific samples. Thus, the aim of this study is to synthesise evidence from longitudinal studies on the association between anxiety, depression and quality of life in a systematic review.
Methods and analysis A systematic review of evidence from longitudinal studies analysing the association between anxiety, depression and quality of life will be conducted, taking into account the current Preferred Reporting Items for Systematic Reviews and Meta-Analysis guidelines. Several electronic databases from relevant fields of research (PubMed, PsycINFO, PSYNDEX, EconLit, NHS EED) will be searched in September 2018 using defined search terms, with an updated search planned. Moreover, reference lists of included studies will be searched manually. Study eligibility will be appraised in a two-step process against pre-defined inclusion/exclusion criteria. Primarily, information on study design and assessment, statistical methods, participant characteristics as well as results regarding our research question will be extracted. The quality of included studies will be assessed using an appropriate tool. Study selection, data extraction and assessment of study quality will be performed by two reviewers. Disagreements will be resolved through discussion or by inclusion of a third party. Results will be synthesised narratively in text and tables. Depending on the number and heterogeneity of the studies included, a meta-analysis will be performed.
Ethics and dissemination As no primary data will be collected, approval from an ethics committee is not required. Results will be disseminated through conference presentations and publication in a peer-reviewed, scientific journal.
To our knowledge, this is the first systematic review synthesising and critically assessing evidence from longitudinal, observational studies on the association between anxiety, depression and quality of life, focusing on samples without specific disorders.
The study's focus on longitudinal evidence from observational studies should strengthen the conclusions drawn from our results and may facilitate causal inference across studies.
Two independent reviewers are involved in the processes of study selection, data extraction and quality assessment.
Due to the possible heterogeneity between studies, conducting a meta-analysis may not be appropriate.
There are several systematic reviews that have synthesised evidence on the association between anxiety and/or depression and quality of life previously. However, existing reviews are limited with respect to the samples analysed. For example, some reviews have focused on certain age categories,13 14 or samples with specific diseases.15 16 Specifically, Creighton et al 13 as well as Sivertsen et al 14 have analysed the association between anxiety or depression and quality of life in older people. Regarding reviews of disease-specific samples, Blakemore et al 15 have analysed the association between anxiety, depression and quality of life in patients with chronic obstructive pulmonary disease (COPD), and Schram et al 16 have focused on depression and quality of life in patients with diabetes. Looking at longitudinal studies in particular, most reviews find a negative association with varying strength of the association. For example, the meta-analysis conducted by Blakemore et al 15 in patients with COPD found, that depression and anxiety were significantly related to reduced health-related quality of life at follow-up with moderate to large effect sizes. In contrast, Schram et al 16 reported no to small, negative effects of depressive symptoms on domain-specific quality of life in samples of patients with diabetes.
To the best of our knowledge, there are no recent systematic reviews specifically analysing evidence from longitudinal studies on the association between anxiety, depression and quality of life across all age categories and focusing on samples without specific diseases or disorders (other than anxiety or depression). Looking at longitudinal studies in particular provides the advantage that individual trajectories can be observed over time and thus, temporal associations between the variables can be assessed. Moreover, focusing on observational studies analysing samples without specific diseases means the effects of interventions or specific illnesses will be limited in terms of impact on this association.
Thus, to add to the present literature, we aim to synthesise current longitudinal, observational studies on the association between anxiety, depression and quality of life, as well as assess the quality of existing studies.
This paper provides the protocol for a systematic review that aims to synthesise and critically appraise longitudinal, observational studies assessing the association between anxiety, depression and quality of life. The studies of interest are those analysing participants from all age categories with anxiety/depression, as well as those that include samples without a specific disease or receiving a specific intervention, applying valid measures for the main variables.
This systematic review protocol was developed taking into account the Preferred Reporting Items for Systematic Reviews and Meta-Analysis Protocols guidelines.17 The study is registered with the International Prospective Register of Systematic Reviews (PROSPERO, registration number: CRD42018108008, under https://www.crd.york.ac.uk/PROSPERO/).
Studies will be assessed for inclusion according to the inclusion/exclusion criteria outlined below. Before the final eligibility criteria are applied, they will be tested against a sample of 100 titles/abstracts and refined should further clarification be required.
Observational studies analysing the longitudinal association between anxiety/depressive symptoms or disorder and quality of life across all age categories (to analyse the association with quality of life according to severity of anxiety/depressive symptoms as well as for those who fulfil the criteria for a clinical diagnosis).
Studies analysing samples with anxiety/depression or samples without a specific disease (other than anxiety/depression).
Studies applying an appropriate measure for anxiety and depression (eg, psychiatric diagnosis according to criteria of the International Classification of Diseases [ICD], the Diagnostic and Statistical Manual of Mental Disorders [DSM], or using a valid self-report screening questionnaire, such as the depression scale from the Patient Health Questionnaire [PHQ-9] or the Hospital Anxiety and Depression Scales [HADS]).
Studies applying an appropriate measure for quality of life (eg, the 36-item Short Form Health Survey).
Publications in German or English language, published in peer-reviewed journals.
Studies not focusing on the association between anxiety or depression and quality of life over time.
Studies only analysing disease-specific samples (meaning conditions other than anxiety/depression) or samples receiving specific interventions.
Study design other than observational.
Assessment of anxiety, depression or quality of life not appropriate (eg, for anxiety/depression not according to ICD/DSM criteria or no valid, self-report screening questionnaire).
Studies not published in peer-reviewed journal or in language other than German or English.
Information sources and search strategy.
As quality of life is an outcome analysed in several fields of research, such as medicine, psychology and health economics, several databases from multiple scientific fields (PubMed, PsycINFO, PSYNDEX, EconLit, NHS EED) will be searched electronically in September 2018. It is planned to update the search prior to the submission of the final review to ensure that it contains the most recent evidence. The electronic database search will be conducted using predefined search terms, including anxiety disorder, depressive disorder, anxi*, depress* (truncated to also capture studies using terms such as anxious or depressed), quality of life, and longitudinal study. The PubMed search strategy is provided in table 1. There will be no restriction on location or time of the publication. Where possible, search terms will be entered as Medical Subject Headings (MeSH), or keywords in the title/abstract. As needed, the search will be modified according to the specific requirements of each database.
Additionally, the reviewers will search the reference lists of included articles for further relevant articles.
All records retrieved in the database search will be imported into the literature management software EndNote to facilitate the management of the references. Should it be possible to conduct a meta-analysis, Stata (StataCorp, College Station, Texas) will be used for the quantitative analysis.
All studies obtained in the electronic and manual search will be assessed for inclusion/exclusion in a two-step process ( title/abstract screening and  screening of full texts). Before the final selection criteria are applied, the criteria will be pre-tested against a sample of 100 titles/abstracts and, if necessary, refined afterwards. The study selection process will be conducted independently by two reviewers (JKH and EQ) using the previously defined and refined selection criteria. Disagreements between the two reviewers will be resolved through discussion or by inclusion of a third party (AH).
The data extraction process will involve two reviewers as well (JKH and EQ). One reviewer will extract relevant data items from the studies in standardised form, and the second reviewer will cross-check the extracted data. Again, in case of disagreements consensus will be reached through discussion or by inclusion of a third party (AH). In cases where relevant data cannot be extracted or require clarification, the authors of the study will be contacted.
Data items to be extracted from the original studies include information on study design, definition and assessment of the main variables of interest, sample characteristics, statistical methods, as well as results regarding the longitudinal association between anxiety, depression and quality of life.
The quality of the individual studies will be assessed using a quality assessment tool appropriate for longitudinal, observational studies, such as the Newcastle-Ottawa Scale18 or the Quality Assessment Tool for Observational Cohort and Cross-Sectional Studies.19 For example, the Quality Assessment Tool for Observational Cohort and Cross-Sectional Studies comprises 14 quality criteria to assess the internal validity of a study, through consideration of potential forms of bias.
Each study included in the final synthesis will be appraised for study quality by two reviewers (JKH and EQ) independently. Should there be disagreement, consensus will be reached through discussion or by inclusion of a third party (AH). Results from the quality assessment will be included in the study synthesis.
Following its completion, the study selection process will be illustrated by means of a PRIMSA flowchart. The results from data extraction and quality assessment processes will be synthesised qualitatively in text and tables. If possible, results will be categorised according to specific disorder/symptoms analysed in the studies (anxiety/depression) or quality of life domain/measure analysed.
Egger et al 20 advise caution when conducting meta-analyses to pool results specifically from observational studies, as this may result in precise but incorrect estimates, due to confounding and bias within the included studies. However, recommendations on conducting a meta-analysis on data from observational studies vary, and to date there is no agreed on, comprehensive methodological outline for such analyses. For the most part, current recommendations centre around whether to, and how one should, pool the data.21 Whether a meta-analysis is performed (in view of the number of studies included as well as their heterogeneity) in our study, and how the analysis is conducted (ie, method used for the standardisation of the outcome measure as well as statistical methods for calculation of the overall effect), will be discussed in the final systematic review, with reference to current recommendations.
Thus, only if appropriate, a meta-analysis will be performed to obtain a pooled, quantitative estimate. Moreover, if a meta-analysis can be performed and a sufficient number of studies can be included in the synthesis, we plan to perform subgroup analyses (eg, by gender, age) or sensitivity analyses (eg, excluding studies with lower quality/higher risk of bias) to explore possible sources of heterogeneity as well as check the robustness of the results.
The present review protocol did not involve individual patients or public agencies.
This systematic review will provide an overview of evidence from longitudinal, observational studies on the association between anxiety, depression and quality of life across all age categories. Moreover, the quality of included studies will be rated. If possible, data will be pooled quantitatively by means of a meta-analysis, and subgroup or sensitivity analyses will be performed.
As stated in the introduction, this review adds to the present literature by additionally including samples without a specific disease (eg, general population samples) and not focusing on a specific age category.
Beyond providing an overview of evidence on the association between anxiety, depression and quality of life, and thus highlighting possible gaps in current research, there are a range of questions that could possibly be answered by this review. For example, our study could ascertain whether specific quality of life domains are particularly affected by specific disorders or symptoms across studies over time. Comer et al 9 found in a cross-sectional analysis in the general population that different anxiety disorders were associated with varying decrements in different health-related quality of life domains. If this type of finding were observed over several longitudinal studies in the course of our systematic review, our study could also inform clinical research. Identification of the specific domains impacted by anxiety/depression, for example, could act as the starting point for the analysis of treatment goals or the analysis of the effectiveness of interventions aiming to improve quality of life. However, as intervention studies, such as randomised controlled trials, will not be included in our review, additional research would need to be undertaken to build on this further. In addition, if studies show different results, possible sources of heterogeneity could be assessed across studies. Heterogeneity might be due to study design and methodology as well as due to sample-specific characteristics. Moreover, subgroups, such as age group or gender, could be analysed.
Finally, longitudinal evidence has several advantages over cross-sectional data. For example, trajectories over time can be analysed within the same individuals. Moreover, applying appropriate methods in longitudinal analysis can take intraindividual heterogeneity into account, which is a key point in the analysis of quality of life.22 Thus, as it draws from longitudinal evidence across several studies, our systematic review provides the basis for stronger, more definitive conclusions that one would not be able to draw from a single study.
To our knowledge, this is the first study aiming to synthesise and critically appraise evidence from longitudinal, observational studies analysing the association between anxiety, depression and quality of life across all age categories. Focusing on observational studies and samples without a specific illness (other than anxiety or depression), should limit the influence of a specific intervention, treatment or disease on the association between anxiety, depression and quality of life. In turn, this should strengthen the conclusion we can draw from our results. Moreover, another strength is the inclusion of two independent reviewers in the processes of study selection, data extraction and quality assessment. One limitation is that conducting a meta-analysis may not be appropriate due to heterogeneity of existing studies. Thus, it may not be possible to obtain a pooled estimate.
No primary data will be collected. Thus,approval by an ethics committee is not required. The results from the systematic review are planned to be published in a peer-reviewed journal and presented at scientific conferences.
. The Newcastle- Ottawa Scale (NOS) for assessing the quality of nonrandomised studies in meta-analyses. http://www.ohri.ca/programs/clinical_epidemiology/oxford.asp (Accessed Aug 2018).
National Heart, Lung, and Blood Institute. Study quality assessment tools. https://www.nhlbi.nih.gov/health-topics/study-quality-assessment-tools (Accessed Aug 2018).
. Systematic reviews of observational studies. In: Egger M , Smith GD , Altman DG , eds. Systematic reviews in health care: meta-analysis in context. London: BMJ Books, 2006.
Contributors The study concept was developed by JKH, H-HK and AH. The manuscript of the protocol was drafted by JKH and critically revised by H-HK, EQ and AH. The search strategy was developed by JKH and AH. Study selection, data extraction and quality assessment will be performed by JKH and EQ, with AH as a third party in case of disagreements. All authors have approved the final version of the manuscript.
Ethics approval As no primary data will be collected, approval by an ethics committee is not required. The results from the systematic review are planned to be published in a peer-reviewed journal and presented at scientific conferences.


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Muskie snipers starting to score
Lucas Punkari
During the early stages of the NorWOSSA season, the Muskie boys' hockey team has had trouble cashing in on their scoring chances in front of the net despite having outshot their opponents by a wide margin.
However, if this weekend's play at the Victoria Inn tournament in Brandon, Man. is any indication, the Muskie snipers may have started to begin to fire more accurately.
Despite being blanked 3-0 by Selkirk in the 'B'-side final Sunday, Fort High regularly hit the twine in their other four games over the weekend, with many different players contributing to the cause.
"Overall, I thought we played pretty good all weekend long and we were very solid," said assistant coach Brad Gushulak, who was in charge behind the bench during the tournament in place of head coach Shawn Jourdain.
"Our last game against Selkirk was probably the most physical game that we have had all year long, and we ended up getting away from our game a little bit there and ended up running around a little bit," he noted.
"But still, I'm pleased with how things went."
The Muskies finished the round-robin portion of the tournament with a 1-1 record in Pool 'A.'
After opening with a 6-2 win over Vincent Massey on Thursday, a 5-4 setback to Stonewall on Friday dropped the black-and-gold down to the 'B'-side bracket and a playoff match-up with their NorWOSSA rivals from Kenora.
"To be honest, it's kind of tough when you go all that way for a tournament and you end up facing someone in your league again as you would like to go against some teams that you haven't faced before," Gushulak reasoned.
Nevertheless, the Muskies skated away with a 3-0 win over the Broncos, with goalie Scott Parsons stopping every shot that came his way.
The black-and-gold then beat St. John's Ravenscourt 5-1 in the 'B'-side semi-finals on Saturday evening, setting the stage for their battle with Selkrik in the final.
"In the first period against St. John's, both sides were losing their cool a little bit over the refereeing," Gushulak recalled.
"During the intermission, we settled our guys down, and from then on we were able to take it to a pretty good team," he added.
While Gushulak said the entire roster contributed to the cause in Brandon, a few performances stood out in his mind.
"We put together a line with Nick Kaun, Braden Webb, and Cole Godbout that worked quite well as they were able to get a couple of big goals for us," he noted.
"Also, you had guys like Ryan Windigo and Braedon Keast, who had a couple of real solid games for us and helped us out in a lot of ways."
The performance in Brandon came on the heels of a 3-2 overtime win against the Dryden Eagles in NorWOSSA action here last Tuesday (Nov. 29).
Tarran Romyn netted the game-winning goal with just 14.5 seconds left in the extra frame.
"I'm not the one who is supposed to be scoring those types of goals," laughed Romyn, a Grade 12 defenceman.
"I just picked up the puck from my teammate [Webb] and took advantage of the situation," he noted. "And once you get into the zone, time just starts to stand still.
"Most teams know that I take the shot, but this time I thought I would try something a little bit different, and I gave the defender a toe drag and I went to my backhand," Romyn recalled.
"We didn't think he had those kind of hands, but we will take it," Gushulak said.
The Muskies peppered the Eagles' net throughout the game, outshooting their rivals by an eye-popping 55-13 margin.
But as noted, the black-and-gold has had a tough time capitalizing on their chances.
"It's extremely frustrating, especially when you don't allow the other team to have that many chances," Gushulak remarked.
"We will be spending some time in the practices on our 3-on-2 rushes in order to get some better shots on those chances, and we will go from there."
Tyler Mosbeck and Robbie Rea scored for the Muskies in regulation time, with Kirk Markowski and Richie Callewaert replying for the Eagles.
"I think we are kind of fortunate to get a point as this was not our best outing," Eagles' coach Darryl Mousseau admitted.
"We just couldn't take care of our zone, and we ended up getting outshot by a 3-to-1 margin.
"But our goaltender [Justin Oltsher] played well and kept us in it," Mousseau added.
With a 3-0-1 record in league play, the Muskies will look to continue their winning ways tomorrow night when they head to Kenora to face St. Thomas Aquinas.
Then the squad will returning home for a 7:30 p.m. clash Saturday with the Red Lake Rams at the '52 Canadians Arena.
"We haven't played either team yet this season, but T.A. has beaten Kenora already this year and Red Lake has won over Dryden, which shows you that anyone can win any game," Gushulak warned.
"We just want to go in there and take an approach like we do with every other team," Romyn said.
"We want to pound them into the ground and take advantage with what we got on our roster," he stressed.
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