text:
No Longer Made
No Longer Made: Snyper One Chronograph Watch
In 2010, I first covered the Snyper watch brand (see aBlogtoWatch coverage here). At the time, I discussed the new sports watch brand from Jean-Francois Ruchonnet, who was well-known for being the designer and "creator" (though Guy Semon was the man who got it to work) of the TAG Heuer Monaco V4. Ruchonnet was a bit of a wild character and, in part of his life, he was a sniper in the French military. His career later took him into design, which is what eventually led him to work on timepieces. Snyper was supposed to be his lifestyle brand that combined his machismo aesthetic style and the era's growing appetite for fun, high-end sports watches. Snyper's debut product was the Snyper One, and it came in a truly dizzying array of styles at loads of price points (yes, there were precious stone-decorated models), many of which can be seen in my debut hands-on article above.
In addition to being visually inspired by a modern military sniper rifle, the Snyper One watch and brand was a sort of allegory for adult male life, at least some people's idealized versions of it. Rifle sniping was meant to serve as a metaphor for a more honed style of "attack" on life's problems. There are people who study situations, wait patiently, and then try to act only at the right moment. If that sounds like you, then Mr. Ruchonnet might suggest that you, too, are a Snyper kinda person.
Visually, the Snyper One watch is very much like an Audemars Piguet Royal Oak Offshore from another (less expensive but still pricey) mother, with the core visual elements being things you'd find on a sniper rifle and its accessory parts. My favorite detail is the crown, which is meant to look like one of the adjusting knobs on the scope. The bold, fun, legible, tactical style of the Snyper One has aged a bit, but it still makes for a cool-looking design that has merit today.
The most novel part of the Snyper One's PVD-coated black steel case is the left flank, which has two polished steel bars. These make an attachment point for an accessory to attach to the case, similar to how a gun or rifle might have an accessory attachment point. Allow me to get into that for a moment. The second time that I met with the Snyper brand after my initial 2010 meeting was six years later, in 2016. I then produced an aBlogtoWatch watch review of the Snyper One watch here (same core piece as the black and yellow reference 10.260.00 I look at in this article, but with a different case and dial finishing). At the time I was able to loan some of the attachments that were available for the One, which included a laser pointer/light and a butane lighter (so check out that article for pictures). Snyper had more attachments in mind to develop including a telemeter (for people sniping play)… and probably some type of substance container (for other types of play).
Now in 2021, as I write this aBlogtoWatch No Longer Made article, the Snyper watch brand appears to be defunct. Maybe it will come back, probably it won't. I think a big reason for that is because watch brands today are very personality-driven, and I don't believe Jean-Francois Ruchonnet has been involved in the watch industry for at least several years. That said, there are still Snyper watches out there, so collecting these rare modern sports watches can be a bit of a thrill.
The watch case itself is heavy and strong-feeling. It's built a bit like a wristwatch tank (and even looks that way). I really love all the over-engineered design, from the complex case shape and pushers to the elaborate graphics on the caseback and dial. The case is about 43mm-wide without the flank and crown, and with them, the watch case is about 53mm-wide. The case is also about 14mm-thick and has a lug-to-lug distance of about 58mm. That means this watch wears large, which I like. But I would also have preferred one more inside strap hole so that the watch could fit on my slightly under seven-inch circumference wrist.
The Snyper One case is water-resistant to 100 meters and has an AR-coated sapphire crystal over the dial and caseback. Don't miss the period-trendy carbon fiber dial. Speaking of the dial, I've always liked the staggered look of the day and date indicator windows that seem like an actual refinement. I also like the strong legibility of the dial with the artistic backdrop of what looks like a closeup section of a Japanese Gundam robot character.
Inside the Snyper One watch is what the brand called its caliber F101 movement, which was essentially a Swiss Made ETA Valjoux 7750 automatic chronograph movement that I believe was decorated by watch movement maker Concepto. Movement presentation is great, actually. The 7750 has a nicely perlage-polished surface, blued steel screws, and a custom automatic rotor that moves very smoothly and is probably mounted on ceramic ball bearings. What's cool (and rare) is that the Snyper brand logo and "F101" label are engraved and colored into the movement itself. This is about as nice a 4Hz, 48-hour power reserve 7750 movement you could hope for (without it fundamentally becoming a different movement). So, the movement is pretty nicely done.
Snyper watches tend to have a lot of what you want to see in a high-end watch, including things such as a lot of custom parts and high-quality materials. The brand clearly did not find enough success in the marketplace between two owners, but I don't think the products are to blame. Often, cool watches miss out because of marketing deficiencies, changes in consumer trends, and relationships that brands have with retailers and other distribution points. If Snyper came out again today, it would find a challenging marketplace because its price point was too high, even though I do think there are consumers out there for something like this at a lower price point. Personally, I have had some good experiences with the Synper One watch and feel that in a lot of ways, it stands the test of time. It is a very polarizing watch with legitimate opinions on both sides of the table, but it is hard to deny that it is fun and distinctive. This particular reference 10.260.00 Snyper One watch was a limited edition of 100 pieces and had an original retail price of $8,200 USD.
Chronograph Watches, No Longer Made Watches
Snyper One Watch Review August 15, 2016
The 2010 Watch Buyer's Holiday Gift Guide October 26, 2010
A Brief History Of ETA: THE Swiss Watch Movement Maker


Compulsory schooling in Malta starts at the age of three and ends at 16. The primary school consists of two kindergarten years (3-4 years), Classes 1-5 (5-10 years). At 11, there is the option of either a Junior Lyceum leading to further studies or Area Secondaries leading to technical skills' teaching.
Vocational, technical and higher-education oriented schools are available. The University of Malta has a variety of courses available for the 18-year old students embarking on their first degree. Post-graduate studies are also available.
If you are interested in learning English in Malta, there are various exciting programmes and schools ready to welcome you aboard. Most schools offer accomodation with Maltese families so to practice your English during the day. Culture and education, what a wonderful mix.


DR DOPING sells medicines, vitamins, nootropics, additives, quetiapine mechanism of action bipolar ii and other healthy products at the international level.
DR DOPING continuously strives to improve your online shopping experience. Initially, our company promoted the benefits of doping Mildronate and stippled epiphyses warfarin and vitamin; hence the name - DR DOPING. We are confident that the DR DOPING offers the best conditions for the ordering of health products available on the market.


The ancient Celtic kingdom of Cornwall is blessed with over 300 miles of spectacular coastline along its distinctive southern shores. Its sparkling aquamarine waters are only disturbed by little villages and harbours, welcoming home colourful fishing boats at the end of the day as the sun slowly sets behind them. All have a worthy place on the bucket list of places to visit, but the ancient harbour of Polruan should definitely be one of your top choices.
Part of the parish of Lanteglos-by-Fowey, it is surrounded by water on three sides; jutting majestically out into the Fowey Estuary on one side and to the sea on the other. Across the water, its bustling harbourside neighbour, Fowey, is a wonder of medieval and Georgian architecture with a vibrant maritime history. Slightly further afield, and only an hour away by car is the cathedral city of Truro as well as St. Austell, home to the renowned Eden Project and the Lost Gardens of Heligan.
This lovely hilly outpost, with its cottages gazing down to the sea below, is famous for its boat building heritage and is still an important centre for fishing as well as other maritime activities. A solitary road into Polruan leads you down to the quay where the Lugger Inn, full of history and intrigue, will greet you for a well-deserved drink in its cosy bar. Rubbing along happily with the village store, it has a charm reminiscent of childhood holidays where the only thing that worried you was where you would buy your next ice cream. Hazy memories are sure to come back to you after just a few hours in this lovely village.
And across the water, where a passenger ferry will leisurely transport you, is the buzzy harbour town of Fowey. Here you can browse along the maze of little streets, wander around the art galleries and pick up some treats from one of the many bakeries or fudge shops. There are plenty of things to do in and around Fowey – beach trips and fishing, boat trips up the river and along the spectacular coast. It has a friendly and welcoming ambience that will make you want to return again and again.
Now we can't think of any other place we would rather go than to a cove with this fabulous name. Tucked away just inside the mouth of the river and surrounded by sweet little houses, you are sheltered enough to not have to worry about sandy sandwiches! Children will love paddling and rock pooling at low tide and older ones can swim out to the pontoon floating in the sparkling aquamarine sea. For those who get a bit bored after a couple of hours in the sun, there is a fabulous ten-minute stroll to St Catherine's Castle.
One of the most picturesque coves in the area to swim, sunbathe and play about in the sand, this remote spot is positively beckoning you to call at its shores. Polridmouth Cove has two gorgeous beaches looking out to St Austell Bay, another fabulous place to visit. Not only that but just above the high tide mark are the lake and grounds belonging to Menabilly House, allegedly the inspiration for the boathouse in Daphne du Maurier's epic novel, Rebecca. Its tranquillity comes at a price, however – it is a fifteen-minute walk to get to from any direction.
Take a day trip to this small sandy beach at low tide in Fowey, perfect for sitting back and watching the Polruan ferry taking its passengers from side to side. Sheltered by Whithouse Point and Quay, it is near enough to the esplanade to pick up an ice cream when the weather gets too hot and there is also a refreshing sea water paddling pool for little ones. Perfect for Famous Five-type adventures, just don't forget the tinned sardines and ginger beer!
What these beaches lack in name, they certainly make up for in convenience. Almost touching the Polruan quay is Front Beach, just a pebble's throw away from most of the properties in the village. Great at low tide when little ones can race around, collecting water for sandcastles when sunny and flying kites when windy - the infamous Winkle Picker shop on the quay is right on hand for crabbing lines and ice creams if you run out. Back beach is situated to the left of the quay and is slightly more quiet, better if you like your sunbathing uninterrupted.
If you are partial to a spot of fishing, the areas around Polruan are a fisherman's dream. Cast off from the quay or castle, go crabbing from the slipway or take a full day's deep-sea fishing. A favourite spot is St Catherine's Castle which you can get to via the base of the rocks at low tide. Be careful though as past half tide, you will have to climb the rocks back up to the castle! All sorts of fish can be caught in this region, from dogfish and mackerel to coalfish and conger! Whichever you manage to reel in, you will return home with fond memories and a catch to remember.
Who doesn't like a day messing about in sailing boats? Fowey Harbour is a haven for all types of sailing and no matter what your level, one of the yachting clubs, in true Cornish style, will welcome you with open arms. Both the Royal Fowey Yacht Club and the Gallants Sailing Club showcase an exciting racing calendar throughout the season for both cursers and dinghies. If you need some pointers, call the Fowey Maritime Centre or the Fowey Harbour Sailing School who will be able to give you a bit of training.
If you fancy a bit more speed, hire a motor boat from the town quay in Fowey during the summer and feel the breeze in your hair, regardless of what the wind is doing. You will be brought your boat, kitted out in life jackets and shown how to steer (just like a car apparently). Make your own way along this wonderful area, travelling down river towards the sea or up river past Fowey's iconic Troy sailing boats towards the docks and RNLI lifeboat. If being the captain of your own vessel is too much for you however, and you prefer to sit back and watch the envious faces on the banks as you sip a glass of champagne, then drop anchor and hand over to a local skipper who will take you on a river or coastal trip from the town quay.
For those of you who prefer a little tranquility on your holiday, why not immerse yourself in the calm and peaceful waters of the river in a canoe? Get back to nature and see the river from a different perspective, looking out for Herons and Kingfishers as you glide through the ripples – the odd deer has even been spotted! The Upper Harbour is the perfect area for this - you can hire a kayak from Fowey Kayak Hire and discover the hidden treasures of the river on your own or take an escorted trip with Fowey River Expeditions, perfect if it's your first time or if you have little ones with you.
Though Polruan has just a couple of shops for essentials (croissants are essentials in our opinion), Fowey over the river has a lovely selection of shops from little boutiques selling gifts and cards, interiors shops selling beautiful things for the home, confectioners selling fudge and chocolate and traditional chandlerys selling all sorts of things for messing about in boats.
When you have spent all your pennies, head for one of the many art galleries and studios where you can not only browse the artwork on display but get to see local artists at their best. Our favourite has to be the Venetian gondolier who in the midst of a mid-life crisis, swapped the canals of Venice for a year in Cornwall – his brilliant Art for Serious People Gallery should be on every visitor's to-do list.
For insider knowledge on the best shops and art galleries in the area, have a peek at our Shopping Guide for inspiration.
If you can bear to tear yourself away, there are plenty of fun things to do in the areas around Polruan and Fowey – the nearby villages and harbourside towns make a great family day out with food festivals, water sports, wildlife boat trips and horse-pulled trams, not forgetting the spectacular huge square sail ships of Charlestown, home to the famous TV series, Poldark.
The historic twin towns of East and West Looe are a great place to take a trip on a glass-bottomed boat - you may see some seals and dolphins playing about in the waves if you are lucky! Don't miss the unspoilt fishing villages of Polperro and Mevagissey for a taste of the traditional side of Cornwall - they really take you back to another era. Daring types can hunt for the mysterious beast of Bodmin Moor; for those who do not want to have an extra guest on their day out, they can instead traverse the coast on the more tranquil Bodmin and Wenford Railway.
Slightly further afield, the cathedral town of Truro is just over an hour away and the old market town of St. Austell is slightly nearer at three-quarters of an hour's drive. Visit one of its many attractions including the fascinating Eden Project, the Lost Gardens of Heligan and Caerhays Castle. If that isn't enough, you can take a day trip down to St. Ives and the most south-westerly point of Great Britain, Land's End, visiting the majestic St. Michael's Mount on the way - just make sure that you check the tide table first!
If we have persuaded you that Polruan and Fowey should be that bit nearer the top of your bucket list, have a peek at our lovely selection of cottages in Polruan and cottages in Fowey for ideas on where to stay.


1. Bavaria is an experienced manufacturer with a longstanding reputation.
First of all, Bavaria is not new to this game by any means. The company was founded all the way back in 1978, so Bavaria has decades of experience creating high quality yachts. Indeed, today the company is one of the largest sailing manufacturers around. Every year, they manufacture as many as 4,000 vessels to ship out to buyers around the globe. So when you take out a Bavaria yacht, you know you can trust in its quality.
2. As you might expect with German manufacturing, Bavaria cuts no corners.
Indeed, that quality is ensured through Bavaria's attention to detail in the manufacturing process. The company utilizes cutting edge technology, and only has a single factory. That means that all their boats are manufactured in the same location using the same equipment. So you will find no quality control issues here; all the vessels are manufactured according to the same high standards. The materials used in these boats are all of the highest grade; you will never see particle board where you expect to see real wood.
Nonetheless, Bavaria never stops to rest on its laurels. This is a company which continues to innovate both in the manufacturing and design departments. New vessels are being added to the Bavaria catalogue on a regular basis.
The components used in Bavaria vessels also are of the highest quality, and come from trusted brands. Electronics are provided by Garmin, the sails are Elvstrom, the deck equipment is Lemwar, Jefa makes the steering systems, and so on.
3. Bavaria vessels are considered very easy to handle, making them ideal even for beginners.
Bavaria's boats are not just known for their high quality materials and sturdy builds; they are known for their easy handling and sophisticated designs as well. Indeed, many of them can be sailed with just one or two crewmembers. That means that if you want to set out to explore the Croatian coastline by yourself, you can. Even so, you will find plenty of technical features to enjoy. That means that Bavarian yachts are an excellent choice for novice and advanced sailors alike.
4. There is a wide selection of comfortable Bavaria charter vessels to choose from.
• Bavaria 55 Cruiser Yacht Charter: This is one of Bavaria's larger yachts, with 5 cabins and 10 berths. This would be a fantastic choice if you want to set sail with a larger group and are looking for plenty of room to accommodate everyone.
• Bavaria 40 Cruiser Avantgarde Yacht Charter: This is a 3 cabin model which contains 6 berths. Many of Bavaria's models are mid-sized like this, offering the perfect amount of space for a smaller group.
• Bavaria Easy 9.7 Yacht Charter: This yacht has 2 cabins and berths for 4. It is a smaller option which is ideal if you are setting out with just one other person or a small group or family.
• Bavaria B/one: This is one of the smallest yachts offered by Bavaria, set up for just one person.
This is just a small sampling of Bavaria's catalogue.
You will find numerous different options no matter what size of vessel you are looking to charter.
When you call to book a charter yacht in Croatia, you will have a number of different manufacturers to choose from.
While there are a lot of great choices, one of the best in the industry today is Bavaria.
So ask about Bavaria vessels, and enjoy your holiday sailing the beautiful Adriatic!


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redpajama_set_name: RedPajamaCommonCrawl


In the pantheon of celebrity golfers, Charles Barkley holds a prominent -- some would argue notorious -- standing. While Barkley's dominance on the hardwood was a thing of beauty, the Round Mound of Rebound's golf swing is anything but. Even a stint with Hank Haney failed to produce results, as the NBA Hall of Famer's hacks continued to resemble a man trapped in a telephone booth with bees. Barkley was so dejected about this game that he promised to build a statue of the person that could fix his woes.
Hard to argue that logic.
Whatever gets him on the course, we're just glad to call Sir Charles a fellow golfer.


text:
\section{Introduction}\label{Introduction}

The differential master equations and the integral operator-sum
Kraus form of the completely positive dynamical maps are the basic
however mutually non-equivalent methods in modern open quantum
systems theory \cite{KrausSEO,BreuPetr,RivasHuelga,N&Ch}. On the
one hand, for the completely positive (CP) dynamical maps,
solution of a master equation can always be expressed as a quantum
operation \cite{KrausSEO} in the operator-sum representation
\cite{BreuPetr,RivasHuelga,N&Ch}. On the other hand, an
operator-sum representation cannot necessarily be presented in a
master-equation form \cite{BreuPetr,RivasHuelga,N&Ch,PRSA}. Hence the
relative advantage of the operator-sum {\it formalism}.

Nevertheless, in the case of the Markovian (and particularly
time-ho\-mo\-gen\-eous semigroup) dynamics
\cite{BreuPetr,RivasHuelga,N&Ch}, when the methods are
interchangeable, there is a {\it physical} advantage of the
master-equations formalism that is worth emphasizing. Actually,
the master equations formalism fully respects the microscopic
details in the Hamiltonian form of the composite system ``open
system + environment''. These details regard the realistic
physical situations and, in principle, open the door for the
better control of the open system's dynamics. However, such
microscopic-model details are typically absent from the
operator-sum representation of the open-system's dynamics. That
is, the Kraus operators are often constructed without the clear
and unambiguous microscopic-model criteria \cite{N&Ch}. Hence the
natural, as yet virtually intact, question arises: whether or not
the widely used Kraus operators \cite{N&Ch} reliably represent the
microscopically modelled quantum noisy channels?

In this paper we answer this question for the quantum depolarizing
channel
\cite{N&Ch,RomLoFranco,BCSQIQC,QCD,ByChManiscalco,Marinescu,KlimSoto,GeomEntDurstPhD}
while the analogous results for the (generalized) amplitude and
phase damping processes can be found in \cite{KJS}.  We make use
of a recently devised method for deriving the Kraus operators from
a local-in-time master equation for the finite-dimensional open
systems \cite{AndCresHall}. Formally, this procedure is equivalent
to solving the master equation, whether Markovian, non-Markovian,
of Lindblad form or not. The method can also be used in the
opposite direction.

We first derive an interaction-picture master equation and hence
derive the \textit{novel} Kraus operators that generalize the
standard ones. The new Kraus operators can be termed
``generalized'' as the proper choice of the model parameters
reduce the new to the standard Kraus operators for the depolarizing  (DP) channel.
Dissent of the novel from the standard Kraus operators is
investigated via temporal behavior of the Bloch sphere volume, the
state trace-distance and entropy production. We find that, in
short time intervals, the generalized DP channel is less
deteriorating than the standard DP channel, while in the
asymptotic limit the channels are indistinguishable. This finding is confirmed by investigating entanglement dynamics for a pair of qubits initially in a maximally entangled state.

Structure of this paper is as follows. In Section \ref{The method}
we overview the basics of the procedure devised in
Ref.\cite{AndCresHall}. Based on a master equation for the
depolarizing channel that is derived in Appendix, in Section
\ref{KrausDP} we derive the  generalized Kraus operators for the
depolarizing channel. In Section \ref{Analysis} we perform
comparison of the standard and the here obtained generalized
depolarizing processes. Section \ref{Discussion} is discussion and
we conclude in Section \ref{Conclusion}.

\section{Overview of the method}\label{The method}
In Ref. \cite{AndCresHall}, the authors  developed a general
procedure for deriving a Kraus decomposition from a known master
equation and vice versa, regarding the  finite-dimensional quantum
systems. The only assumption is that the master equation is local
in time.

The so-called Nakajima-Zwanzig projection method
\cite{BreuPetr,RivasHuelga} gives the following master equation
for the system's density operator $\hat{\rho}_S(t)$, ($\hbar=1$):
\begin{equation}\label{NZmaster}
\frac{d\hat{\rho}_S(t)}{dt}=-i[\hat{H},\hat{\rho}_S(t)]+\int_{0}^{t}
\mathcal{K}_{t,s}[\hat{\rho}_S(s)]ds,
\end{equation}
where $\hat{H}$ represents the system's self-Hamiltonian (that
includes the so-called Lamb-shift term) and $\mathcal{K}_{t,s}$ is
the memory kernel which accounts for the non-unitary effects  due
to  the environment.

Certain processes can be written  in a local-in-time form
\cite{BreuPetr,RivasHuelga,ChatShi,ShiTakash}:
\begin{equation}\label{Localmaster}
\dot{\hat{\rho}}_{S}(t)=\Lambda_{t}(\hat{\rho}_{S}(t)),
\end{equation}
where $\Lambda_{t}$ is a linear map which preserves hermiticity,
positivity and  unit trace of $\hat{\rho}_{S}(t)$ and has the
property:
\begin{equation}\label{Traceproperty}
tr\Lambda_{t}(\hat{\rho}_{S}(t))=0.
\end{equation}

Alternatively, dynamics can be presented in a non-differential,
``integral'' form \cite{BreuPetr,RivasHuelga,AndCresHall}:
\begin{equation}\label{Integralform}
\hat{\rho}_{S}(t)=\phi_{t}(\hat{\rho}_{S}(0)),
\end{equation}
where $\phi_{t}$ is a completely positive and trace preserving,
linear map.

It can be shown \cite{AndCresHall}, that the linear maps
$\Lambda_{t}$ and $\phi_{t}$ are connected via the matrix
differential equation:
\begin{equation}\label{IntDiffconnection}
\dot{F}=LF,
\end{equation}
where the $F$ matrix represents the $\phi_t$ map while the matrix
elements of $L$ are given by:
\begin{equation}\label{MatrixelemL}
L_{kl}=tr[\mathcal{G}_k\Lambda(\mathcal{G}_l)].
\end{equation}
In eq.\eqref{MatrixelemL}, $\{\mathcal{G}_k\}$ is  an orthonormal
basis  of the Hermitian operators acting on the system's Hilbert
state space. For the time independent $\Lambda_t$, i.e. $L$,
eq.\eqref{IntDiffconnection} has the unique solution:
\begin{equation}\label{Integralformsolution}
F=e^{Lt}.
\end{equation}

Complete positivity of the map $\phi_t$ (and hence of the matrix
$F$) is equivalent to the positivity of the, so called, Choi
matrix, $S$ \cite{AndCresHall,Choi} whose elements are defined
as \cite{AndCresHall}:
\begin{equation}\label{Choimatrix}
S_{nm}=\sum_{s,r}F_{sr}tr[\mathcal{G}_r\mathcal{G}_n^{\dag}\mathcal{G}_s\mathcal{G}_m],
\end{equation}

\noindent where $F_{sr}$s are entries of the $F$ matrix,
eq.\eqref{IntDiffconnection}.

With the use of eq. \eqref{Choimatrix}, eq.\eqref{Integralform}
takes the form:
\begin{equation}\label{AlternDecomp}
\phi(\hat{\rho}_S(0))=\sum_{nm}S_{nm}\mathcal{G}_n\hat{\rho}_S(0)\mathcal{G}_m^{\dag},
\end{equation}
which, after diagonalization of the  $S$ matrix:
\begin{equation}\label{ChoiDiag}
S=UDU^{\dag},
   \end{equation}
gives rise to a Kraus decomposition. The eigenvalues $d_i$ and the
eigenvectors of the $S$ matrix constitute the diagonal matrix $D$
and the unitary matrix $U=(u_{ij})$, respectively; columns of the
unitary matrix $U$  are  the normalized eigenvectors of the $S$
matrix. Then the Kraus operators:
\begin{equation}\label{KrausoperViaDandU}
E_i=\sum_j\sqrt{d_i}u_{ji}\mathcal{G}_j
   \end{equation}
yield the Kraus decomposition of the  dynamical map $\phi_t$:
\begin{equation}\label{KrausDecompphiB}
\phi_t(\hat{\rho}_S(0))=\sum_{k}\hat{E}_k(t)\hat{\rho}_S(0)\hat{E}_k^{\dag}(t).
\end{equation}

Hence the chain of the construction is established: from a master
equation to calculate $L$, then via relation
\eqref{Integralformsolution} to obtain the matrix $F$ and, due to
eq.\eqref{Choimatrix} and eq.\eqref{ChoiDiag}, to calculate the
Kraus operators, eq.\eqref{KrausoperViaDandU}.

\section{The depolarizing channel}\label{DPchannel}

While microscopic derivation for the phase and amplitude damping
master equations can be found in the  literature
\cite{RivasHuelga}, to the best of our knowledge, this is not the
case for the master equation concerning the depolarizing process,
eq.\eqref{RomLoFrancoDPmaster}. One possible form (due to the
unitary freedom) of the standard depolarizing Kraus operators is
given by\cite{N&Ch,RomLoFranco,BCSQIQC,Marinescu}:
\begin{align}\label{RomLoFrancoDPKraus}
\hat{E}_{0}&=\sqrt{1-\frac{3p(t)}{4}} \hat{I},&
\hat{E}_{i}&=\frac{\sqrt{p(t)}}{2}\hat{\sigma}_{i}, (i=x,y,z)
\end{align}

\noindent with the associated, {\it phenomenological} master
equation in the interaction picture:

\begin{align}\label{RomLoFrancoDPmaster}
\frac{d\hat{\rho}_S(t)}{dt}&=\frac{\gamma}{8}\sum_i(
   2\hat{\sigma}_i\hat{\rho}_S(t)\hat{\sigma}_i-\hat{\sigma}_i^2\hat{\rho}_S(t)-
   \hat{\rho}_S(t)\hat{\sigma}_i^2)\,,
\end{align}
where  the Pauli sigma-operators  $\hat{\sigma_i}$ ($i=x,y,z$)
pertain to the qubit's degrees of freedom and
 $\gamma$ is so-called damping
constant, while $p(t)=1-e^{-\gamma t}$.

In this section, for a microscopic (Hamiltonian)  model we derive
the Kraus operators starting from a Markovian (Lindblad-form)
master equation, which generalizes eq.\eqref{RomLoFrancoDPmaster}
and therefore leads to a generalization of the standard Kraus
operators, eq.\eqref{RomLoFrancoDPKraus}. Thereby we implicitly
test derivability of eq.\eqref{RomLoFrancoDPKraus} from
eq.\eqref{RomLoFrancoDPmaster} that here will not be made
explicit. Analogous derivation regarding the GAD and PD channels
can be found in \cite{KJS}.

The microscopic model for  the depolarizing  process we are
interested in is given by the following Hamiltonian
\cite{RomLoFranco}:
\begin{equation}\label{A1}
    \hat{H}=\hat{H}_S\otimes\hat{I}_E+\hat{I}_S\otimes\hat{H}_E+\hat{A}_x\otimes\hat{B}_x
    +\hat{A}_y\otimes\hat{B}_y+\hat{A}_z\otimes\hat{B}_z.
\end{equation}

The terms in eq.\eqref{A1} [in the physical units of $\hbar = 1$]:
the self-Hamiltonians $\hat{H}_S=\frac{\omega_0}{2}\hat{\sigma}_z$
and $\hat{H}_E=\sum_i\int^{\omega_{max}}_0 d\omega
\hat{a}^{\dag}_{\omega i}\hat{a}_{\omega i}$ for the $S$ system
and the environment (thermal bath) $E$, respectively, while the
interaction terms
$\hat{A}_i\otimes\hat{B}_i=\hat{\sigma}_i\otimes\int^{\omega_{max}}_0
d\omega h(\omega) (\hat{a}^{\dag}_{\omega i}+\hat{a}_{\omega i})$,
$i=x,y,z$, with the coupling functions $h(\omega)$ accounting for
the interaction strength per frequency. From eq.\eqref{A1} we can
learn that the environment $E$ effectively acts as a set of three
independent bosonic environments, $E_i, i=x,y,z$, with the
standard, Bose-operators,  commutation relations $[\hat{a}_{\omega
i},\hat{a}_{\omega' j}^{\dag}]=\delta(\omega-\omega')\delta_{ij}$,
$i,j=x,y,z$.

Our approach is minimalist in the sense that we do not introduce
assumptions that might force us to leave the standard physical
models. Therefore we are concerned with the time-homogeneous $L$,
cf. eq.\eqref{MatrixelemL}, as well as with the Ohmic spectral
density.

\subsection{Kraus operators for the generalized depolarizing channel}\label{KrausDP}

In Appendix we derive the following, Markovian (Lindblad-form),
master equation for the microscopic model eq.\eqref{A1} of the
depolarizing channel:

\begin{equation}\label{GeneralizedMJ}
       \begin{split}
\displaystyle\frac{d\hat{\rho}_S(t)}{dt}&=
-i\Delta[\hat{\sigma}_z,\hat{\rho}_S(t)]+ \gamma_{zz}(0,
T)(\hat{\sigma}_{z}\hat{\rho}_S(t)\hat{\sigma}_{z}-\hat{\rho}_S(t))+\\&
+\frac{\gamma_{xx}(\omega_0, T)+\gamma_{yy}(\omega_0,
T)}{2}(\hat{\sigma}_{x}\hat{\rho}_S(t)
\hat{\sigma}_{x}-\hat{\rho}_S(t))+\\& +\frac{\gamma_{xx}(\omega_0,
T)+\gamma_{yy}(\omega_0, T)}{2}
(\hat{\sigma}_{y}\hat{\rho}_S(t)\hat{\sigma}_{y}-\hat{\rho}_S(t))\,.
\end{split}
\end{equation}

\noindent Definitions of the constants appearing in
eq.\eqref{GeneralizedMJ} can be found in Appendix where we
strongly emphasize  that the microscopic model eq.\eqref{A1}
allows for the Markovian dynamics eq.\eqref{GeneralizedMJ}
\textit{only if} the spectral density is of the Ohmic type and
$\omega_0/\omega_{c} \ll 1$, while the high temperature limit
gives rise to the constraint $k_B T/\hbar\omega_0 \gg 1$ and
$\omega_{c}$ is the cutoff frequency.

Eq.\eqref{GeneralizedMJ}  straightforwardly reduces to the
standard one, eq.\eqref{RomLoFrancoDPmaster}, for $\Delta=0$ (no
Lamb shift term) and $2
\gamma_{zz}(0,T)=\gamma_{xx}(\omega_0,T)+\gamma_{yy}(\omega_0,T)$.
From eq.\eqref{A1} it follows the same interaction-strength per
frequency $h(\omega)$, which givers rise the same interaction
strength, $\alpha$ for all environments--see Appendix.

To ease the calculation we  introduce: $\mathbf{x}=\Delta$,
$\mathbf{y}=\gamma_{zz}$ and
$\mathbf{z}=\frac{\gamma_{xx}+\gamma_{yy}}{2}$ with which the
equation (16) reads:
\begin{equation}\label{DPmastereqxyz}
       \begin{array}{r c l}
\displaystyle\frac{d\hat{\rho}_S(t)}{dt}&=&
-i\mathbf{x}[\hat{\sigma}_z,\hat{\rho}_S(t)]+
\mathbf{y}(\hat{\sigma}_{z}\hat{\rho}_S(t)\hat{\sigma}_{z}-\hat{\rho}_S(t))\\&
+&\mathbf{z}(\hat{\sigma}_{x}\hat{\rho}_S(t)\hat{\sigma}_{x}-\hat{\rho}_S(t))+
\mathbf{z}(\hat{\sigma}_{y}\hat{\rho}_S(t)\hat{\sigma}_{y}-\hat{\rho}_S(t))\,,
\end{array}
\end{equation}

Then, the use of eq.\eqref{MatrixelemL} gives rise to:

\begin{equation}\label{DPmatL}
L=\left(
\begin{array}{cccc}
 0 & 0 & 0 & 0 \\
 0 & -2 (\mathbf{y}+\mathbf{z}) & -2 \mathbf{x} & 0 \\
 0 & 2 \mathbf{x} & -2 (\mathbf{y}+\mathbf{z}) & 0 \\
 0 & 0 & 0 & -4 \mathbf{z}
\end{array}
\right)\,.
\end{equation}

Multiplication of the matrix $L$ by
$\frac{1}{2(\mathbf{y}+\mathbf{z})}$ and introduction of the new
parameters, $\theta =\frac{\mathbf{x}}{\mathbf{y}+\mathbf{z}}$,
$\Omega= -\frac{2 \mathbf{z}}{\mathbf{y}+\mathbf{z}}$, $\tau
=2(\mathbf{y}+\mathbf{z})t$, give

\begin{equation}\label{DPmatLthetaOmegatau}
L'=  \frac{L}{2(\mathbf{y}+\mathbf{z})}=\left(
\begin{array}{cccc}
 0 & 0 & 0 & 0 \\
 0 & -1 & -\theta  & 0 \\
 0 & \theta  & -1 & 0 \\
 0 & 0 & 0 & \Omega
\end{array}
\right)\,
\end{equation}
while $\Omega\in(-2,0)$, $\tau\in(-\infty,\infty)$.

Now eq.\eqref{Integralformsolution} can be written as:

\begin{equation}\label{DPmatF}
 F=e^{L'\tau},
\end{equation}

\noindent that makes easy the calculation of the exponential
matrix $F$:

\begin{equation}\label{DPmatF1}
F=\left(
\begin{array}{cccc}
 1 & 0 & 0 & 0 \\
 0 & e^{-\tau } \cos (\theta  \tau ) & -e^{-\tau } \sin (\theta  \tau ) & 0 \\
 0 & e^{-\tau } \sin (\theta  \tau ) & e^{-\tau } \cos (\theta  \tau ) & 0 \\
 0 & 0 & 0 & e^{\tau  \Omega }
\end{array}
\right)\,.
\end{equation}

\noindent as well as of the  Choi matrix, cf.
eq.\eqref{Choimatrix}:
\begin{equation}\label{DPmatS}
  S=\left(\scalefont{0.8}{
\begin{array}{cccc}
 e^{-\tau } \cos (\theta  \tau )+\frac{e^{\tau  \Omega }}{2}+\frac{1}{2} &
 0 & 0 & i e^{-\tau } \sin (\theta  \tau ) \\
 0 & \frac{1}{2}-\frac{e^{\tau  \Omega }}{2} & 0 & 0 \\
 0 & 0 & \frac{1}{2}-\frac{e^{\tau  \Omega }}{2} & 0 \\
 -i e^{-\tau } \sin (\theta  \tau ) & 0 & 0 & -e^{-\tau } \cos (\theta  \tau )+
 \frac{e^{\tau  \Omega }}{2}+\frac{1}{2}
\end{array}}
\right)\,.
\end{equation}

Diagonalization of the $S$ matrix, eq.\eqref{DPmatS}, gives the
eigenvalues:
\begin{subequations}\label{DPEigenvalues}
\begin{align}
\frac{1}{2} \left(1-e^{\tau  \Omega }\right), \\
\frac{1}{2} \left(1-e^{\tau  \Omega }\right), \\[0.5ex]
\frac{1}{2} e^{-\tau } \left(-2+e^{\tau }+e^{\tau +\tau  \Omega
}\right),\\ \frac{1}{2} e^{-\tau } \left(2+e^{\tau }+e^{\tau +\tau
\Omega }\right)\,,
\end{align}
\end{subequations}

\noindent and  the  respective non-normalized eigenvectors:
\begin{subequations}\label{DPEigenvectors}
\center{
\begin{align}
 \{0,0,1,0\},\\  \{0, 1, 0, 0\},\\ \{i (\cot[\theta  \tau ]-\csc[\theta  \tau ]),0,0,1\},
 \\\{i (\cot[\theta  \tau ]+\csc[\theta  \tau ]),0,0,1\}\}\,.
\end{align}}
\end{subequations}

With the use of eq.\eqref{KrausoperViaDandU}, from
eqs.\eqref{DPEigenvalues}-\eqref{DPEigenvectors} follow the
desired Kraus matrices:
\begin{equation}\label{DP1}
\mathbb{E}_1=\left(
\begin{array}{cc}
 0 & -\frac{1}{2} i \sqrt{1-e^{\tau  \Omega }} \\
 \frac{1}{2} i \sqrt{1-e^{\tau  \Omega }} & 0
\end{array}
\right)
\end{equation}

\begin{equation}\label{DP2}
\mathbb{E}_2=\left(
\begin{array}{cc}
 0 & \frac{1}{2} \sqrt{1-e^{\tau  \Omega }} \\
 \frac{1}{2} \sqrt{1-e^{\tau  \Omega }} & 0
\end{array}
\right)
\end{equation}

\begin{equation}\label{DP3}
\mathbb{E}_3=\left(
\begin{array}{cc}
 \frac{1}{2} \left(1-i \tan \left(\frac{\theta  \tau }{2}\right)\right)
 \sqrt{\frac{e^{\tau  \Omega }-2 e^{-\tau }+1}{\tan ^2\left(\frac{\theta  \tau }{2}\right)+1}} & 0 \\
 0 & \frac{1}{2} \left(-1-i \tan \left(\frac{\theta  \tau }{2}\right)\right)
 \sqrt{\frac{e^{\tau  \Omega }-2 e^{-\tau }+1}{\tan ^2\left(\frac{\theta  \tau }{2}\right)+1}}
\end{array}
\right)
\end{equation}

\begin{equation}\label{DP4}
\mathbb{E}_4=\left(
\begin{array}{cc}
 \frac{1}{2} \left(1+i \cot \left(\frac{\theta  \tau }{2}\right)\right)
 \sqrt{\frac{e^{\tau  \Omega }+2 e^{-\tau }+1}{\cot ^2\left(\frac{\theta  \tau }{2}\right)+1}} & 0 \\
 0 & \frac{1}{2} i \left(\cot \left(\frac{\theta  \tau }{2}\right)+i\right)
 \sqrt{\frac{e^{\tau  \Omega }+2 e^{-\tau }+1}{\cot ^2\left(\frac{\theta  \tau }{2}\right)+1}}
\end{array}
\right)
\end{equation}

It is easy to show that the  Kraus matrices
\eqref{DP1}-\eqref{DP4} satisfy the completeness relation $
\sum_{k}\hat{\mathbb{E}}_k(t)^{\dag}\hat{\mathbb{E}}_k(t)=\hat{I}$
and also reduce to the standard ones in
eq.\eqref{RomLoFrancoDPKraus} with the following choice of the
parameters: $\theta=0$ (equivalently $\mathbf{x}=0$, neglecting
the Lamb shift) and $\Omega=-1$ (equivalently
$\mathbf{y}=\mathbf{z}$).  That is, for this choice of the
parameters values, from eqs. \eqref{DP1}-\eqref{DP4} easily follow
approximations of the Kraus operators eqs.\eqref{DP1}-\eqref{DP4},
respectively:

\begin{equation}\label{DP1Stand}
\mathbb{E}_1=\left(
\begin{array}{cc}
 0 & -\frac{1}{2} i \sqrt{1-e^{-\tau }} \\
 \frac{1}{2} i \sqrt{1-e^{-\tau }} & 0
\end{array}
\right)\,,
\end{equation}

\begin{equation}\label{DP2Stand}
\mathbb{E}_2=\left(
\begin{array}{cc}
 0 & \frac{1}{2} \sqrt{1-e^{-\tau }} \\
 \frac{1}{2} \sqrt{1-e^{-\tau }} & 0
\end{array}
\right)\,,
\end{equation}

\begin{equation}\label{DP3Stand}
\mathbb{E}_3=\left(
\begin{array}{cc}
 \frac{1}{2} \sqrt{1-e^{-\tau }} & 0 \\
 0 & -\frac{1}{2} \sqrt{1-e^{-\tau }}
\end{array}
\right)\,,
\end{equation}

\begin{equation}\label{DP4Stand}
\mathbb{E}_4=\left(
\begin{array}{cc}
 -\frac{1}{2} i e^{-\frac{\tau }{2}} \sqrt{3+e^{\tau }} & 0 \\
 0 & -\frac{1}{2} i e^{-\frac{\tau }{2}} \sqrt{3+e^{\tau }}
\end{array}
\right)\,.
\end{equation}

The matrices \eqref{DP1Stand}-\eqref{DP3Stand}  are the
$\hat{\sigma}_z=|0\rangle\langle0|-|1\rangle\langle1|$
representation of the standard Kraus operators in
eq.\eqref{RomLoFrancoDPKraus} while the matrix $\mathbb{E}_4'=2i
\mathbb{E}_4$ exhibits a unitary freedom for the Kraus operator
$\mathbb{E}_4$ in eq.\eqref{RomLoFrancoDPKraus}.

\subsection{Comparison of the generalized and the standard depolarizing channels}\label{Analysis}

The Kraus matrices eqs.\eqref{DP1}-\eqref{DP4} represent a
generalization of the standard DP channel
eq.\eqref{RomLoFrancoDPKraus} (i.e.
\eqref{DP1Stand}-\eqref{DP4Stand}). In this section we compare
these two kinds of the noisy channels by comparing their dynamical
effects regarding the Bloch-sphere volume, the von Neumann entropy
and  the trace distance. In order to give precise meaning of the
task, we first provide definitions of these standard
quantum-information quantities.

The Bloch sphere is a geometrical representation of the
single-qubit state space where every pure state  can be written as
\cite{N&Ch}

\begin{equation}\label{DensOperBloch}
   \hat{\rho}=\frac{1}{2}\left(\hat{I} +\vec{n} \cdot \hat{\vec{\sigma}}
   \right)
\end{equation}

\noindent where the vector $\vec{n}$ of the unit length, $\vert
\vec{n}\vert = 1$, corresponds to the points on the Bloch sphere
determined by the spherical coordinates $\vec{n}=(\sin v \cos u,
\; \sin v  \sin u, \; \cos v)$ and uniquely determine a pure
quantum state of a single qubit; $v\in[0,\pi]$, $u\in[0,2\pi]$.
The points inside the sphere correspond one-to-one to the mixed
states for every $\vert\vec{n}\vert < 1 $. The noisy channel
effects can be geometrically presented as the Bloch-sphere
deformation \cite{N&Ch}. Denoting the volume of the [deformed]
Bloch sphere $V(t)$ in an instant of time $t$, the relative speed
of the volume-change is defined as:

\begin{equation}\label{Volumevariation}
   \kappa(t)=\frac{1}{V_0}
\frac{dV(t)}{dt}
\end{equation}

\noindent where $V_0$ is the volume at $t=0$.

The von Neumann entropy of a quantum state $\hat\rho$ in an
instant $t$ of time

\begin{equation}\label{vonNeumannEntropy}
    S(t) = - \mathrm{tr}(\hat{\rho}(t) \ln\hat{\rho}(t)),
\end{equation}

\noindent increases for all kinds of the noisy channels
\cite{N&Ch}. Increase of the von Neumann entropy is an alternative
(non-geometrical) description of the Bloch-sphere deformation that
brings the information-theoretic description of the regarded
processes \cite{N&Ch} that cannot be uniquely deduced from
eq.\eqref{Volumevariation}.

The so-called trace distance

\begin{equation}\label{TraceDistance}
 T(\hat{\rho},\hat{\sigma}) := \frac{1}{2}||\hat{\rho} - \hat{\sigma}||_{1} =
 \frac{1}{2} \mathrm{tr}\left[\sqrt{(\hat{\rho}-\hat{\sigma})^\dagger (\hat{\rho}-\hat{\sigma})}
 \right]
\end{equation}

\noindent quantifies how much the quantum states $\hat\rho$ and
$\hat\sigma$ differ from each other \cite{N&Ch}. Hence comparison
of the regarded channels is possible with the use of
eq.\eqref{TraceDistance} by setting $\hat\rho(t)$ and
$\hat\sigma(t)$ as the states produced by the generalized and by
the standard DP channel, respectively.

For the generalized DP channel, i.e. from eqs.
(\ref{DP1}-\ref{DP4}) follows:
\begin{subequations} \label{DPEffects}
\begin{align}
\phi_{\tau}(\hat{I})&=\hat{I}\,, \label{DPEffectsA} \\
\phi_{\tau}(\hat{\sigma}_x)&=e^{-\tau}(\hat{\sigma}_x\cos\theta\tau+\hat{\sigma}_y\sin\theta\tau)\,, \label{DPEffectsB}\\
\phi_{\tau}(\hat{\sigma}_y)&=e^{-\tau}(\hat{\sigma}_y\cos\theta\tau-\hat{\sigma}_x\sin\theta\tau)\,, \label{DPEffectsC}\\
\phi_{\tau}(\hat{\sigma}_z)&=e^{\tau\Omega}\hat{\sigma}_z\,,
\label{DPEffectsD}
\end{align}
\end{subequations}
\noindent that is
\begin{equation}\label{DPDensOperBlochthetaOmegatau}
\begin{split}
  \phi_{\tau}(\hat{\rho})&=\frac{1}{2}[\hat{I}+e^{-\tau}\sin v\cos(u+\theta\tau)\hat{\sigma}_x+\\&
  +e^{-\tau}\sin v\sin(u+\theta\tau)\hat{\sigma}_y+e^{\tau\Omega}\cos
  v\hat{\sigma}_z]\,,
  \end{split}
\end{equation}
\noindent and equivalently:
\begin{equation}\label{DPDensOperBlochxyz}
\begin{split}
 \phi_{t}(\hat{\rho})&=\frac{1}{2}[\hat{I}+e^{-2(\mathbf{y}+\mathbf{z})t}\sin v
  \cos(u+2\mathbf{x}t)\hat{\sigma}_x+\\&
  +e^{-\tau}\sin v\sin(u+2\mathbf{x}t)\hat{\sigma}_y+e^{-4\mathbf{z}t}\cos
  v\hat{\sigma}_z]\,.
  \end{split}
\end{equation}
\noindent Expressions \eqref{DPDensOperBlochthetaOmegatau} and
\eqref{DPDensOperBlochxyz} are solutions of the master equation
eq.\eqref{GeneralizedMJ}. It is well known that for Pauli channel
Pauli matrices are eigenvectors of the channel, but, as can be
seen from eqs.\eqref{DPEffects}, it is not the case for
generalized depolarizing (GDP) channel.

From eq.\eqref{DPDensOperBlochxyz} we analytically obtain the time
dependence

\begin{equation}\label{DPBlochvolumechange}
   V(\tau)=\frac{4\pi}{3}e^{\tau(\Omega-2)},
\end{equation}

\noindent and the relative change

\begin{equation}\label{DPBlochspeedchange}
   \kappa(t)=-4(2\mathbf{z}+\mathbf{y})e^{-4
(2\mathbf{z}+\mathbf{y})t}
\end{equation}

\noindent of the deformed-Bloch-sphere volume. The von Neumann
entropy and the trace distance are numerically calculated for the
initial state defined by $u=0=v$, eq.\eqref{DensOperBloch}.
Comparison with the standard DP channel follows from placing
$\mathbf{x}=0$ and $\mathbf{y}=\mathbf{z}$ in
eq.\eqref{DPDensOperBlochxyz}.

Fig.\ref{FigVDPRelVDP}(left) exhibits  faster deformation of the
Bloch sphere for the standard than for the generalized DP channel,
while Fig.\ref{FigVDPRelVDP}(right) depicts the volume relative
change, eq.\eqref{DPBlochspeedchange}.

From Fig.\ref{FigChangeTrDistDP}(top left) we can see that, for
long times, the trace distance  tends to be  equal for both
channels.

Time dependence of the entropy and the rate of its change are
depicted on Fig.\ref{FigEntroDP}.

\bigskip

\begin{figure}
\centering $\begin{array}{cc}
 \includegraphics[width=0.35\textwidth]{FigDP3D}&
  \includegraphics[width=0.4\textwidth]{FigRelVDP}
  \end{array}$
  \caption{(left) The depolarizing channel for: $T=50$,
$\alpha=0.02$, $\omega_0=1$ and $\omega_c=15$. The big sphere is
for $t=0$, the small sphere is obtained for the standard DP
channel ($\mathbf{x}=0$ and $\mathbf{y}=\mathbf{z})$, while the
ellipsoid concerns the generalized DP channel, both for  $t=0.05$.
(Right) The relative change of the Bloch sphere volume. The dashed
line is for the standard DP channel and the thick one for the
generalized DP channel, respectively.} \label{FigVDPRelVDP}
\end{figure}

\begin{figure*}
\centering
 $\begin{array}{rl}
    \includegraphics[width=0.4\textwidth]{FigChangeTrDistDP} &
    \includegraphics[width=0.4\textwidth]{FigChangeTrDistDP1}\\
    \multicolumn{2}{c}{\includegraphics[width=0.4\textwidth]{FigChangeTrDistDP2}}
\end{array}$
\caption{(Top left) The temporal change of the trace distance;
(Top right) The rate of change of the trace distance; (Bottom) The
trace distance for the states obtained for the two channels. The
dashed lines are for the standard DP channel, while the thick
lines are for the generalized DP channel. The parameters: $T=50$,
$\alpha=0.02$, $\omega_0=1$ and
$\omega_c=15$.}\label{FigChangeTrDistDP}
\end{figure*}

\begin{figure}
\centering $\begin{array}{cc}
  \includegraphics[width=0.4\textwidth]{FigEntroDP}&
   \includegraphics[width=0.4\textwidth]{FigChangeEntroDP}
   \end{array}$
\caption{(Left) The entropy  change (both lines start from zero,
which is not shown here)(Right) The rate of the entropy change.
The dashed lines are for the standard DP channel, while the thick
lines are for the generalized DP channel. The parameters: $T=50$,
$\alpha=0.02$, $\omega_0=1$ and $\omega_c=15$.} \label{FigEntroDP}
\end{figure}

\begin{figure}
\centering
  \includegraphics[width=0.35\textwidth]{FigDP3DComparison}

\caption{The GDP process ellipsoids for different parameters
values, starting from the largest one, respectively: $T=50,
\alpha=0.005, \omega_c=15$; $T=100, \alpha=0.005, \omega_c=50$;
$T=50, \alpha=0.02, \omega_c=15$; $T=50, \alpha=0.02,
\omega_c=50$; $T=100, \alpha=0.02, \omega_c=15$. The parameters
$\omega_0=1$ and $t=0.05$ are kept fixed. The ellipsoids
pertaining to ($T=50, \alpha=0.02, \omega_c=15$) and  ($T=50,
\alpha=0.02, \omega_c=50$) are mutually almost indistinguishable.}
\label{FigDP3DComparison}
\end{figure}

\begin{figure*}
\centering
 $\begin{array}{rl}
    \includegraphics[width=0.4\textwidth]{FigRelVDPComparison} &
    \includegraphics[width=0.4\textwidth]{FigChangeTrDistDPComparison}\\
    \multicolumn{2}{c}{\includegraphics[width=0.4\textwidth]{FigEntroDPComparison}}
\end{array}$
\caption{(Top left) The relative change of the Bloch sphere
volume. (Top right) The trace-distance change. (Bottom) The
entropy change. For all figures the meaning of the lines is the
same: thinner dashed line  ($T=50$, $\alpha=0.005$ ,
$\omega_c=15$), dot-dashed line  ($T=100$, $\alpha=0.005$ i
$\omega_c=50$), thinner  solid  line ($T=50$, $\alpha=0.02$ i
$\omega_c=15$), thicker dashed line ($T=50$, $\alpha=0.02$ i
$\omega_c=50$), thicker solid line ($T=100$, $\alpha=0.02$ i
$\omega_c=15$).  The parameter $\omega_0=1$ is kept fixed.
}\label{FigComparison}
\end{figure*}

\bigskip

Expectably, operation of the GDP channel is sensitive to the
values of the parameters $T$, $\alpha$ and $\omega_c$. Qualitative
similarity of the GDP-channel operations for certain values of the
parameters is depicted in Figs. \ref{FigDP3DComparison} and
\ref{FigComparison}.

From Figs. \ref{FigDP3DComparison} and \ref{FigComparison}
increase in temperature $T$ and the coupling constant $\alpha$
gives rise to the more efficient action of the channel. To this
end, relatively the most relevant contribution is due to the
strength of the interaction-which, of course, cannot be
arbitrarily large for the weak-coupling Markovian processes such
as the GDP channel. The contribution of the cutoff frequency comes
essentially from the better satisfied formal  condition for
Markovianity, i.e. form the inequality $\omega_0/\omega_c \ll 1$,
and hence is of the secondary importance.

\section{Discussion}\label{Discussion}

Microscopic derivation of the Kraus operators is poorly addressed
in the literature. Nevertheless, microscopically derived Kraus
operators provide the physically richer and mathematically the
more accurate description of the realistic physical processes.
Hence we can expect that the parameters control can in principle
give rise to the open system's partial control. Bearing in mind
that the error-correction protocols \cite{N&Ch} are adapted to the
concrete forms of the Kraus operators, the knowledge of the exact,
generalized Kraus operators eqs.\eqref{DP1}-\eqref{DP4} can help
in devising the more reliable and more accurate error-correction
protocols
\cite{Terhal,Raussendorf,ShorPreskill,KrausGisinRen,BBPSSW}. To
this end, the results will be presented elsewhere. Here we just
emphasize that our results regard the short-time intervals of
interest for the realistic quantum information/computation
protocols/algorhithms.

Practical implementation of the quantum information-theoretic
protocols \cite{N&Ch} {\it assumes} that the quantum ``hardware''
operates as it is theoretically described. This non-trivial
assumption is here theoretically addressed in the context of the
one-qubit quantum noise channels. The results regarding the
amplitude damping and the phase damping quantum channels are
presented in \cite{KJS}.

On the other hand, to the best of our knowledge, the generalized
Kraus operators for the depolarizing channel, eqs.
\eqref{DP1}-\eqref{DP4}, are here derived for the first time
exhibiting a subtle yet physically important dissent from the
standard and widely used ones, eq.\eqref{RomLoFrancoDPKraus}. On
this basis we can recognize the following research lines of
interest.

From Figs. \ref{FigDP3DComparison} and \ref{FigComparison} we can
learn that higher temperature and strong interaction give rise to
the more efficient and faster action of the generalized
depolarizing channel that, at least in principle, can be
experimentally tested. On the other hand, transition to the
Schr\"{o}dinger picture:
\begin{equation}\label{SchredKraus}
   \hat{\mathbb{E}}_i^S (t)= \hat{U}(t)\hat{\mathbb{E}}_i (t),
\end{equation}
where $\hat{U}(t)= e^{-\frac{i}{\hbar}\hat{H}_qt}$ and $\hat{H}_q$
is the qubit's self-Hamiltonian, provides a study of the
external-fields influence on the single-qubit's dynamics.

Temporal change of the Bloch sphere, Section \ref{Analysis},
reveals that the standard channel is  more detrimental than the
here introduced generalized channel for the short time intervals.

However, for long times we observe the full match of the standard
and the here derived generalized DP channel. This can be seen in
the asymptotic limit ($\tau\to\infty$) as follows. On the one
hand, the state in eq.\eqref{DPDensOperBlochthetaOmegatau} clearly
satisfies $\lim_{\tau\to\infty} \phi_{\tau}(\hat{\rho}) =
\hat{I}/2$, which is the defining feature of the standard DP
channel, eq.\eqref{RomLoFrancoDPKraus}. On the other hand, in the
asymptotic limit, all the Kraus operators,
eqs.\eqref{DP1}-\eqref{DP4}, \eqref{DP1Stand}-\eqref{DP4Stand} and
eq.\eqref{RomLoFrancoDPKraus}, give the unique set of Kraus
operators:

\begin{equation}\label{DPasym1}
\mathbb{E}_1=\frac{1}{2}\left(
\begin{array}{cc}
 0 & - i  \\
  i  & 0
\end{array}
\right),
\end{equation}

\begin{equation}\label{DPasym2}
\mathbb{E}_2=\frac{1}{2}\left(
\begin{array}{cc}
 0 &1 \\
 1 & 0
\end{array}
\right),
\end{equation}

\begin{equation}\label{DPasym3}
\mathbb{E}_3=\frac{i}{2}\left(
\begin{array}{cc}
1 & 0 \\
 0 & -1
\end{array}
\right),
\end{equation}

\begin{equation}\label{DPasym4}
\mathbb{E}_4= \frac{i}{2}\left(
\begin{array}{cc}
1 & 0 \\
 0 & 1
\end{array}
\right),
\end{equation}

\noindent that obviously satisfy the completeness relation.

With the use of eqs.\eqref{A12}-\eqref{A14}, the condition $2
\gamma_{zz}(0,T)=\gamma_{xx}(\omega_0,T)+\gamma_{yy}(\omega_0,T)$
(Section \ref{KrausDP}) for reducing the GDP to the DP channel,
numerically gives the unique (physically relevant, i.e.
non-negative)  value $\omega_{\circ} = 0$ for every temperature
$T$. Then neglecting the Lamb-shift term makes the
interaction-picture equation \eqref{RomLoFrancoDPmaster} valid
also in the Schr\" odinger picture.

As an application, we consider entanglement dynamics for a pair of
qubits regarding the following typical scenario in quantum
information-processing. Alice and Bob share a pair of the
initially entangled qubits (denoted $1$ and $2$) and locally and
independently perform operations on their respective qubits. The
qubits are sufficiently remote from each other and hence do not
mutually interact while being subjected to the mutually
independent, local GDP channels. The task is to investigate
dynamics of entanglement for the pair $1+2$.

Being spatially remote, the qubits are dynamically independent.
Therefore the Kraus operators for the combined $1+2$ system are
the tensor-product Kraus operators for the individual qubits.
Transition to the Schr\" odinger picture is fulfilled by the
unitary operations, $U_i(t)=\exp(-\imath t H_{i}/\hbar)$ for the
qubits $i=1,2$, independently of each other. Given the
combined-system's state (in the Schr\" odinger picture):
\begin{equation}
\rho(t) = \sum_{i,j} U_1(t)K_{1i}(t)\otimes U_{2}(t) K_{2j}(t)
\vert \Psi(0)\rangle\langle\Psi(0)\vert  K_{1i}^{\dag}
U_1^{\dag}(t)\otimes K_{2j}^{\dag}(t) U_2^{\dag}(t),
\end{equation}

\noindent we  calculate the so-called concurrence for a
bipartite mixed state $\rho(t)$ that is defined as \cite{Wooters}:
\begin{equation}
C(\rho(t)) = \max \{0, \Lambda(t)\},
\end{equation}

\noindent where

\begin{equation}
\Lambda(t) = \sqrt{\lambda_1(t) }- \sqrt{\lambda_2(t) }- \sqrt{\lambda_3(t) }
- \sqrt{\lambda_4(t) },
\end{equation}

\noindent with the eigenvalues $\lambda_1
> \lambda_2 > \lambda_3 > \lambda_4$ of

\begin{equation}\label{VutersRho}
\rho(t) (\sigma_{1y}\otimes\sigma_{2y}) \rho^{\ast}(t)
(\sigma_{1y}\otimes\sigma_{2y}).
\end{equation}

\noindent In \eqref{VutersRho}, $\sigma_{iy}$ is the $y$-Pauli-matrix for the
$i$th qubit, while $\ast$ denotes the standard complex
conjugation. For the initial entangled state we choose
$\vert\Psi\rangle = [\vert 0\rangle_1\vert 0\rangle_2 + \vert 1\rangle_1 \vert 1\rangle_2
]/\sqrt{2}$, where we omit the symbol of the tensor-product and $\vert i\rangle, i=0,1$ represent the eigenbasis of the $\sigma_z$ Pauli matrix.

\begin{figure}
\centering
 \includegraphics[width=0.4\textwidth]{CcDPandGDP}
  \caption{Concurrence dynamics for the standard DP channel (solid line) and the GDP channel (dotted line). The parameters: $\omega_{1}=0.1$, $\omega_{2}=0.2$, $T=10$, $\alpha=0.02$.} \label{CcDPandGDP}
\end{figure}

Bearing in mind the definition of the so-called entanglement-of-formation \cite{Wooters}:
\begin{equation}
\mathcal{E}(\rho) = H\left({1+\sqrt{1 + C^2(t)}\over 2}\right)
\end{equation}

\noindent where $H(x) = -x \log_2 x - (1-x) \log_2(1-x)$, from Fig. \ref{CcDPandGDP}, we conclude about the occurrence of the ``entanglement sudden death" \cite{Yu Eberly} also for the GDP channel. That is, for the concurrence $C(t)=0$ for some $t$, the entanglement of formation $\mathcal{E}=0$ in the same instant of time--the loss of the initial entanglement in the instant of time $t$. Fig. \ref{CcDPandGDP} justifies the above finding that the GDP channel is less deteriorating (slower occurrence of the entanglement sudden death) than the standard DP channel.

\section{Conclusion}\label{Conclusion}

Microscopic derivation of the Kraus operators is a new emerging
line of research of foundational as well as of interest for
application in modern open quantum systems theory. The
microscopically derived Kraus operators for the generalized
depolarizing channel are here presented for the first time while
carrying the clear signatures of the microscopic-models details.
Implications for the possible open-system's control and for an
elaboration of the known error-correction protocols will be
presented elsewhere.

redpajama_set_name: RedPajamaArXiv


text:
Out of Eden
Geoffrey Barraclough
by Anthony Eden
Houghton Mifflin, 716 pp., $8.50
Considering the prominent position he held from 1940 to 1945, Anthony Eden's war memoirs are curiously unexciting. Of course, so much has been written on Anglo-American-Soviet relations and the "Big Three" that it would have been enrealistic to expect startling disclosures at this late stage. Even so, much in this volume of 700 pages is small, and sometimes rather flat, beer. Its interest lies mainly in what—unconsciously rather than deliberately—it allows us to see of the undercurrents that muddied the wartime waters. It tells us much of Eden himself, and his limitations as a person and as a Foreign Secretary; it tells us a great deal about Churchill and a little about Stalin and Roosevelt; and it casts an oblique light on what for most people is the outstanding feature of the diplomacy of the period—the genesis of the "cold war."
"The Foreign Office," Churchill once exclaimed in a moment of exasperation with Eden, is "always on small points, like chrome and ships in Sweden." This chance outburst provides a key to the essential difference between the two men. Churchill, in common with Roosevelt and (be it said) with Stalin also, had a vision of a new international order which would deliver the world from the self-destroying cycle of crisis and war—a vision based, like Wilson's in 1917, not on a "balance of power" but on "a community of power." Eden was "always on small points." His conception of diplomacy was entirely conventional, a matter of compensations and national interest narrowly conceived. The Western powers, he told Churchill, should refuse "concessions about what Stalin wants except in return for concessions in respect of what we want." There may have been nothing wrong with this argument in principle; but it was a curious application of it for Eden to tell Molotov that Britain wanted "more voice in Bulgaria and Hungary than we had accepted in Rumania." In 1944 Britain had no conceivable interest in Bulgaria or Hungary and could have done nothing about it, if it had. To Molotov Eden's demand must have sounded like an echo from the days of Disraeli, Bismarck, and Gorchakov, a threat of a return to the conditions of 1878. No wonder that, immediately afterwards, Churchill complained that Eden "had dispelled the good atmosphere he had created."
This was, indeed, the crux. There were two ways, and only two ways, of conducting inter-allied negotiations. The one was to create an atmosphere of good will, and this entailed recognizing Russia's preponderant role in eastern Europe; the other was to fight a step by step engagement, contesting every yard of territory. It was the second that Eden represented. "I do not think," he advised Churchill on the eve of the Yalta Conference, "we should indicate straight away that we regard any proposals put forward by the Russians as well founded." The ordinary reader, unversed in the ways of diplomacy, will regard this extraordinary suggestion with astonishment. Why, if the Russian proposals were well founded, miss the opportunity to gain good will by accepting them? But the more significant fact is that this was a game at which two could play. If Eden made it his business to "secure the position of the anti-communist Serbs" in Yugoslavia, was it so surprising that Molotov made it his business to secure the position of the anti-western Poles in Poland? There is little profit in the game of the pot calling the kettle black; but it is hard to think of any surer recipe than Eden's for rousing suspicion and ill will on the Russian side.
It is the orthodox view today that the cause of post-war troubles was the "softness" of Churchill and Roosevelt towards Stalin at Yalta and earlier. Eden, characteristically, endorses the verdict. He himself, he claims, was quick to foresee Soviet ambitions, and the implication is that if more attention had been paid to his advice, they might have been successfully parried. But Churchill fell "under Stalin's spell," and Roosevelt refused pointblank to "gang up" with the British against Russia. The truth, of course, is not so simple. Churchill and Roosevelt, with a wider vision than Eden, planned to work with, not against Russia; and we should not too easily assume that this policy was foredoomed to failure. It is easy to accuse the Russians of bad faith; but even as late as Yalta, on Eden's evidence, they were "relaxed" and "friendly," and Stalin scrupulously observed his agreement with Churchill when the British intervened in Greece. If there was a sharp deterioration in the relations between Russia and the West between the Yalta and the Potsdam Conferences, it is by no means clear that one main reason may not have been the intense suspicion of Russian designs which Eden evinced.
Certainly there is no sign, in anything written in this volume, that Eden was aware of the likely effects on the Russians of the policies he advocated. Nowhere is this clearer than in regard to Poland. As early as 1943 the Russians had made it plain that they would accept nothing less than "the Curzon line, with minor adjustments." Was it good policy, considering Britain's inability to take positive action, to resist this demand, or to treat it (as Eden did) as negotiable against a quid pro quo? It is true, of course, that its agreements with the Polish government in exile made the position of the British government exceedingly uncomfortable; but it is also true that Eden's moralistic emphasis on his obligations to the London Poles looked in Moscow like a cloak for an attempt to reconstruct the pre-war cordon sanitaire against the Soviet Union. And, to judge from a number of remarks scattered through this book, this suspicion was not entirely false. In 1944 and 1945, Eden admits, the British set out "to shape their policies to counter Soviet designs" in the Balkans; if they were unable to do as much in Poland, it was only because "the result was beyond our reach to decide."
It would be wrong, of course, to draw a sharp antithesis between Churchill and Eden, any more than between Roosevelt and Stettinius or between Stalin and Molotov. What distinguished them was methods rather than aims. Nevertheless the fact remains that Roosevelt and Churchill saw the issues, particularly the Polish question, in a different light. It was, said Roosevelt, for Britain, the United States, and Russia to decide "what was a just solution, and Poland would have to accept." It is easy to condemn Roosevelt's judgment as harsh and unfeeling; but the outcome for the Poles might have been a good deal happier if it had been unflinchingly applied. The trouble with Eden's policy was the discrepancy between means and aims. "To go on trying" to squeeze out concessions for the London Poles looked well in the record; but it created a tension from which Poland was the first to suffer. "It was important," Eden said, for "Mikolajczyk to be able to show his people that he had done everything in his power." Roosevelt and Churchill, looking further, saw that there was something even more important for the future peace of the world; and that was for Britain and the United States to convince the Russians that they had done everything in their power to ensure that Russia's legitimate demand for security in the west would be acknowledged and respected.
It was old Cordell Hull who put his finger on Eden's main characteristic as Foreign Secretary. "My, Mr. Eden," he said, "I had no idea you were such a politician." If there is a difference between a politician and a statesman, this observation was both shrewd and accurate. Eden's conception of diplomacy turned on small strategies and maneuvers. This, without doubt, as well as the strain of wartime events, accounts for the frequent outbursts of irritation he aroused in Churchill. "Sulky," "bad-tempered," and "febrile" were some of the epithets Churchill used of Eden; and he might have added vain and pompous as well. Surrounded by nonentities like Kingsley Wood and Sir John Anderson, Churchill perforce looked on Eden as his "only really intimate friend among his colleagues," his "chief lieutenant," and his "right arm." But temperamentally the two men were poles apart. Above all else, Eden was essentially a backward looking man, who pinned his faith on "reconstruction within the framework of that traditional civilization which is our common heritage," and had little inkling how fundamentally the world had changed since 1939. The impression left by this book is of a man who failed to measure up to the magnitude of events, as he was to do again in 1956. In the new world, which did not respond to the conventional rules of diplomacy, he was out of his depth; lions were prowling in the garden Eden, and the only remedy he could think of was to put salt upon their tails.
More by Geoffrey Barraclough
Date Palms and Elephant Tusks
Elias Defended
Return of the Natives
Geoffrey Barraclough (1908–1984) was a British historian.
Simon Raven
Two for the Money
Larry McMurtry
The Star Attraction
Eric L. McKitrick
Reconstructing Andrew Johnson
Peter Gay
Napoleon Against Himself
Martin Malia
Catherine Was Great


text:
Romanische, französische Literaturen
Guinevere - A Medieval Puzzle
Images of Arthur's Queen in the Medieval Literature of England and France
lieferbar (3-5 Tage)
2005. Buch. X, 441 S. Softcover
Carl Winter. ISBN 978-3-8253-5012-3
Format (B x L): 16,5 x 24,5 cm
Das Werk ist Teil der Reihe: Anglistische Forschungen; 345
Despite her prominence, Guinevere is still largely reduced to her role in the Arthurian Triangle. Proceeding from a review of scholarship to a new look at all extant material, this book challenges the traditional concept, uncovering and explaining the extreme diversity of concepts and currents. It testifies to mysogyny, mariolatry, and pagan magic, to archetypes, stereotypes, and double standards, but primarily to an intense interest in a woman's fate. A mute consort acquires a voice, a mind, and the power to act. Encompassing all of a woman's scope and much of a man's, Guinevere is a mirror of past legend and contemporary reality, and a medieval vision of female emancipation and the beginnings of individuality.


April 16, 2019 Updated: April 16, 2019 3:01 p.m.
GLENDALE, Ariz. (AP) — A suspect in a fatal road rage shooting in Glendale has turned himself in to police.
Police say 22-year-old Daddy Alexander Felix of Phoenix surrendered Monday.
He's accused of killing 18-year-old Phoenix resident Alexander Pettygrove on Saturday.
Police say a passenger in Pettygrove's vehicle — 18-year-old Sam Luviano-Gonzalez — was shot in the leg and is recovering.
Police found a vehicle that had crashed into a wall outside of a gas station with two 18-year-old victims inside and both appeared to have been shot.
They identified the driver as Pettygrove, who was pronounced dead at a hospital.
Witnesses told police there was an altercation involving occupants of the two cars before the shooting.
They say an SUV pulled up beside Pettygrove's sedan at an intersection and angry words were exchanged.


text:
Spain's News
Spanish News in English
Unicef regrets the "improvisation" of the State to take in migrant children
Mizar rescues a group of migrants of sub-Saharan origin in Fuerteventura.
It is not the first time that they have warned of this drama, but they do not understand how, year after year, Spain is unable to provide a dignified welcome to the kids and adolescents who leave their families in Africa and risk their lives to reach Europe. Unicef has presented this Wednesday an investigation into the situation in which minors who arrived in the Canary Islands during 2020 and 2021 are living. The conclusion points out that 80% of them are living in macro-centers where their psychological needs are not taken into account, nor do they take them to school and that, in some cases, they suffer violence. "The receiving communities are saturated, we need a state redistribution and coordination mechanism to be able to guarantee the rights of these children," they have insisted from the entity.
Over the course of several months, various members of Unicef in Spain they have visited about thirty protection centers where the children who risked their lives on the high seas live. Since 2019, more than 3,800 children have reached the Canary Islands by boat. It is the deadliest route and it is estimated that more than a thousand people died in 2020. What happened to those who survived the route? Some 2,500 still live in centers for minors on the Islands, only 4% were transferred to other autonomous communities and less than 3% have obtained their residence permit, a document to which they are entitled and which allows them to have rights in Spain.
"The diagnosis is clear. Canary Islands She was not prepared. In 2020 arrivals grew 650% and in 2021 they have once again exceeded 150% ", explained Pablo Ceriani, author of the report and member of the UN Committee on the protection of migrants.Humanitarian reception has been insufficient, inadequate and ineffective. The child protection system was saturated and children have been unprotected, "he explained.
In fact, according to the author, two sides of the coin have been seen. 20% of the centers open for this emergency in the Canary Islands do things well. "They are small centers, with less than 20 places, with individualized aids, therapeutic and multidisciplinary teams where children can go to school and have papers: in short, they achieve integration", he says. However, in 80% of places, human rights are not guaranteed. No school, no specialized activities, no sufficiently trained or specialized workers. "They cannot respond adequately to the reception and reception of children," the author insisted.
"We have found cases of children who had an enormous anguish because they still suffered for their friends who had died in the boat. Children who hurt themselves or who spend the day crying. In these resources they have nothing to do, they are locked up and their mental and emotional health is crumbling. And they all come here to work and train, "described Sara Collantes, a UNICEF migration specialist in Spain and a member of the research team.
A situation similar to that which occurs in Ceuta, where after the arrival of 1,500 children during the month of June, there are 1,128 in centers for minors and about 500 who live on the streets. "It is urgent to be able to meet their needs, the care that is being provided to them is not necessary because the city cannot cope," Carmen Molina, director of awareness and childhood policies of the organization, has told from the autonomous city.
That is why, from the entity, they insist that it is necessary for the Central Government to invest resources but, above all, to redistribute minors around the province. A decision that, today, is voluntary and born of the solidarity of the autonomous governments, but which is negligible compared to those that are still there. They trust that the Government will approve as soon as possible the national strategy for the reception of migrant children, where several social entities demand a state pact on the distribution of minors. "What cannot be is that we continue improvising: Spain is and will be the door to the border with Europe, and children will not stop arriving crossing the sea. They must be welcomed, and welcomed well," said Gustavo Suárez Pertierra, president of the 'UNICEF Spain.
Tags: embrace, improvisation, Kids, the Mint, Unicef
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text:
Romaine lettuce to blame for E. coli outbreak in 11 states
About 5 to 10% of people diagnosed with this illness develop hemolytic uremic syndrome, or HUS.
Although none of the confirmed E. coli cases have yet been linked to Fresh Foods, the company is concerned that its romaine supplier may have been involved in the outbreak.
The CDC said Friday that 35 people across 11 states have become ill from chopped romaine lettuce from Yuma.
The CDC reports that early information points to chopped romaine lettuce from the Yuma, Arizona region as the potential culprit.
Epidemiologic evidence collected to date indicates that chopped romaine lettuce is the likely source of this outbreak. "This takes an average of two to three weeks".
The federal response to this outbreak stands in contrast to the E. coli outbreak this past winter, when investigators at the CDC and the Food and Drug Administration did not make definitive consumer recommendations. The restaurants reported using bagged, chopped romaine lettuce to make salads.
More news: The Walking Dead: Maggie justified turning to dark side
"Consumers anywhere in the United States who have store-bought chopped romaine lettuce at home, including salads and salad mixes containing chopped romaine lettuce, should not eat it and should throw it away, even if some of it was eaten and no one has gotten sick", the CDC said. Cases of illness showing E. coli symptoms have been reported in Connecticut, Idaho, Illinois, Michigan, Missouri, New Jersey, New York, Ohio, Pennsylvania, Virginia, and Washington. "Anyone experiencing symptoms of this illness should see a healthcare provider".
In addition, the agency recommends asking grocery stores and restaurants to confirm their chopped romaine is not from Yuma. Sixty-nine percent of ill people are female.
Any lettuce that came from that area (and there's a LOT!) has been pulled from shelves, people are urged to dispose of any recently purchased from the store that originated in that region.
The symptoms of Shiga toxin-producing E. coli infections vary, but usually include severe and painful stomach cramps and bloody diarrhea. The damaged red blood cells clog the filtering system of the kidney, which can lead to life-threatening kidney failure.
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Head coach Quin Snyder didn't seem overly concerned as he said: "Everything that we know right now, he's fine ". On Wednesday, the former Louisville standout was at it again before Utah's game against Portland.
Higher interest rates, loans lift Bank of America profits
It increased, as 66 investors sold BAC shares while 561 reduced holdings. 162 funds opened positions while 585 raised stakes. Teacher Retirement System of Texas boosted its holdings in shares of Bank of America by 1.9% during the 3rd quarter.
John Stamos and wife welcome baby boy
The 32-year-old McHugh posted the first public photo of her baby bump on Instagram in January. Actor John Stamos has finally made his dreams of becoming a father come true.
Mourinho Blames Man United's Players For West Brom Defeat
Not for the first time this season at Old Trafford United started sluggishly and could have been a goal down after 12 minutes. In my opinion, it affected for sure some of the people that were too happy just with that", United manager explained.
Ethereum Gold Trading Up 10.3% This Week (CRYPTO:ETG)
One COMSA [ETH] token can now be bought for approximately $0.54 or 0.00006834 BTC on popular exchanges including YoBit and Zaif. Investors can then use their newly-acquired Bitcoin or Ethereum to buy Loopring using one of the aforementioned exchanges.
Tesla and SpaceX: Can Elon Musk Run Both?
Guggenheim reaffirmed a "buy" rating and issued a $300.00 price objective on shares of Tesla in a report on Monday, February 27th. Institutions purchase large blocks of a company's outstanding shares and can exert considerable influence upon its management.
Airstrike in Syria was the only alternative says British PM May
The United States responded with 58 missile strikes that destroyed 20% of the Syrian airforce. He reiterated that Canada condemns the use of chemical weapons in Ghouta.
Telegram messenger app blocked in Russian Federation
The Roskomnadzor communications watchdog on Monday ordered internet and mobile phone operators to block access to Telegram within 24 hours.
Russian rouble and stocks slide on new sanctions fears, lower oil
McKenzie, director of the Joint Staff at the Pentagon, said the allied airstrikes "took out the heart" of Assad's chemical weapons arsenal.
Inmates Dead in SC After Massive Prison Brawl
Lee County Fire and Rescue a sent crews to assist in the "mass causality incident", the agency said in a Facebook post. COLUMBIA, S.C. (AP) — State police have been called to one of South Carolina's most unsafe prisons.
At least seven dead after 'mass casualty incident' at SC prison
Corrections officials did not immediately say what led to the incident and why it took hours to stop the series of altercations. Emergency services from several SC counties responded to the "mass casualty incident", Lee County Fire wrote in a tweet .
Markets recover early loss; Sensex closes 100 points above
Shares of one of India's biggest software firms, Infosys fell 3.86% to Rs 1,123.90 on less-than-expected FY19 revenue guidance. Among other Asian markets, Hong Kong's Hang Seng shed 1.44%, while Shanghai Composite Index was down 1.05% in early trade.
Daryl Worley Is Released By The Eagles After Being Arrested
John Oliver's HBO Show Bought Russell Crowe's 'Cinderella Man' Jockstrap for $7000
NXP Semiconductors NV (NXPI) Received Consensus Rating of "Hold" by Brokerages
Baaghi 2 enters the Rs 150 crore club
United States pastor goes on trial in Turkey after arrest in coup crackdown
18:44Three Policemen Injured in Clashes With Protesters in Yerevan
Samsung smartphone for students to avoid exam distractions
Broncos cut RB Anderson
Shenmue 1 and 2 are heading to PC and consoles this year
Shire sells oncology unit to Servier for $2.4bn


Instead of a Gift, How About a Family Vacation?
January's birthstones, garnet and rose quartz, are known for their spectacular colors, restorative properties and symbolism of friendship and trust. With abundant natural and synthetic supplies available, it's no wonder these birthday gemstones are popular choices for affordable statement pieces, carvings and pendants.
Garnets have been traded as gemstones since the Bronze Age, and can be found in a variety of colors; however, red is the most common.
The word garnet is ultimately derived from the Latin garanatus, or "seedlike," referring to the deep red arils of the pomegranate. Large deposits in central Europe, India and Australia have made garnet a popular choice for religious figures, royalty and nobility across centuries and cultures.
Garnet is one of the twelve stones in the Breastplate of Aaron, and has been venerated by many cultures for its ability to protect the wearer from battle wounds. Legends of garnets protecting travelers even extend to dreams: place a garnet under your pillow to stave off nightmares.
There are many varieties of quartz – titanium, iron and manganese deposits lend rose quartz its pinkish color. Rose quartz beads have been in use since the days of the Mesopotamians. Ancient Egyptians believed rose quartz crystals to promote healing, prevent aging and attract love.
Both garnet and rose quartz were used by the Romans to create signet rings. A wax seal stamped with a particular signet could be used to transfer property and execute important documents, similar to modern-day handwritten signatures.
Not only are garnet and rose quartz great choices for jewelry: they have important industrial uses, too.
Garnet sand is a common abrasive due to the hardness of the stone. Garnet sandpaper is favored by woodworkers for finishing raw surfaces. When mixed with a high-pressure water jet, it's strong enough to cut through steel. Garnet sand can also be used to filter water.
Quartz is a common mineral found all over the world in sand and most types of rock; you're more likely to find it in your backyard than in a bag of Scrabble letters. Quartz is one of the main ingredients in manufactured glass. Quartz is also piezoelectric, which means that if you pass electricity through it, it vibrates at a specific frequency. For this reason, synthetic quartz crystals are widely used in powering clocks and electronics.
Both stones are popular choices for jewelry makers, who love the durability and abundance of these easy-to-care-for gems.
The juxtaposition of naturally-cut Andean garnets and a polished long-link chain make this a great choice for modern and bohemian wardrobes alike. Adriana de Gadea's designs are inspired by the play of forms and colors and made using local materials. Best of all, your purchase of this beautiful piece can provide 140 sachets of oral rehydration salts, which help children combat dehydration.
With a focus on the harmonization of body and mind, each of Eveli Przepiorka's rings features a unique pink quartz in its natural state, accentuated by a single winding piece of silver that wraps itself around the stone.
Neeru Goel, one of the UNICEF Market's most popular artists, designed these .925 sterling silver hook earrings with six garnets accentuating each center quartz stone. For those in search of a set for special occasions, Neeru's collection also includes a matching pendant necklace. Proceeds from this purchase can provide polio vaccines for 51 children.
If you're looking for a functional, timeless and elegant gift, consider this ballpoint pen crafted by Balinese artist Dewi Putera. The intricate design features an oval pattern and an inlaid faceted garnet on the screw-on cap. One replaceable Parker ink cartridge is included.
This adjustable bracelet is made from braided poly cord with a prominent round rose quartz. Samart Takham's design is simple enough for every-day wear and can even double as an anklet. Best of all, your purchase can provide 12 doses of measles vaccine.
Click here to learn how to sponsor a child with UNICEF USA. And learn more about which charities to donate to today.
February Gemstones: Amethyst and Black Onyx Birthstone Gifts Fit For Royalty!
Next articleFebruary Gemstones: Amethyst and Black Onyx Birthstone Gifts Fit For Royalty!
Finding a Special Gift for Dad for the Holidays.


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