I came across these today and LOVED watching them!
The first video is Donald Trump on Late Night with David Letterman from 1987.
I've seen a lot of Trump interviews on Letterman, but I'd never seen this one before! Maybe it will be new to you as well.
I think it's probably one of his first times on Letterman.
Watch how calm and cool Trump is compared to Letterman trying to poke and prod.
And I say that as a Letterman fan! I actually have always thought Dave was hilarious, it's a real shame to see how political he got in his later years and how far-left he is now. Too bad.
But Trump handled him with aplumb!
Donald Trump on late night in 1987.
And then this one is Donald Trump on Jay Leno from Dec 7, 1999.
I also had never seen this interview.
The most amazing thing in this one is Jay Leno introduced him as the next President of the United States....in 1999!


text:
Metromix's Top 10 D.C. Concerts of 2011
It's been another memorable year of music in D.C., Maryland and Virginia—filled with giant stadium and arena rock shows, memorable R&B and pop performances, and some of the biggest hip-hop shows ever to take stage. Adding to the excitement for 2011 was a brand-new, state-of-the-art music venue in Silver Spring that will surely bring top-notch acts to the area for decades to come. Metromix's intrepid concert photographer Alfredo Flores offers his take on 2011's ten best concerts based on visual spectacle, stage presence, artist rapport with audience, and, of course, the music. This year's edition of his top picks contains a couple of firsts—two acts making multiple stops in the D.C.-area in the same calendar year, as well as one music gathering so impressive that two Top 10 entries came out of that show. Click through the gallery to see Alfredo's best concert picks!
Labels: mhttp://www.blogger.com/img/blank.gifmx top 10 concerts 2011
Mark Osborne and Ryan Creighan Wedding
Photo Credit:Alfredo Flores,www.alfredoflores.net
Labels: prv mark and ryan
Ryan Zimmerman's "A Night at the Park"
Photo Credit:Alfredo Flores, Washington Life Magazine
Bating a Thousand: Washington Nationals third baseman Ryan Zimmerman hit a few home runs for Multiple Sclerosis at the ziMZ Foundation's premier event "A Night in the Park." Guest bid on auction items autographed by athletes including the 2010 Washington Capitals and Derek Jeter. Magician David Blaine wowed the crowd and Amos Lee and Mutlu provided the night's smooth soundtrack. Inspired by his mother, Cheryl, and her battle with MS, Zimmerman and the ziMS Foundation will continue to step up to the plate.
Labels: wl 150 nats zimmerman
Photo by Vincent Peters
IF JOHN LEGEND WAS your boyfriend, he'd want to slow dance and he doesn't care who sees this public displays of affection. And if that weren't enough, his favorite love song is about spending time with you — in the morning. At least that's the scenario he presents on "Good Morning," off of his latest CD, "Evolver." "Yeah, that song is pretty self-explanatory," said Legend in a husky morning baritone. "It's a real-life song — a lot of people know what I'm talking about — and I think that's why people like it a lot." Legend's live shows highlight the romance, packed with moments where he steps away from his grand piano to dance with female audience members, goes down on one knee while serenading senoritas and gives them roses, hugs and kisses on the cheek. Legend's songs are about love, lovemaking and heartbreak, every one exposing intimate details. "They're about personal experience, just living, experiencing life," he said. "I try to tell a story in my songs that people can relate to." Legend's breakout single, 2005's narrative ballad "Ordinary People," was just that — a song about everyday emotion and love and how it falters, how hard it is to keep strong. Other "Evolver" songs depict pursuing an elusive girl ("Green Light" featuring Outkast's Andre 3000); giving up his previous playboy ways ("This Time"); getting dumped, but in a good way ("It's Over," with Kanye West); and finding his one true love ("No Other Love," showcasing U.K. singer Estelle, who tours with Legend). But in addition to his smooth vocals, bravado and style, there's plenty of substance to Legend's craft — and to him as a person. His various philanthropic efforts include the Show Me Campaign to fight poverty through fostering sustainable development in Africa, where he's adopted the village of Mbola in western Tanzania. And during the Democratic National Convention this summer he performed the resounding debut of his Barack Obama-inspired "If You're Out There" — a call-to-action anthem, which discusses poverty and what young people, such as Legend, 29, can do about it, while also putting pressure on politicians to make a difference. "It was great to be there for such a historic moment," he said. "To be part of it was an honor. I felt the song was really appropriate for the moment that we're in and I was really happy to perform it there." Legend was born John Stephens, and taking a stage name such as Legend is bold. But it's not arrogance on JL's part; just confidence in his music. His accolades speak for themselves: multiple-platinum albums, five Grammys, and after toiling around as a session writer and musician for the likes of Lauryn Hill, Alicia Keys and Jay -Z, he now has his pick of the litter when it comes to who he wants to perform for him — bringing in West, Brandy and Andre on "Evolver." "I feel like I only want to work with people who are the best at what they do," he said. "Everything I do I want it to be high quality, every bit of music that I'm part of I want it to be exceptional. I want it to be great."Legend's also getting into the record biz — the first artist signed to his HomeSchool Records is Estelle — and transitioning to being a demographic-transcending artist, having performed on TV's "Dancing With the Stars" and "A Colbert Christmas," and opened the Latin Grammys with Colombian rocker Juanes in the Spanish-language debut of "If You're Out There." All of that work is just part of his restless work ethic and creativity. "Hopefully, I'll be alive for another 60 to 70 years and I'll experiment with all kinds of things musically, and that's the fun in what I do," Legend said. "Every time I do a new album or enter a new project it's an adventure. Every time I do a new tour — I get the opportunity to expand my palate." May there be many more good mornings to come. » DAR Constitution Hall, 311 18th St. NW; Tue., Dec. 9, 8 p.m., $71; 202-628-1776.
Labels: john legend piano kanye west green light evolver express
Kane's ButterBall with LMFAO at The Fillmore Silver Spring
The party rockers came out in full force in all shapes and sizes—from the young, to the extremely young—all ready to shake that and sport their Afro wigs, zebra pants, fluffy shoes, giant sunglasses, and other wacky outfits that the party band of all party bands, LMFAO, rock. The unlikely combination of the uncle and nephew duo--DJ Redfoo and SkyBlu, both decendants from Motown Records founder Berry Gordy—have taken meticulous steps in their skyrocketing to stardom with hits like "Get Crazy," "I'm in Miami," and "Party Rock Anthem." An extremely hard working ethic, tremendous self-promotion and grasping to this generation's online audience, has led to nearly eight million Facebook followers, untold riches, fame and fortune for the band, and an adulation that has already led to sold out shows in arenas. So fans of the electro pop duo in the D.C. area were in for a special treat, not only did they get to hang out with the crew with Hot 99.5's The Kane Show as part of their ButterBall festivities, but also to see LMFAO in a relatively intimate setting at the new Fillmore Silver Spring. As far as neon-bright outfits, hair ensemble, and overall fun atmosphere it was right up there with the fanatics that come out for Lady Gaga shows, and if the hits keep on coming, LMFAO will continue their path to superstardom.
Labels: mmx lmfao kane
TWO FULL PAGES SPREAD
Pharrell Williams introduces his new liqueur Qream
DREAMY NIGHT: Three-time Grammy-winning artist Pharrell Williams launches his high-end liqueur Qream with an all-out bash featuring 400 of the city's young and connected overachievers. Guests enjoyed hor d'oeuvres and Peach Creme and Strawberry Creme served on the rocks -- two popular Qream flavors. Go-go legend Chuck Brown stopped by along with hip-hop artist Pusha T, ABC7 anchors Cynne Simpson, Rebecca Cooper and Britt McHenry, and Mix 107.3 FM's Brooke Ryan.
CLICK HERE TO VIEW TWO FULL PAGE MAGAZINE PRINT OF PHOTOS
CLICK HERE FOR ONE PAGE SPREAD ON PHARRELL WILLIAMS
CLICK HERE FOR ONE PAGE SPREAD ON PASSPORT TO STYLE
Labels: pharrell
D.C. Fight Night 2011
D.C. heavyweights and boxing legends joined forces for the 22nd annual Fight Night DC at the Washington Hilton. Fight Night brings together 2,000 champions of business and government, entertainment and community service, as well as legends from the boxing and sports world, for an evening of fun, auctions, and live professional boxing. The event is the signature fundraiser for Fight For Children, an independent non-profit organization committed to improving education in D.C. schools.
Labels: mmx fight night
Violence is Scary IV
Photo Credit:Alfredo Flores, www.alfredoflores.net
Labels: v is scary iv shoot
"Pink Rocks the Runway" Fashion Show
Pink Rocks the Runway is an annual breast cancer awareness fashion event that combines couture fashion and live rock music. This year's show was headlined by celebrity designer Christian Siriano of "Project Runway." In addition, the evening celebrated and featured breast cancer survivors under 40 with a pink couture gown fashion show.
Labels: mmx 2011 pink jams
FULL PAGE SPREAD
A Standing Ovation for D.C. Teachers
"A" Plus: The second annual event honoring 663 "highly effective" D.C. public school teachers featured headline-making presenters, including Alfre Woodard, Mayor Vincent Gray, Grammy-winner John Legend and "godfather of go go" Chuck Brown, who was so excited he felt like "bustin' losse!" Seven teachers and one principal were singled out for excellence awards of $10,000 each this year, with 20 teachers and five principals receiving "Rubenstein" awards of $5,000 apiece donated by event chairman David Rubenstein. Emcee Jim Vance noted this year's festivities raised over $800,000 for awards organizer D.C. Public Education Fund
CLICK HERE TO VIEW FULL PAGE MAGAZINE PRINT OF PHOTOS
Labels: wl john legend ovation
Washington Post Express, October 22, 2008
Written by Express contributor Alfredo Flores
IT SEEMS LIKE just about everyone is claiming to be a maverick these days, but few have the cojones to actually be one. Calle 13 is a hip-hop/Latin-rock phenomenon that's been pegged as the future of the burgeoning reggaeton genre, but the witty duo constantly pokes fun at what's wrong with the Spanish-language pop, earning them irritation from fellow artists. For instance, last year, when Calle 13 exploded on the Latin music scene with eclectic beats and cheeky Spanglish lyrics about political injustices and pop culture figures, it earned the group an armful of awards, including Latin Grammys for best urban album and song and Billboard Latin Music's best reggaeton album, which left their competitors fuming. Fellow Puerto Rican rapper and leading lady of reggaeton Ivy Queen said that the Grammys "should have been shared better." "If you're envious of us, go ahead, cry! Gua-gua-gua, keep crying," Residente taunts in Spanish in the new Calle 13 single, "Que Lloren" from the band's new CD, "Los de Atras Vienen Conmigo." Residente (Rene Perez), along with his half-brother and multi-instrumentalist Visitante (Eduardo Cabra), both 30, have become champions of the fringe alternative-music scene of Puerto Rico, an island whose rich sonic mix of salsa, merengue, bomba and plena has been given the backseat to reggaeton, which mixes Jamaican dancehall, bachata, bling bling, sexual overtones and violent lyrics. The bar has been set high for the Boricua twosome ever since they announced their arrival with 2006's "Atrevete-te-te," a song whose video is loaded with dancing Marilyn Monroe lookalikes, with Visitante playing his clarinet from the rooftops as Residente checks out the ladies, raps that he doesn't care if they all look the same and that they all like Coldplay and Green Day. In a sit down interview in New York following the band's promotional concert at the Nokia Theater Times Square earlier this month, Residente explained in Spanish, "We're walking a fine crystal line where we're tempted to produce music that's for the masses, but that doesn't interest me. I am who I am and I feel proud to be a representative of the Puerto Rican alternative. We need to represent the music they listen to in the barrios, and alternative music is the music of the barrios." The album's title translates to "The Ones Left Behind Are Coming With Me" and the disc is dedicated to the often overlooked working class — the dishwashers, the cooks, the janitors, people who identify with Calle 13's music, with Residente giving shout outs to the names of projects, barrios and favelas all over Latin America. Residente also mentioned that those "behind" could also be his 10-member rock band, arranged by Visitante. The album has a little bit of everything — '80s synthesizer pop ("Electro Movimiento"), New Orleans second-line rhythms and Dixieland ("Gringo Latin Funk"), big band tuba ("Que Lloren"), Balkan whirl ("Fiesta de Locos"), African guitars ("Esto Con Eso"), Afro-Uruguayan Candombe tropical sounds ("La Perla" featuring a rare rap from salsero Ruben Blades about a seaside slum in Old San Juan) and rock accordion (the hit single "No Hay Nadie Como Tu" featuring Cafe Tacuba). The album's messages are positive — don't try to become something you're not just to fit in, just be yourself, stand up to those who put you down, find true love. But it's not all po-faced, as these jesters have a little fun in the process. The album is loaded with smart, funny lyrics: "She doesn't eat dulce de leche because of the cholesterol / She says Spanish words mixed with English / She's a gringa wanna be / Can't even name the capital of her country" ("Gringo"); "In our urban genre you're a tourist / Next to your personal stylist and make up artist / Wearing fur coats in an island where it isn't cold" ("Lloren"); and "For you I'd do anything / Fight 300 ninjas in Korea" ("Electro"). In the love song "Nadie," Calle 13 may have found their perfect musical mate by collaborating with Tacuba, a like-minded Mexican alternative band. The super alterna-groups blend cello, booming bass, jazz horns and accordion with a peppy and upbeat cumbia beat featuring the unmistakable nasally voice of Tacuba's Ruben Albarran and Residente's melodic rap delivery, with PG-13 (the latter's sister Ileana) providing the sultry vocals in the chorus. The song sets the tone for the overall lighter and less racy mood compared to last year's "Residente o Visitante" album. Residente feels that much such harsh language wasn't necessary this time around. "There just wasn't a need to be as hard in this disc," he said. "This is a much more colorful, more tranquil album. We came into this wanting to do something way different from our second disc and our first. We wanted to find a better way to communicate with more people."
Labels: Calle 13 nokia theater times square los de atraz vienen conmigo
Noche De Gala and After Party
ARTISTS IN ACTION: Always a highly anticipated event early in the fall social season, this year's Noche De Gala once again benefitted the National Hispanic Foundation for the Arts, which grants scholarships to young people of Hispanic heritage with dreams of working in television and film. Making the celebration even more poignant was three-time NHFA scholarship recipient, Rashaad Ernesto Green, whose first film, "Gun Hill Road," is currently showing in theaters. Green spoke movingly of NHFA's efforts to help him achieve his lifelong dream of filmmaking. Later, guests headed to Cuba Libre for a rollicking after party with a Latin twist.
Labels: noche de gala wash life
Jill Scott's Summer Block Party at Verizon Center
Host Doug E. Fresh sure knew how to throw down an old school party with this simple one phrase—"We're bringing it back to the Capital Centre!" For the current generation of Washingtonians, the only big arena concert venue in the area has been the Verizon Center. But prior to that, for previous generations, it was the Capital Centre in Landover, Maryland that hosted all the great R&B acts. But at Jill Scott's Summer Block Party, the Verizon Center truly felt like the old Cap Centre! Venerable R&B and hip hop icons DJ Jazzy Jeff and Mint Condition started the night out with a bit of nostalgic music, while crooner Anthony Hamilton brought the house down with his superb vocals. Jill Scott then closed it off with her powerful vocals and tremendous stage presence. All in all, the vibe and feel of the night had people reminiscing about old school block parties in their neighborhoods. It was a night most in attendance won't soon forget!
Labels: jill scott mmx
Thelonious Monk Gala
QUEEN OF SOUL: Aretha Franklin revisited her first love--jazz-in a surprise performance for the Thelonious Monk Institute of Jazz's All-Star 25th Anniversary Benefit, held during the annual International Jazz Competition. Former Secretaries of State Madeleine Albright and Colin Powell presented Franklin with the Maria Fisher Founder's Award for her contributio ns to Monk Institute.
Labels: thelonious monk gala
NOW at the Corcoran
Photo Credit:Alfredo Flores, DC Modern Luxury Magazine
DC Magazine at the Corcoran celebrated the launch of NOW at the Corcoran with a cocktail reception at the museum. NOW at the Corcoran is a program of unique exhibitions of new work by emerging and mid-career artists.
Location: 500 17th St NW, Washington DC
Labels: corcoran now dc mag 75
Baptism at St. Stephen Martyr Church
Labels: af ed baptism dos cinco
Enrique Iglesias' Euphoria Tour at the Patriot Center
While there may not be any NBA basketball in the foreseeable future, there was plenty of Miami heat at the Patriot Center as two of The Magic City's biggest Latin music stars took the stage at the sold out show! Joining heartthrob Enrique Iglesias was fellow Miami-native Pitbull, who got the late-arriving crowd pumped by doing renditions of club hits such as LMFAO's "Party Rock." The crowd surely needed a bolt of energy as many arrived well past the 7 p.m. start time due to traffic, forcing the show to be delayed some 40 minutes. Nevertheless, star-in-the-making Prince Royce got the show started by crooning the predominant female crowd with soft jams such as his Spanish version of "Stand By Me." Pitbull then took the energy level even further by tirelessly working the crowd, jumping, fist-pumping, pointing, posing, and then displaying the rapid-fire rhymes and sick hooks from his new album "Planet Pit," featuring the club hit "Give Me Everything." Fully engaged, the crowd went into absolute pandemonium the second Enrique Iglesias took the stage. His show did not disappoint as he showed off his charm- giving his adoring fans hugs and handshakes- and displaying his soulful vocals that went well with the driving dance and pop beats from his "Euphoria" album.
Labels: mmx enrique
Andy Warhol: Shadows Opening Party at the Hirshhorn Museum
The Hirshhorn Museum and Sculpture Garden hosted the grand opening party for Andy Warhol: Shadows art exhibit, kicking off an entire season of Warhol in D.C. The soirée included a reception flowing inside the museum lobby, out on to the plaza and a very special transformation of the Hirshhorn's Lerner Room into an artist's lounge. Influenced by Warhol's favorite colors: "aubergine, chartreuse, carmine red, yellow, midnight blue … and white," the funky chic lounge inspired a "Warholian" outlook on the present day. Andy Warhol: Shadows will run from September 25th to January 15, 2012.
Labels: mmx warhol
PHOTO STORY Metromix's Top 10 D.C....
PHOTO SHOOT Mark Osborne and Ryan Creighan Weddin...
PHOTO STORY Ryan Zimmerman's ...
ARTS & EVENTSGood Mornings: John LegendWritten by...
PHOTO STORY Kane's ButterBall with LMFAO at The F...
PHOTO STORY TWO FULL PAGES SPREAD ...
PHOTO SHOOT Violence is Scary IV Photo Credit:Alf...
PHOTO STORY "Pink Rocks the Runway" Fashion S...
PHOTO STORY FULL PAGE SPREAD ...
ARTS & EVENTSAlterna-Rican: Calle 13 Washington Po...
PHOTO STORY Jill Scott's Summer Block Party at Ve...
PHOTO STORY NOW at the Corcor...
PHOTO SHOOT Baptism at St. Stephen Martyr Church P...
PHOTO STORY Enrique Iglesias' Euphoria Tour at th...
PHOTO STORY Andy Warhol: Shadows Opening Part...

redpajama_set_name: RedPajamaCommonCrawl


WELCOME TO MEDICAL MISSIONS 101!
Medical Missions 101 is an online community dedicated to educating, encouraging and engaging those who share our passion for medical missions. As a community we are pursuing best evidence based practice in medical missions and global health; we invite you to join us. New content is added each week! If you find our content helpful, please share it!
Experience has shown me that asking the wrong questions can lead to unnecessary failure on the mission field but what are the right questions? This set of questions deals with matching with a health facility or program as a medical missionary.
I have seen countless medical missionaries fail from different organizations because they landed a place that's not a good fit. They land at a mission station where all attempts to thrive individually and/or as a family fail. They then revert to survival mode where it becomes difficult to sustain the joy in life or the joy in serving. It may take only a few months or sometimes a full three year term, but serving in a place that is not a good fit will inevitably wear people down and bring them home.
Christian Mission hospitals and health programs account for about 50% of all healthcare delivered in Sub-Saharan Africa (Olivier, et al., 2015). That figure is probably closer to 70% of the truly functional healthcare services delivered. Sadly however, many of these facilities have closed, those that remain are fighting for survival.
Do you share our passion for medical missions?
This channel was created to tell the story of medical missions, health programs, mission hospitals, and the missionaries who serve in them. Each week we will upload new episodes from the field and from here at home as we interview mission leaders facilitating health programs and medical missions around the world.
For healthcare professionals and students - gain knowledge and skills for global health service.
Healthcare professionals and students - learn the basics of simple skin suturing in resource-limited settings.
Healthcare professionals and students - enhance your diagnostic skill set with ultrasound.
For all learners - respond with life-saving and dignity-upholding measures.
CHSC is very pleased to offer ONLINE access to the teaching sessions from our Medical Missions 101 Gathering.
When Healthcare Hurts is the first book to look seriously at the challenges of patient safety and developmental safety in global health missions, and it defines evidence based guidelines through which these issues can be addressed.


text:
/*============================================================================*
* Title       : Artsy (Social Application w/ Art Auction)
* Description : "Artsy" is a social application idea that would provide artists
                 a place to showcase their artworks and offer them the chance
                 to make a profit from it.
* Filename    : BidPage.cs
* Version     : v1.0
* Author      : Gensaya, Carl Jerwin F.
* Yr&Sec&Uni  : BSCS 3-3 PUP Main
* Subject     : Advance Programming
*============================================================================*/

using System;
using System.Drawing;
using System.Threading;
using System.Timers;
using System.Windows.Forms;

namespace Artsy {

   public class BiddingPage : Panel {

    /*============================================================================*
     *  Function   : DisplayBidPage
     *  Params     : None
     *  Returns    : Void
     *  Description: This serves as the GUI panel for Artsy's bidding page.
     *=============================================================================*/

      public BiddingPage() {
         DisplayBidPage();
      }

      void DisplayBidPage() {
         BackColor = Color.White;
         DoubleBuffered = true;
         AutoScroll = true;

         _BiddedItem = new PictureBox() {
            Size = new Size(250, 300),
            Location = new Point(50, 100),
            Image = Image.FromFile("res/log2.jpg"),
            SizeMode = PictureBoxSizeMode.StretchImage
         };
         this.Controls.Add(_BiddedItem);

         _ItemFullInfo = new Label() {
            Size = new Size(250, 100),
            Font = new Font("Segoe UI", 12),
            Location = new Point(50, 400),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "Art full info w/ bidding."
         };
         this.Controls.Add(_ItemFullInfo);

         _ItemName = new Label() {
            Size = new Size(350, 50),
            Font = new Font("Segoe UI", 15, FontStyle.Bold),
            Location = new Point(330, 100),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "ITEM NAME: Bidding time left."
         };
         this.Controls.Add(_ItemName);

         _BidPrice[0] = new Label() {
            Size = new Size(350, 40),
            Font = new Font("Segoe UI", 12),
            Location = new Point(330, 170),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "Current bid: $$$$$$$$ [Total Bids Count: #]"
         };
         this.Controls.Add(_BidPrice[0]);

         _PriceBox = new TextBox() {
            Size = new Size(180, 40),
            Font = new Font("Segoe UI", 12),
            Location = new Point(500, 230),
         };
         this.Controls.Add(_PriceBox);

         _BidPrice[1] = new Label() {
            Size = new Size(150, 30),
            Font = new Font("Segoe UI", 12),
            Location = new Point(330, 230),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "Your bid:"
         };
         this.Controls.Add(_BidPrice[1]);

         _BidPrice[2] = new Label() {
            Size = new Size(100, 40),
            Font = new Font("Segoe UI", 12),
            Location = new Point(330, 300),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "Buy now: $$"
         };
         this.Controls.Add(_BidPrice[2]);

         _BuyOption[0] = new Button() {
            Size = new Size(110, 40),
            Font = new Font("Segoe UI", 12),
            Location = new Point(450, 300),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "Add to Cart"
         };
         this.Controls.Add(_BuyOption[0]);

         _BuyOption[1] = new Button() {
            Size = new Size(110, 40),
            Font = new Font("Segoe UI", 12),
            Location = new Point(570, 300),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "Buy it Now"
         };
         this.Controls.Add(_BuyOption[1]);

         _BidRemind = new Label() {
            Size = new Size(350, 190),
            Font = new Font("Segoe UI", 12),
            Location = new Point(330, 360),
            TextAlign = ContentAlignment.MiddleCenter,
            Text = "Bidding Reminders"
         };
         this.Controls.Add(_BidRemind);

         _Tab = new Label() {
            Size = new Size(900, 100),
            Text = "TAB HOLDER ETC",
            BackColor = Color.Gray,
            Font = new Font("Segoe UI", 20, FontStyle.Bold),
            TextAlign = ContentAlignment.MiddleLeft
         };
         //this.Controls.Add(_Tab);

         _SellerInfo = new Label() {
            Size = new Size(150, 150),
            Text = "Seller Info",
            Font = new Font("Segoe UI", 15, FontStyle.Bold),
            TextAlign = ContentAlignment.MiddleCenter,
            Location = new Point(700, 100)
         };
         this.Controls.Add(_SellerInfo);

         _SellerMess = new Label() {
            Size = new Size(150, 280),
            Text = "OP's message regarding the item being sold.",
            Font = new Font("Segoe UI", 15, FontStyle.Bold),
            TextAlign = ContentAlignment.MiddleCenter,
            Location = new Point(700, 270)
         };
         this.Controls.Add(_SellerMess);

         _ItemAction[0] = new Label() {
            Size = new Size(120, 30),
            Location = new Point(50, 520),
            Text = "Message OP",
            Font = new Font("Segoe UI", 10, FontStyle.Bold),
            TextAlign = ContentAlignment.MiddleCenter
         };
         this.Controls.Add(_ItemAction[0]);

         _ItemAction[1] = new Label() {
            Size = new Size(120, 30),
            Location = new Point(180, 520),
            Text = "Report Item",
            Font = new Font("Segoe UI", 10, FontStyle.Bold),
            TextAlign = ContentAlignment.MiddleCenter
         };
         this.Controls.Add(_ItemAction[1]);

         _Back = new Label() {
            Size = new Size(120, 20),
            Location = new Point(60, 20),
            Text = "◄ BACK",
            Font = new Font("Segoe UI", 15),
            TextAlign = ContentAlignment.MiddleCenter,
            ForeColor = ColorTranslator.FromHtml("#212121"),
            BackColor = ColorTranslator.FromHtml("#954535")
         };
         _Back.MouseEnter += new EventHandler(__MouseEnter);
         _Back.MouseLeave += new EventHandler(__MouseLeave);
         _Back.Click += new EventHandler(__MouseClick);
         this.Controls.Add(_Back);
      }

      void __MouseEnter(object source, EventArgs e){
         var sender = source as Label;
         sender.BackColor =  ColorTranslator.FromHtml("#212121");
         sender.ForeColor = ColorTranslator.FromHtml("#954535");
      }
      void __MouseLeave(object source, EventArgs e){
         var sender = source as Label;
         sender.ForeColor =  ColorTranslator.FromHtml("#212121");
         sender.BackColor = ColorTranslator.FromHtml("#954535");
      }
      void __MouseClick(object source, EventArgs e){
         this.Hide();
      }

      /*==============================INITIALIZATION==============================*/
      Label _Tab;
      PictureBox _BiddedItem;
      Label _ItemFullInfo;
      Label _ItemName;
      Label _SellerMess;
      Label[] _BidPrice = new Label[3];
      Label _SellerInfo;
      TextBox _PriceBox;
      Label _Back;
      Button[] _BuyOption = new Button[2];
      Label[] _ItemAction = new Label[2];
      Label _BidRemind;
   }

}

redpajama_set_name: RedPajamaGithub


Heveningham is a village and civil parish in Suffolk Coastal, Suffolk, England. In 2001 there were 110 people living in Heveningham. Heveningham is pronounced Henningum. Heveningham has a church called St Margaret.


text:
The KOMODO Story
Sustainable fashion brand KOMODO has been supporting SOS for many years. In this guest blog, they explain why sustainability is important in fashion and why they support SOS's work.
KOMODO are known as The Original Ethical brand since 1988. We are aware this is a bold claim, so let us tell you a little bit about ourselves.
From the very beginning, using ethically sourced materials and eco modes of production have always been at the core of what we do. Sustainability has been woven into the brand right from the very beginning. This all started rather organically back in 1988, when our founder Joe had the idea of using discarded old jeans to create a patchwork jacket that he literally cut and pasted together on his kitchen table. The result was so good that it resulted in a great little business that saw him making them up in an East End workshop and selling them in West End jean shops. This coincided with his days as a backpacker, where on his travels, he would collaborate with Tibetan and Balinese artisans and create beautifully handcrafted pieces, bringing them back to London to sell. The process was always very natural, and gradually grew and grew until suddenly we were creating a buzz, selling at all the trade shows, and our little ethical brand was suddenly snowballing!
KOMODO has always believed it is our duty, not just as creatives but as humans, to protect the world we all live in. This planet is our one and only home, and one of the most important ways we can nurture the environment is to invest in responsibly and ethically made clothing that will last. We truly believe that the simple everyday actions we all make will have a huge and positive impact on our futures. Eco fabrics are always our starting point, our brand is centred around natural, organic and recycled fibres. We adopted hemp early on, developing it into various forms, and it is still a plant we love making clothes out of today. We invested in using organic cotton right from the very beginning, and today use it in many forms including twill, poplin and knits, to bring you everyday staples that are designed to get better with age. Our wool is always cruelty free, and we produce all our knitwear in our solar powered factory in Kathmandu. More recently we have added modal, Tencel, and cupro. We also re-cycle everyday plastic bottles and turn them into various alternatives, including waterproof raincoat material, faux fur and fleecing.
Alongside the actual garments, it's so important to us to give back to charities like SOS. Joe has become a close friend of the charity, and in turn we are part of 1% for the planet organisation, meaning we give 1% of all our profits to them and have done for several years. We produce an orangutan centred print or graphic for them nearly every season to help raise awareness for the wonderful work they do and for the plight of these beautiful creatures and their habitats.
We feel that whilst it's our privilege to dress up in style, it must be our responsibility to ensure that style is sustainable, otherwise you can only look good on the outside. Eco fashion is quickly becoming something everyone wants to be part of, and while we're so glad about this, we know there is a lot of greenwashing and it's imperative to us that we are not part of this, and that we keep learning and keep growing. We have made sure to stick with our family run small factories in Bali, Kathmandu and India. Loyalty is a rare commodity in fashion, but we like it! We would love for you to join us, consider the choices you make when buying clothes, and ask yourself who made what you're wearing every day, and where they made it.
When Joe Komodo met Lucy Wisdom
It was 1992 and the village of Seminyak in Bali was becoming a hive of creative activity. Like most of my friends there, I was working hard to build my little brand. My girlfriend, Yoga-Jane, was giving lessons in our garden and Lucy, being from London, stayed for a cup of tea.
Her story was different from all the others as she was spending most of her time up in the wilds of Sumatra to help local people protect orangutans. Lucy had a magnetic charm and a sharp north London wit, and we naturally became friends and she would visit us often. I was glad to help out by shipping things she made back home and supporting her efforts to raise awareness of her great project.
Orangutan translates as 'person of the forest', and Lucy was a big sister to them. She was an inspiration to so many people and her legacy is a testament to her enduring positive energy.


violation. This code is applicable to National Certified Counselors and those who are seeking certification from NBCC.
The National Board for Certified Counselors, Inc. and Affiliates (NBCC), an independent not-for-profit credentialing body for counselors, was incorporated in 1982 to establish and monitor a national certification system, to identify those counselors who have voluntarily sought and obtained certification, and to maintain a register of those counselors. NBCC's certification program recognizes counselors who have met predetermined standards in their training, experience, and performance on the National Counselor Examination for Licensure and Certification (NCE), the most portable credentialing examination in counseling. NBCC has approximately 40,000 certified counselors. These counselors live and work in the US and over 50 countries. NBCC examinations are used by more than 40 states, the District of Columbia, and Guam to credential counselors on a state level.
This course is recommended for health care professionals, especially Counselors certified by the National Board for Certified Counselors, but also including psychologists, other counselors, social workers, and nurses who seek knowledge about ethical standards for the practice of counseling. It is appropriate for all levels of participants' knowledge.
Explain the relationship between the NBCC Code of Ethics and any other Codes to which a given professional might be subject.
Apply the NBCC Code of Ethics to the practice of counseling.
List and explain the ethical obligations of Certifed Counselors to themselves, their employers, their professional associates, the individuals they serve, the Counseling profession, and society.
Identify the Certified Counselor's ethical obligations to the client whose condition indicates that there is a clear and imminent danger to the client or others.
Apply the NBCC Code of Ethics to the participation of Certified Counselors in research and publication activities.
Define the ethical obligations of Certified Counselors providing professional supervision.
Explain the ethical obligations of Certified Counselors with regard to the use of measurement and evaluation techniques.


text:
\section{Introduction}

In~\cite{nonlocality} the authors describe a scheme for generalized quantum measurements that allows for the probabilities of each outcome to be monitored continuously. The ability to halt the quantum measurement at a desired confidence then plays a crucial role in proving the inability of two parties sharing Local Operations and Classical Communication (LOCC) to successfully identify orthogonal product states. The continuous scheme, however, relies on attaching a large number of ancilla and the ability to perform large unitary operations on their joint system. In~\cite{weakuniversal, generalizedstochastic} the authors develop an alternative scheme that instead uses only diffusive weak measurements~\cite{simplemodel} with closed-loop feedback.

In this paper we aim to characterize a continuous measurement procedure that lies between the two schemes above.  We are motivated by both theoretical and experimental considerations.  Many quantum mechanical systems either have naturally slow measurement times, or can only be probed weakly.  For example, the beautiful experiments of Haroche and Raimond \cite{rydberg} use a stream of Rydberg atoms to repeatedly probe the state of a microwave mode in a superconducting cavity.  Homodyne and heterodyne measurements are widely used in optics, and produce a continuous output current.  Superconducting qubits (or similar solid-state devices) can be measured by a weak dispersive coupling to a microwave cavity, which can be measured in turn by homodyne measurement.  Magnetic resonance force microscopy \cite{MRFM} can do single-spin measurement by a continuous measurement procedure.  Moreover, latency is sufficiently low in modern experiments that it is possible to do continuous feedback in real time, as has already been demonstrated in the microwave cavity/Rydberg atom system \cite{rydberg-feedback}.

For most quantum systems, there is no direct way to implement a given generalized measurement.  Generalized measurements are performed by unitary coupling to an ancillary system, followed by projective measurement on the ancilla.  In this paper, we explore the type of measurements that can be built up from a particular type of fixed weak interaction.  Moreover, the type of protocols we explore in this paper are examples of closed-loop quantum control, where continuous measurement is fed back not just to control the Hamiltonian of the system but the continuous measurement itself.  Such feedback can be used, for example, to improve the accuracy of phase measurements \cite{Phase}.  Here, we use it to decompose generalized measurements, but no doubt other types of protocols can be done using similar techniques.

We consider a situation where the system to be measured can only be probed weakly, and the experimenter has only limited control over the system itself.  The interaction between the probe and the system is fixed.  The experimenter, however, has complete control over both the preparation and measurement of the probes.  We wish to characterize what combinations of probe states, measurements, and interaction Hamiltonians yield continuous decompositions of generalized measurements.

Our results will concern only qubit probes. Although this may seem like a rather narrow class of experiments it is a natural setting for two-outcome measurements. Our results will apply equally well to the procedure outlined in~\cite{nonlocality, weakuniversal} that allows for any general $n$-outcome measurement to be decomposed into a series of two-outcome measurements.

This paper is organized as follows: In Section~\ref{sec:theory} we introduce the discretized steps of a continuous measurement (weak measurements), and review how they can be constructed from a qubit probe. In Section~\ref{sec:result} we state and prove our main result about interaction Hamiltonians between the probe and system. In Section~\ref{sec:unitaries} we briefly discuss the role of unitary pulses in our scheme. In Section~\ref{sec:qubit} we exhibit how our model can be used to decompose a generalized diagonal measurement on a system qubit using a qubit probe and the interaction Hamiltonian $Z \otimes Z$. We summarize these findings in Section~\ref{sec:conclusions}.

\section{\label{sec:theory}Reversible weak measurements and random walks}
In~\cite{weakuniversal, infinitesimal} the authors show how to to decompose an instantaneous quantum measurement into a continuous process. Such a decomposition must respect two properties: that the state of the system being measured evolve smoothly, and that the entire continuous process allow for multiple possible outcomes. If we consider a discretized version of the process, then a sequence of diffusive weak measurements~\cite{simplemodel} satisfies both of these requirements. A weak measurement is parametrized by a ``strength" parameter $\delta} \def\D{\Delta$ and its associated operators have the general form
\[ M_k \propto \mathds{1} + \delta} \def\D{\Delta \hat{\varepsilon}_k, \]
where $\hat{\varepsilon}_k$ is an operator of bounded norm. Since the outcome of each measurement is a random function of the state, a sequence of weak measurements forms a stochastic process.

However, if a stochastic process is to be a faithful decomposition of a quantum measurement, then the result of the process must depend only on the state being measured and not on the total time of the evolution. In~\cite{weakuniversal} the authors show that this can be accomplished by casting the weak measurement steps as corresponding to a $1$-dimensional random walk indexed by the pointer variable $x$. In this case the result of the process does not depend on total time but, instead, on the drift of the pointer $x$ which is, in turn, dependent on the state. Each weak measurement step updates $x$ to $x \pm \delta} \def\D{\Delta$ depending on the result. Any dependence on the duration of the process is accounted for by constructing step operators $M_{\pm}(x)$ that cancel when applied in opposite directions. More precisely, the scheme requires the steps to be \emph{reversible} (Def. ~\ref{def:reversibility}).
\begin{defin}[Reversibility condition] \label{def:reversibility}
We say a one-parameter family of weak measurement operators $\{ M_{\pm}(x) \}_x$ satisfies the \emph{reversibility condition} if
\begin{equation}
	\label{eqn:reversibility}
	M_{\mp}(x\pm \delta} \def\D{\Delta) M_{\pm}(x) \propto \mathds{1}
\end{equation}
for all $x$ in a given interval.
\end{defin}
The step operators $M_{\pm}(x)$ in Eq.~\eqref{eqn:reversibility} are chosen such that the first operator updates $x$ to $x \pm \delta} \def\D{\Delta$ and the second returns it to $(x \pm \delta} \def\D{\Delta) \mp \delta} \def\D{\Delta = x$. If the product of the two is proportional to the identity, then the operators have no net effect on the system state up to a normalization constant.

\begin{figure}
	\includegraphics[width=1.0\linewidth]{RCM-02.png}
	\caption{The random walk construction begins at $x=0$ with a weak measurement given by the step operators $M_{\pm}(0)$. Each successive step of the walk contributes to the total walk operator, Eq.~\eqref{eqn:totalwalkoperator}, via one of the two step operators and updates $x$ accordingly. The walk terminates at either endpoint $\pm X$ where the total walk operator is designed to match the desired instantaneous measurement.}
	\label{fig:graphicalmodel}
\end{figure}

To help clarify this construction, we provide a graphical representation in Figure~\ref{fig:graphicalmodel}. As one can see, there is one random walk performed by the pointer $x$ and another random walk performed by the evolution of the system state under each weak measurement. The path of both walks is uniquely parametrized by $x$.

The evolution operator $M(x)$ describing the total evolution of the state under the random walk above is given by the product of step operators from the initial state at $x=0$ to the current value of the pointer variable $x$,
\begin{equation}
	M(x) \propto \left\{ \begin{array}{lcr} \displaystyle\prod_{j=0}^{\lfloor |x| / \delta} \def\D{\Delta \rfloor} M_+(j \delta} \def\D{\Delta) & \hspace{0.25in} & x>0 \bigskip \\ \displaystyle\prod_{j=0}^{\lfloor |x| / \delta} \def\D{\Delta \rfloor} M_-(-j \delta} \def\D{\Delta) & & x<0 \end{array} \right.
	\label{eqn:totalwalkoperator}
\end{equation}

The particular instantaneous two-outcome measurement to which this decomposition corresponds is given by the endpoints of the random walk in the continuous limit,
\begin{equation}
	M_1 = \lim_{\delta} \def\D{\Delta \rightarrow 0} M(X) \hspace{0.25in} \text{and} \hspace{0.25in} M_2 = \lim_{\delta} \def\D{\Delta \rightarrow 0} M(-X) \label{eqn:matchingendpoints}
\end{equation}

Altogether, these operators define a \emph{continuous decomposition}.
\begin{defin}[Continuous decomposition] \label{def:continuousdecomposition}
We call a one-parameter family of weak measurements $\{ M_{\pm}(x) \}_x$ a \emph{continuous decomposition} of $\{ M_1, M_2 \}$ if $\{M_{\pm}(x)\}_x$ satisfy the reversibility condition (Def.~\ref{def:reversibility}) and the endpoints, as given by Eq.~\eqref{eqn:matchingendpoints}, match $M_1$ and $M_2$. We call $\{M_{\pm}(x)\}_x$ the \emph{step operators} of this decomposition.
\end{defin}

We are interested in performing the step operators described above via a projective measurement on a weakly interacting probe. In Figure~\ref{fig:circuit} we illustrate preparing a probe state, allowing the probe and system to interact for a short time $\delta} \def\D{\Delta$, and then measuring the probe to update $x$. In the continuous limit, this feedback loop is considered to occur instantaneously. We call the circuit in Figure~\ref{fig:circuit} a \emph{probe feedback loop}.

\begin{defin}[Probe feedback loop]\label{def:probefeedbackloop} We say that a one-parameter family of $2$-outcome measurements $\{M_{\pm}(x)\}_x$ is generated by a \emph{probe feedback loop} if the measurement operators are of the form
\[ M_{\pm}(x) = \bra{\Phi^{\pm}(x)} e^{i \delta} \def\D{\Delta H_{PS}} \ket{\sigma(x)}, \]
where $\ket{\sigma(x)}$ is the \emph{probe} qubit, $\ket{\Psi^{\pm}}$ are two orthogonal states associated with the destructive measurement of the \emph{detector}, and $H_{PS}$ is the interaction Hamiltonian between the probe $P$ and system $S$.
\end{defin}

\begin{figure}
\includegraphics[width=1.0\linewidth]{RCM-01.png}
\caption{In the above figure, we model a single step of the stochastic process for a pointer value $x$. Time flows from left to right. The value of the pointer variable is classical and thus symbolized by a double line. All single lines are quantum states. The probe state $\ket{\sigma (x)}_P$ is a qubit prepared according to the pointer variable. The probe and system $\ket{\psi}$ are then permitted to interact via the Hamiltonian $H_{PS}$. The probe is destroyed in the projective measurement $\{ \bra{\Phi^{\pm}(x)} \}$ which can depend on $x$ although we sometimes drop this dependence in our notation. The result of the measurement is then used to update the pointer variable for the following step.}
\label{fig:circuit}
\end{figure}

In some cases, the probe feedback loops are interlaced with weak unitary pulses on the system. These unitaries play a special role in that they encapsulate some notion of the experimentalist's power. In our efforts to characterize all measurements possible with a given $H_{PS}$, we will assume that an ``all-powerful" experimentalist will have the ability to perform any unitary pulse. We will examine constraints on the pulses available to the experimentalist in Section~\ref{sec:pulseconstraints}.

The most general Hamiltonian we can write for the interaction of a qubit probe and an arbitrary quantum system is the following
\begin{equation}
	H_{PS} = \mathds{1} \otimes H_S + X \otimes H_X + Y \otimes H_Y + Z \otimes H_Z
	\label{eqn:generalHamiltonian}
\end{equation}
where $\mathds{1}$, $X$, $Y$, $Z$ are the usual Pauli matrices on the probe state $P$, and $H_S$, $H_X$, $H_Y$, $H_Z$ are corresponding Hermitian matrices on the system $S$. Recalling our expression for the step operator above, we note that it now explicitly depends on the geometry of the probe state, the projective measurement, and the Pauli matrices. Combined with our requirement for step operator reversibility, Eq.~\eqref{eqn:reversibility}, we can begin to investigate which decompositions are possible.

Before we begin, however, we note that we are afforded one advantage through our random walk construction because the reversibility condition need not be met exactly. Consider that a classical random walk must take $O(N^2)$ steps to converge with fixed probability (where $N=X/\delta} \def\D{\Delta$). This implies that the total walk operator in Eq.~\eqref{eqn:totalwalkoperator} will accumulate $N^2 O(\delta} \def\D{\Delta)$ terms, $N^2 O(\delta} \def\D{\Delta^2)$ terms, and so on. However, since the contribution of $N^2 O(\delta} \def\D{\Delta^3)$ terms vanishes as $\delta} \def\D{\Delta \rightarrow 0$ regardless of whether or not the step operators are exactly reversible, we only require that the reversibility condition of Eq.~\eqref{eqn:reversibility} be met only up to $O(\delta} \def\D{\Delta^2)$. As we discover below, this requirement is still the source of the most stringent limitations on possible decompositions.

\section{\label{sec:result}Main result}

We consider Hamiltonians where the strength of various terms cannot be engineered. We call these ``fixed'' in the sense that at every weak-measurement step, the interaction between the probe and the system is identical. We allow for weak unitary pulses to be applied in between each step and we later specify which types of pulses are necessary to achieve the class of measurements we find below.

\begin{thm} \label{thm:MAIN}
Any continuous decomposition (Def.~\ref{def:continuousdecomposition}) with step operators generated by a probe feedback loop (Def.~\ref{def:probefeedbackloop}), can only match a $2$-outcome measurement of the form
\[M_1 = U_1 \left( \alpha \; \Pi_S + \beta \; \Pi_{S^{\perp}} \right) V, \]
\[M_2 = U_2 \left( \sqrt{1-\alpha^2} \; \Pi_S + \sqrt{1-\beta^2} \; \Pi_{S^{\perp}} \right) V, \]
where $U_1$, $U_2$, and $V$ are unitary matrices and $\Pi_S$, $\Pi_{S^{\perp}}$ are projectors onto orthogonal subspaces of the system space.
\end{thm}

\subsection{Proof of Theorem~\ref{thm:MAIN}}

We begin with the following observation about weak-measurements generated from qubit probes; when interpreted on the Bloch sphere, there is a geometric constraint between the probe and the detector.
\begin{defin}[Probe basis] \label{def:probebasis}
For any qubit probe $\ket{\sigma(x)}$ and projective qubit measurement $\bra{\Psi^{\pm}}$, we define a real orthonormal basis for the Bloch sphere $\{\vec{n}_1(x), \vec{n}_2(x), \vec{n}_3(x)\}$. We call this a \emph{probe basis} if
\begin{itemize}
\item $\vec{\sigma}$, the Bloch vector associated with the probe state $\ket{\sigma(x)}$, is no further than distance $\delta} \def\D{\Delta$ from $\vec{n}_1$,
\item $\vec{n}_2$ is the Bloch vector associated with $\ket{\Phi^+}$,
\item and $\vec{n}_3 = \vec{n}_1 \times \vec{n}_2$.
\end{itemize}
\end{defin}
The constraint itself is expressed in the following lemma, the proof of which can be found in the Appendix.
\begin{lem}[Probe basis of a weak measurement] \label{lem:probebasis}
Any diffusive weak measurement given by a probe feedback loop (Def.~\ref{def:probefeedbackloop}) with a probe basis $\{\vec{n}_1(x), \vec{n}_2(x), \vec{n}_3(x)\}$ must have $\vec{n}_2 \cdot \vec{\sigma} \sim O(\delta} \def\D{\Delta)$. Thus, an orthonormal basis for the Bloch sphere that approximates the probe and detector always exists.
\end{lem}

The lemma above yields an important tool for our analysis. Recall that for a given interaction Hamiltonian we seek to characterize the weak measurement step operators achievable via \emph{any} probe feedback loop parametrized by $x$. However, we can instead fix a probe basis for the probe feedback loop and consider a family of interaction Hamiltonians $H'_{PS}(x)$ which give rise to the same set of weak measurement step operators. This transformation is performed with the following identifications
\begin{eqnarray*}
	H_{PS} & = & X \otimes H_X + Y \otimes H_Y + Z \otimes H_Z \\
	& = & \vec{n}_1(x) \cdot \vec{P} H_1(x) + \vec{n}_2(x) \cdot \vec{P} H_2(x) + \vec{n}_3(x) \cdot \vec{P} H_3(x)
\end{eqnarray*}
where $\vec{P} = \left[ X \; , \; Y \; , \; Z \right]^T$ and
\[ \left[ \begin{array}{c} H_1(x) \\ H_2(x) \\ H_3(x) \end{array} \right] = \left[ \begin{array}{c|c|c} & & \\ \vec{n}_1(x) & \vec{n}_2(x) & \vec{n}_3(x) \\ & &  \end{array} \right] \left[ \begin{array}{c} H_X \\ H_Y \\ H_Z \end{array} \right] . \]
We will also later abbreviate the vectors above as $\vec{H}'(x) = \left[ H_1(x) \; , \; H_2(x) \; , \; H_3(x) \right]^T$ and $\vec{H} = \left[ H_X \; , \; H_Y \; , \; H_Z \right]^T$. Finally, we define an interaction Hamiltonian in the probe basis
\[ H'_{PS}(x) = X \otimes H_1(x) + Y \otimes H_2(x) + Z \otimes H_3(x), \]
and this yields an advantageous rewriting of the weak measurement step operators
\begin{eqnarray*}
	M_{\pm}(x) & = & \bra{\Psi^{\pm}(x)} \exp \left( i \delta} \def\D{\Delta H_{PS}\right) \ket{\sigma(x)} \\
	& = & \bra{\pm} \exp \left( i \delta} \def\D{\Delta H'_{PS}(x) \right) \ket{0} + O(\delta} \def\D{\Delta).
\end{eqnarray*}
In this basis, the detector states $\ket{\Phi^{\pm}}$ are $\ket{\pm}$, the $\pm1$ eigenstates of $X$, and the initial state $\ket{\sigma(x)}$ is close to $\ket{0}$, the $+1$ eigenstate of $Z$. This choice also allows us to ignore the $\mathds{1}_P \otimes H_S$ term in the general Hamiltonian Eq.~\eqref{eqn:generalHamiltonian} since $ \bra{\pm} \mathds{1} \ket{0} = \bra{\pm} Z \ket{0}$ and any contribution from $H_S$ can be rewritten as part of $H_3$.

In lemma~\ref{lem:probebasis} we required that the probe and detector Bloch vectors be orthogonal only up to $O(\delta} \def\D{\Delta)$. We will therefore allow the probe states to be perturbed from $\ket{0}$ in our analysis. This causes an adjustment in our expression for the weak measurement step operators, parametrized by two functions $c(x)$ and $\psi(x)$,
\begin{eqnarray*}
	\ket{\sigma(x)} & = & \cos \left( \delta} \def\D{\Delta c(x) \right) \ket{0} + \sin \left( \delta} \def\D{\Delta c(x) \right) e^{i \psi (x)} \ket{1} \\
	& \approx & \ket{0} + \delta} \def\D{\Delta c(x) e^{i \psi(x)} \ket{1} \\
	& = & \ket{0} + \delta} \def\D{\Delta \ket{\Delta (x)},
\end{eqnarray*}
where in the last line we've implicitly defined $\ket{\Delta (x)} = c(x) e^{i \psi(x)} \ket{1}$. Equivalently, this contributes a term of $O(\delta} \def\D{\Delta)$ to our step operators
\begin{equation}
	M_{\pm}(x) = \bra{\pm} e^{i \delta} \def\D{\Delta H'_{PS}(x)} \ket{0} + \delta} \def\D{\Delta \bra{\pm} e^{i \delta} \def\D{\Delta H'_{PS}(x)} \ket{\Delta(x)} .
\end{equation}
Grouping together $O(1)$, $O(\delta} \def\D{\Delta)$ and $O(\delta} \def\D{\Delta^2)$ terms in the above expression yields
\begin{eqnarray*}
	\lefteqn{M_{\pm}(x) = \bra{\pm} \Big( \ket{0} + \delta} \def\D{\Delta \ket{\Delta(x)} \Big) \cdot \mathds{1}} \\
	& & + i \delta} \def\D{\Delta  \bra{\pm} H'_{PS}(x) \Big( \ket{0} + \delta} \def\D{\Delta \ket{\Delta(x)} \Big) - \frac{\delta} \def\D{\Delta^2}{2} \bra{\pm} H'_{PS}(x)^2 \ket{0} \\
	& = & \frac{\mathds{1}}{\sqrt{2}} + \delta} \def\D{\Delta \left( i \bra{\pm}H'_{PS}(x) \ket{0} \pm \frac{c(x) e^{i \psi(x)} \mathds{1}}{\sqrt{2}} \right) \\
	& &  - \frac{\delta} \def\D{\Delta^2}{2} \Big( \bra{\pm} H'_{PS}(x)^2 \ket{0} - 2i \bra{\pm}H'_{PS}(x) \ket{\Delta(x)} \Big) \\
	& = & \frac{\mathds{1}}{\sqrt{2}} + \delta} \def\D{\Delta M^{(1)}_{\pm}(x) - \frac{\delta} \def\D{\Delta^2}{2} M^{(2)}_{\pm}(x),
\end{eqnarray*}
where we've implicitly defined $M^{(1)}_{\pm}(x)$ and $M^{(2)}_{\pm}(x)$ to collect the $O(\delta} \def\D{\Delta)$ and $O(\delta} \def\D{\Delta^2)$ terms. We can now write the reversibility condition in terms of the above:
\begin{eqnarray}
	\lefteqn{M_{\mp}(x \pm \delta} \def\D{\Delta) M_{\pm}(x)} \nonumber \\
	& = & \left( \frac{\mathds{1}}{\sqrt{2}} + \delta} \def\D{\Delta M^{(1)}_{\mp}(x \pm \delta} \def\D{\Delta) - \frac{\delta} \def\D{\Delta^2}{2} M^{(2)}_{\mp}(x \pm \delta} \def\D{\Delta) \right) \nonumber \\
	& & \cdot \left( \frac{\mathds{1}}{\sqrt{2}} + \delta} \def\D{\Delta M^{(1)}_{\pm}(x) - \frac{\delta} \def\D{\Delta^2}{2} M^{(2)}_{\pm}(x) \right) \nonumber \\
	& = & \frac{\mathds{1}}{2} + \frac{\delta} \def\D{\Delta}{\sqrt{2}} \left( M^{(1)}_{\mp}+ M^{(1)}_{\pm} \right) \label{eqn:O2toappendix} \\
	& &  - \frac{\delta} \def\D{\Delta^2}{2\sqrt{2}} \left( \mp 2 \partial_x M^{(1)}_{\mp}+ M^{(2)}_{\mp} + M^{(2)}_{\pm} - 2\sqrt{2}M^{(1)}_{\mp}M^{(1)}_{\pm} \right), \nonumber
\end{eqnarray}
where we've dropped $x$-dependence in the last line for legibility. First, for the $O(\delta} \def\D{\Delta)$ term we find
\[ M^{(1)}_{\mp}+ M^{(1)}_{\pm} = i \sqrt{2} \langle H'_{PS} \rangle_0 = i \sqrt{2} H_3. \]
We provide the calculations for the following $O(\delta} \def\D{\Delta^2)$ terms in Appendix~\ref{sec:miscellaneouscalculations}:
\begin{eqnarray}
	\lefteqn{M^{(2)}_{\mp}+ M^{(2)}_{\pm} - 2\sqrt{2}M^{(1)}_{\mp}M^{(1)}_{\pm}} \nonumber \\
	& = & \sqrt{2} \left( 2 (H_2^2 +H_3^2) + i \left[ H_1 \pm H_3, H_2 \right] \right. \nonumber \\
	& &  \left. - i \left\{ H_1, H_2 \right\} \pm \left[ H_3, H_1 \right] + 4ce^{i \psi} H_2 \right) . \label{eqn:d2terms}
\end{eqnarray}
Expanding the $\partial_x M^{(1)}_{\mp}$ term is a bit more complicated since it corresponds to an infinitesimal rotation of the probe basis at each value of $x$. We can define an axis of rotation on the Bloch sphere with three components $\vec{\O}(x) = \left[ \O_1(x)\ \; , \; \O_2(x) \; , \; \O_3(x) \right]$, so that $\partial_x \vec{H}'(x) = \vec{\O}(x) \times \vec{H}'(x)$. This implies that
 \[ \mp 2 \partial_x M^{(1)}_{\mp} = \sqrt{2} \left[ i, -1, \mp i \right] \cdot \left( \vec{\O} \times \vec{H}' \right), \]
where we've ignored any term proportional to the identity operator as these automatically satisfy the reversibility condition. Altogether these reductions yield the expression of interest for the reversibility condition,
\begin{eqnarray*}
    \lefteqn{M_{\mp}(x \pm \delta} \def\D{\Delta) M_{\pm}(x)} \\
    & = & \frac{\mathds{1}}{2} + i \delta} \def\D{\Delta H_3 - \frac{\delta} \def\D{\Delta^2}{2} \Big\{ \left[ i, -1, \mp i \right] \cdot \left( \vec{\O} \times \vec{H}' \right) \\
    & & + 2 (H_2^2 +H_3^2) + i \left[ H_1 \pm H_3, H_2 \right]  \\
    & & - i \left\{ H_1, H_2 \right\} \pm \left[ H_3, H_1 \right] + 4ce^{i \psi} H_2 \Big\} .
\end{eqnarray*}

Finally, we group terms into four types: constant-Hermitian $A$, stochastic-Hermitian $B$ (that is, with a factor of $\pm$), constant-anti-Hermitian $i\bar{A}$, and stochastic-anti-Hermitian $i\bar{B}$:
\begin{equation}
	M_{\mp}(x \pm \delta} \def\D{\Delta) M_{\pm}(x) = \frac{\mathds{1}}{2} + i \delta} \def\D{\Delta H_3 - \frac{\delta} \def\D{\Delta^2}{2} \left\{ A \pm B + i\bar{A} \pm i \bar{B} \right\}, \label{eqn:revers}
\end{equation}
where
\begin{eqnarray*}
    A & = & - [ \vec{\O} \times \vec{H}' ]_2 + 4c \cos \psi H_2 + 2H_2^2 + 2H_3^2 + i \left[ H_1, H_2 \right], \\
    B & = & - i \left[ H_2, H_3 \right], \\
    \bar{A} & = & [ \vec{\O} \times \vec{H}' ]_1 + 4c \sin \psi H_2- \left\{ H_1, H_2 \right\}, \\
    \bar{B} & = & -[ \vec{\O} \times \vec{H}' ]_3 - i\left[ H_1(x), H_3(x) \right].
\end{eqnarray*}
If the reversibility condition is to be satisfied then these, along with the $O(\delta} \def\D{\Delta)$ term, must each be individually proportional to $\mathds{1}$. This can be done either by restrictions on the Hamiltonian terms, or by canceling the terms through unitary pulses applied when the random walk changes direction.

To eliminate the $H_3$ term, we must either set $H_3 \propto \mathds{1}$, or perform a weak unitary pulse of the form $U_1 = \exp \left( 2i\delta} \def\D{\Delta H_3\right)$. As it happens, setting $H_3 \propto \mathds{1}$ does not change the analysis that follows and thus we'll assume instead that the experimentalist performs the pulse $U_1$ at each reversal of the walk direction.

Next, we assume that both the $\bar{A}$ and $\bar{B}$ term can be eliminated via a series of weak unitary pulses of the form $U_2 = \exp \left( i \delta} \def\D{\Delta^2 H \right)$ where $H$ is some Hermitian operator containing linear combinations and products of $H_X$, $H_Y$, and $H_Z$. This leaves only $A$ and $B$ terms:
\begin{eqnarray}
    A & = & - [ \vec{\O} \times \vec{H}' ]_2 + 4c \cos \psi H_2 + 2H_2^2 + i \left[ H_1, H_2 \right] \label{eqn:A} \\
    B & = & - i \left[ H_2, H_3 \right] \label{eqn:B} .
\end{eqnarray}
Since $B$ is traceless it cannot be proportional to the identity and must be set to $0$. However, $A$ is not traceless and we must consider a more complicated solution, one where $A$ is equal to $\alpha \mathds{1}$  for some constant $\alpha$:
\[ \O_3H_1 - \O_1H_3 + 4c \cos \psi H_2 + 2H_2^2 + i \left[ H_1, H_2 \right] = \alpha \mathds{1} . \]
Using lemma~\ref{lem:commutatoridentity} in the appendix we find that
\[ \O_3H_1 - \O_1H_3 + 4c \cos \psi H_2 + 2H_2^2 = \alpha \mathds{1} \]
also implies that $\left[ H_1, H_2 \right] = 0$. Together with the commutation relation from $B$, this means that we can now express all Hamiltonian terms in one common diagonal basis:
\begin{equation}
	\label{eqn:allHPS}
	H_{PS} = \sum_j \left( Xx_j(x) + Yy_j(x)+ Zz_j(x) \right) \ketbra{j(x)} .
\end{equation}
In order to satisfy the condition $A \propto \mathds{1}$, we must consider the diagonal components of $H_2$ as they appear in equation Eq.~\eqref{eqn:A}:
\begin{equation}
	\label{eqn:riccati}
	\partial_x y_j(x)  = q_0(x) + q_1(x) y_j(x) +  q_2(x) y_j^2(x),
\end{equation}
where $q_0(x)=\alpha$ from above, $q_1(x) = -4 c(x) \cos \psi(x)$, and $q_2(x) = 2$. This differential equation is a special instance of the Riccati equation, the solution to which can be found in~\cite{murphyODEs}. In particular, if any solution, $y^{(1)}(x)$, is known, then the general solution is of the form
\[ y_j(x) = y^{(1)}(x) + \frac{\Phi(x)}{C_j - \int q_0(x) \Phi(x) dx} .\]
for $\Phi(x) = \exp \int 2q_0(x)y^{(1)}(x) + q_1(x) dx$. The important feature of this solution is that there is only one free parameter $C_j$ available to match any boundary condition.

To complete the proof, we focus on which instantaneous measurements $M_1$, $M_2$ are achievable at the end points of a continuous decomposition. First, note that all terms in Eq.~\eqref{eqn:allHPS}, including the diagonal basis $\ketbra{j(x)}$, are assumed to be $x$-dependent. Consider the unitary $U(x)$ which diagonalizes $H_1(x)$, $H_2(x)$, and $H_3(x)$. Each of these is a linear combination of $H_X$, $H_Y$, and $H_Z$, and since they are all linearly independent, $U(x)$ must also diagonalize $H_1$, $H_2$, and $H_3$. Whatever unitary does \emph{this}, however, cannot depend on $x$, and therefore, the basis $\ketbra{j}$ is \emph{not} $x$-dependent. Only the $y_j(x)$, $x_j(x)$, and $z_j(x)$ coefficients depend on $x$. This means that every step operator is diagonal in the same basis, and we can write the general form
\begin{eqnarray*}
	M_{\pm}(x) & = & \frac{\mathds{1}}{\sqrt{2}} \mp \frac{\delta} \def\D{\Delta}{\sqrt{2}} \sum_j \left( y_j (x) - c(x) \cos \psi(x) \right) \ketbra{j}\\
	& &  + \frac{\delta} \def\D{\Delta}{\sqrt{2}} \sum_j i \left( z_j(x) \pm x_j(x) \pm c(x) \sin \psi(x) \right) \ketbra{j}.
\end{eqnarray*}
Thus the endpoint measurement operators must also be diagonal in the $j$ basis, and the first of these has the form
\begin{eqnarray*}
	M_1 & \propto & \underset{\delta} \def\D{\Delta \rightarrow \infty}{\lim} \prod_{j=0}^{\lfloor X/\delta} \def\D{\Delta \rfloor} M_+ (j \delta} \def\D{\Delta) \\
	& \propto & \underset{\delta} \def\D{\Delta \rightarrow \infty}{\lim} \prod_{j=0}^{\lfloor X/\delta} \def\D{\Delta \rfloor} \diag{1 - \delta} \def\D{\Delta y_j(x) + i \delta} \def\D{\Delta \left( z_j(x) + x_j(x) \right)} \\
	& = & \diag{\exp \left( \int_0^X -y_j(x) + i \left( z_j(x) + x_j(x) \right) dx \right)}.
\end{eqnarray*}
where the notation $\diag{\cdot}$ represents a diagonal matrix with entries indexed by $j$. Both $x_j(x)$ and $z_j(x)$ only contribute a total phase to each of the diagonal elements. If we let $w_j^{(1)} = \int_0^X z_j(x) + x_j(x) dx$ and $W_1 = \diag{\exp i w_j^{(1)}}$ then
\[ M_1 \propto W_1 \cdot \diag{\exp \left( - \int_0^X y_j(x) dx \right)} .\]
Following a similar procedure, we find that
\[ M_2 \propto W_2 \cdot \diag{\exp \left( \int_{-X}^0 y_j(x) dx \right)} \]
with $W_2$ defined accordingly. Recall however, that these must form a complete measurement, and so they must satisfy $M_1^{\dag}M_1 + M_2^{\dag}M_2 = \mathds{1}$. This condition restricts the parameter $C_j$. Consider the $j^{\text{th}}$ diagonal entry of $M_1^{\dag}M_1$ (up to an overall normalization identical for all $j$),
\begin{eqnarray*}
	\lefteqn{\exp \left( - 2 Y(X) - 2 \int_0^X \frac{\Phi(x)}{C_j - \int q_0(x) \Phi(x) dx} dx \right)} \\
	& = & \exp \left( - 2 Y(X) - 2 \int_{\Phi^{-1}(0)}^{\Phi^{-1}(X)} \frac{d \Phi}{C_j - \int q_0(\Phi^{-1}(x)) d\Phi} \right) \\
	& = & e^{-2Y(X)} \left( \frac{C_j - Q(0)}{C_j - Q(X)} \right)^2,
\end{eqnarray*}
where we've implicitly defined
\[ Y(X) = \int_0^X y^{(1)}(x) dx \hspace{0.25in} Q(X) = \int q_0(x) \Phi(x) dx \]
It is not difficult to see, then, that the expression for each diagonal element is quadratic in $C_j$ and has only two solutions. In fact, even if one were to consider two arbitrary boundaries for the random walk, i.e.: that $-X$ be replaced with $X_1$ and $+X$ be replaced with $X_2$, then the expression for $C_j$ would still be quadratic and, again, yield only two possible solutions.

Finally, we can group the diagonal elements of $M_1$ and $M_2$ in terms of the two possible values of $C_j$ and express the entire measurement operators as the linear combination of two orthogonal projectors. In order to complete the proof of Theorem~\ref{thm:MAIN}, we must justify the appearance of unitaries $U_1$, $U_2$ and $V$. First, the rotation $V$ is simply any rotation that an experimentalist applies to the system $S$ before beginning the measurement procedure. For this reason, it does not depend on the measurement outcome. On the other hand, $U_1$ and $U_2$ are rotations applied to the system \emph{after} the continuous measurement procedure is complete and do not have to be identical. We can absorb the diagonal unitary matrices $W_1$ and $W_2$, which accumulate over the continuous procedure, into the definitions of $U_1$ and $U_2$ respectively.

\subsection{\label{sec:pulseconstraints}Constraints on weak unitary pulses}
In the section above we made repeated use of weak unitary pulses to reduce the equations for reversibility. However, some of these were generated by Hamiltonians with products of $H_1$, $H_2$, and $H_3$. Since we aim for our result to apply for a general but fixed interaction Hamiltonian $H_{PS}$ or, equivalently, three system Hamiltonian terms $H_X$, $H_Y$, $H_Z$, it may be too demanding to assume that a set of unitary pulses generated by their products would also be readily available. For this reason, we now consider satisfying the reversibility condition again, but only allowing weak unitary pulses generated by linear combinations of $H_X$, $H_Y$, and $H_Z$.

We examine this more restricted set of solutions to the reversibility condition by reintroducing the constraints on $H_1$, $H_2$, and $H_3$ that we removed in the previous section by using a weak unitary pulse generated by their products. In particular, we reintroduce the conditions on $\bar{A}$ and $\bar{B}$ that required
\[ \left\{ H_1, H_2 \right\} \propto \mathds{1}, \hspace{0.5in} \text{and} \hspace{0.5in} \left[ H_1, H_3 \right] \propto \mathds{1}. \]
The second of these is already automatically satisfied. The first yields the following relationship between the eigenvalues of $H_1$ and $H_2$:
\begin{equation}
	x_j(x)y_j(x) = \gamma(x) \hspace{0.15in} \forall j,x,
	\label{eqn:linearpulseconstraint}
\end{equation}
for some $\gamma(x)$ independent of $j$.

So far our result has only placed a restriction on the number of distinct singular values that the measurement can have. Here we'll actually be able to prove something about the singular values in the interaction Hamiltonian. We define $\vec{\lambda}_j = (\lambda_j^{(x)}, \lambda_j^{(y)}, \lambda_j^{(z)})$ as the triplet of the $j^{th}$ eigenvalues of $H_X$, $H_Y$, and $H_Z$, and $\vec{\lambda}'_j(x) = \left( x_j(x), y_j(x), z_j(x) \right)$ as the triplet of the $j^{th}$ eigenvalues of $H_1(x)$, $H_2(x)$, and $H_3(x)$. In the solution from the previous section, each $\vec{\lambda}'_j(x)$ corresponds to some $\vec{\lambda}_j$ via a rotation. This rotation takes each triplet from the original basis, to the probe basis in the same way that $\ket{\sigma(x)}$ and $\bra{\Psi^{\pm}(x)}$ were rotated from the original basis to $\ket{0}$ and $\bra{\pm}$.

Since we found only two solutions for $y_j(x)$, we must also restrict $\vec{\lambda}_j$ to lie in one of two planes in the original basis. Furthermore, the restriction in Eq.~\eqref{eqn:linearpulseconstraint} requires that the vectors $\vec{\lambda}'_j(x)$ be constrained to lie on one of two lines parellel to $z$ in the probe basis. In the original basis, this restricts all vectors $\vec{\lambda}_j$ to lie on one of two parallel lines, $l_1$ and $l_2$.

If we have only two assignments for $\vec{\lambda}_j$ then any probe basis is possible so long as the resulting $y_j(x)$ match those of the solution to the Ricatti equation above. However, with three or more assignments, we can only allow probe bases related by a rotation around the axis parallel to the lines $l_1$,$l_2$. The constraint Eq.~\eqref{eqn:linearpulseconstraint} limits this even further and we require that $y_1(x) = - y_2(x)$, meaning also that $\vec{\lambda}_j = c_j \vec{\lambda}_0 \pm \vec{\lambda}_1$ for some constant $c_j$, $\vec{\lambda}_0$ parallel to the lines $l_1$, $l_2$, and arbitrary $\vec{\lambda}_1$. This last expression for $\vec{\lambda}_j$ completely restricts the interaction Hamiltonian one should use if only pulses generated by linear combination of Hamiltonian terms are available.

\section{\label{sec:unitaries}Interleaving unitaries}
In~\cite{weakuniversal} the authors generalize their result for positive measurement operators to general measurement operators by taking the polar decomposition of the endpoint measurement operators. In other words, $M_1 = V_1 ( M_1^{\dag}M_1)^{1/2}$, and similarly for $M_2$. They then construct a one-parameter family of unitary operations $\{ V(x) \}_x$ that yield $V_1$ at $x=X$, $V_2$ at $x=-X$ and $\mathds{1}$ at $x=0$. The step operators are first constructed so as to correspond to the positive operators $(M_{1,2}^{\dag}M_{1,2})^{1/2}$ and padded by unitary operators chosen from the family $V(x)$ as follows
\begin{equation}
	M_{\pm}(x) = V(x \pm \delta} \def\D{\Delta) \tilde{M}_{\pm}(x) V^{\dag}(x)
	\label{eqn:stepwithpadding}
\end{equation}
where $\tilde{M}_{\pm}(x)$ is the step operator for the positive part of the polar decomposition. We will show that in our analysis, padding the step operator with this family of unitary operators is equivalent to a shift in the $H_1(x)$ term.
Consider expanding the $\tilde{M}_{\pm}(x)$ term in Eq.~\eqref{eqn:stepwithpadding} in terms of $\delta} \def\D{\Delta$,
\begin{eqnarray*}
	M_{\pm}(x) & = & \frac{1}{\sqrt{2}} V(x \pm \delta} \def\D{\Delta)V^{\dag}(x) \\
	& & + i \delta} \def\D{\Delta V(x \pm \delta} \def\D{\Delta) \bra{\pm} H'_{PS}(x) \ket{0} V^{\dag}(x) .
\end{eqnarray*}
Recall that $V(x)$ forms a \emph{continuous} family of unitary operators, and if we let $V(x) = e^{i \delta} \def\D{\Delta G(x)}$, then
\[ V(x \pm \delta} \def\D{\Delta) V^{\dag}(x) = \mathds{1} \pm i \delta} \def\D{\Delta \partial_x G(x) .\]
This means we can summarize Eq.~\eqref{eqn:stepwithpadding} as
\[ M_{\pm}(x) = \frac{\mathds{1}}{\sqrt{2}} +  \frac{\delta} \def\D{\Delta}{\sqrt{2}} \left\{ \pm i \left( \tilde{H}_1+\partial_x G \right) \mp \tilde{H}_2 + i \tilde{H}_3 \right\} \]
where $\tilde{H}_{1,2,3} = V H_{1,2,3} V^{\dag}$. Thus, for this scheme we still recover the result of Thm.~\ref{thm:MAIN} but with a small modification. Namely, an experimentalist now has the power to introduce a shift to the $H_1$ term which contributes directly to the unitary terms $U_{1,2}$ that appear in the endpoint measurements $M_{1,2}$. It is important to note, however, that we've not affected the singular value decomposition of $M_{1,2}$, for which there are still only two distinct singular values.

\section{\label{sec:qubit}General diagonal measurement of a qubit}
While the results of this paper restrict the class of measurements that can be achieved in general by this model, it is sufficient to realize any $2$-outcome measurement on a qubit. We now consider performing a generalized diagonal measurement on a qubit via a continuous decomposition (Def.~\ref{def:continuousdecomposition}). A generalized diagonal measurement takes the form
\[ M_1 = W_1 \cdot \left[ \begin{array}{cc} \alpha&0 \\ 0&\beta \end{array} \right],  \hspace{0.25in} M_2 = W_2 \cdot \left[ \begin{array}{cc} \sqrt{1-\alpha^2} & 0 \\ 0 & \sqrt{1-\beta^2} \end{array} \right] \]
where $W_1$ and $W_2$ are unitary matrices. We will effectuate the continuous decomposition via a sequence of probe feedback loops (Def.~\ref{def:probefeedbackloop}) and the interaction Hamiltonian $Z_P \otimes Z_S$. Expressing the interaction Hamiltonian in the probe basis yields
\[ H'_{PS}(x) = X \otimes \left( n_3^x(x) Z \right) + Y \otimes \left( n_3^y(x) Z \right) + Z \otimes \left( n_3^z(x) Z \right)\]
Thus $H_2(x) = n_3^y(x) Z$ and its diagonal values are $y_1(x) = n_3^y(x)$, $y_2(x) = - n_3^y (x)$. In this case, if we return to the Riccati equation~\eqref{eqn:riccati}, we see that we must choose values of $q_0 (x)$ and $q_1(x)$ such that both $n_3^y(x)$ and $-n_3^y(x)$ are solutions. If we add and subtract the Riccati equations for the positive and negative solutions, we get instead the following two equations:
\[ 2 \left( n_3^y (x) \right)^2 = q_0 (x), \hspace{0.25in} \text{and} \hspace{0.25in} \partial_x n_3^y(x) = q_1(x) n_3^y(x), \]
and we see that our solution must be
\[ n_3^y(x) = \exp \left( \int q_1(x) dx \right), \]
with $q_0(x) = 2 \exp \left( 2 \int q_1(x) dx \right)$. The step operators take the form
\begin{eqnarray*}
	M_{\pm}(x) & = & \frac{\mathds{1}}{\sqrt{2}} \mp \frac{\delta} \def\D{\Delta}{\sqrt{2}} \left( n_3^y(x) Z - c(x) \cos \psi(x) \mathds{1} \right) \\
	& & + \frac{i \delta} \def\D{\Delta}{\sqrt{2}} \left( \left( \pm n_3^x(x) + n_3^z(x) \right) Z \pm c(x) \sin \psi(x) \mathds{1} \right)
\end{eqnarray*}
where, as we've defined before, $c(x) e^{ i \psi(x)}$ is the warping of the probe basis, and $q_1(x) = - 4 c(x) \cos \psi(x)$. We can simplify this operator by choosing $\psi(x) = 0$ and $n_3^z(x) = 0$. This forces $n_3^z(x) = \sqrt{1 - \left( n_3^y(x) \right)^2}$. The simplified step operator is
\begin{equation}
	M_{\pm}(x) = \frac{\mathds{1}}{\sqrt{2}} \mp \frac{\delta} \def\D{\Delta}{\sqrt{2}} \left( e^{-4 \int c(x) dx} Z - c(x) \mathds{1} \right) \pm \frac{i \delta} \def\D{\Delta n_3^x(x) Z}{\sqrt{2}}.
\end{equation}
This gives the first endpoint operator the following form
\begin{eqnarray*}
	M_1 & \propto & W_1 \cdot \text{diag} \left\{ \exp \left( \int_0^X \left( -e^{-4 \int c(x) dx} + c(x) \right) dx \right), \right. \\
	& & \hspace{0.69in}  \left. \exp \left( \int_0^X \left( e^{-4 \int c(x) dx} + c(x) \right) dx \right) \right\},
\end{eqnarray*}
and the second, the form
\begin{eqnarray*}
	M_2 & \propto & W_2 \cdot \text{diag} \left\{ \exp \left( \int_{-X}^0 \left( e^{-4 \int c(x) dx} - c(x) \right) dx \right), \right. \\
	& & \hspace{0.69in} \left. \exp \left( \int_{-X}^0 \left( -e^{-4 \int c(x) dx} - c(x) \right) dx \right) \right\} .
\end{eqnarray*}
If we choose the probe basis warping $c(x)$ to be
\[ c(x) = \frac{1}{2} \left( \tanh \left( x-a\right) + \tanh \left( x-b \right) \right), \]
then we recover the endpoint operators
\[ M_1 \propto W_1 \cdot \diag{e^{\int_0^X \tanh(x-a) dx}, e^{\int_0^X \tanh(x-b) dx} }, \]
\[ M_2 \propto W_2 \cdot \diag{e^{- \int_{-X}^0 \tanh(x-a) dx}, e^{- \int_{-X}^0 \tanh(x-b) dx} }, \]
where $W_1 = e^{i \theta Z}$ and $W_2 = e^{-i \theta Z}$ for some value $\theta$ resulting from the integration of $n_3^x(x)$. An appropriate choice of $a$ and $b$ will yield the desired generalized diagonal measurement:
\[a = \ln \sqrt{\frac{\tanh X + (2 \alpha - 1)}{\tanh X - (2 \alpha -1)}}, \hspace{0.15in} b = \ln \sqrt{\frac{\tanh X + (2 \beta - 1)}{\tanh X - (2 \beta -1)}}. \]
The following choice of probe basis corresponds to the values of $\vec{n}_3$ described above:
\[ \left[ \begin{array}{c|c|c} & & \\ \vec{n}_1 & \vec{n}_2 & \vec{n}_3 \\ & & \end{array} \right] = \left[ \begin{array}{ccc} 0&-c(x)&\sqrt{1-c(x)^2} \\ 0 & \sqrt{1-c(x)^2}& c(x) \\ 1&0&0 \end{array} \right] \]
Figure~\ref{fig:ZZ} shows a simulation of this scheme for $\alpha = 0.8$, $\beta=0.2$ where the initial state of the system qubit is $\ket{\Psi}=\ket{+}$.
\begin{figure}
	\begin{subfigure}[b]{\linewidth}
		\fbox{\includegraphics[scale=0.9, clip=true, trim= 175 270 175 310]{simple-Blochwalk.pdf}}
		\caption{}
	\end{subfigure}

	\begin{subfigure}[b]{0.49\linewidth}
		\fbox{\includegraphics[scale=0.45, clip=true, trim= 170 265 195 290]{simple-pointerwalk.pdf}}
		\caption{}
	\end{subfigure}
	\begin{subfigure}[b]{0.49\linewidth}
		\fbox{\includegraphics[scale=0.45, clip=true, trim= 170 265 195 290]{simple-amplitudewalk.pdf}}
		\caption{}
	\end{subfigure}
	\caption{\emph{(color online)} (a) At the beginning of the process the system qubit is in the $\ket{+}$ state, indicated by the green vector. The continuous measurement procedure causes the state to walk along the blue curve on the surface of the sphere, sometimes reversing direction and doubling back along it. At the end of the process, the system qubit reaches the state $M_2 \ket{+} / p_2$. (b) The random walk undertaken by the pointer variable $x$, illustrated by the blue line, ends when the value of $x$ reaches either of the boundaries illustrated by the red lines. (c) The amplitudes of the state evolve towards their post-measurement values. }
	\label{fig:ZZ}
\end{figure}

\section{\label{sec:conclusions}Conclusions}
In this work, we've shown that a continuous decomposition of a two-outcome quantum measurement using a probe qubit and a constant interaction Hamiltonian can only yield measurements with two distinct singular values. In the qubit-to-qubit scheme of section~\ref{sec:qubit} this corresponds to a biased diagonal measurement of the system qubit. Of course, if we consider the recipe outlined in~\cite{weakuniversal} and use our decomposition of two-outcome measurements in sequence to give rise to an $n$-outcome measurement, then this larger measurement procedure can contain operators with $n$ distinct singular values.

The restriction to two singular values is a direct consequence of the reversibility condition (Eq.~\eqref{eqn:reversibility}). This condition, however, is a critical piece of the construction as it guarantees that the desired ``strong'' quantum measurement is faithfully produced at the endpoints. Without the reversibility condition, the continuous measurement procedure is not guaranteed to halt.

In some sense, our scheme is a restricted version of the large-ancilla continuous measurements in~\cite{nonlocality}. Although high-dimensional unitary rotations are not a limiting requirement for quantum computers, for individual quantum systems they can still be restrictive. Our scheme reduces the requirement on the number of probes that need to be simultaneously entangled with the system to one probe for a brief interaction time. It also characterizes the possible measurements in terms of the interaction Hamiltonian of the quantum system. The scheme presented here could be of use for generalized measurements in some types of qubits, such as superconducting qubits~\cite{superconductingnotes}.

Although we've only analyzed \emph{probe} feedback here, we can also analyze \emph{Hamiltonian} feedback, where control parameters of the Hamiltonian become functions of the pointer variable $x$. In this case, the reversibility condition restricts not only the probes and detector states but also the values assigned to the controls in the interaction Hamiltonian. A detailed analysis of this scheme is forthcoming.

Finally, one could also extend this continuous feedback scheme beyond qubit probes to qudit probes or even continuous probe states. One natural reason for doing so would be to embed the control parameters of a Hamiltonian, as described above, into free parameters of the probe state. Alternatively, schemes such as~\cite{generalizedstochastic} provide a framework for decomposing $n$-outcome measurements into a single stochastic process and this framework could be extended to the interacting probe case using higher-dimensional probes.

\begin{acknowledgments}
JF and TAB thank Daniel Lidar and Ognyan Oreshkov for useful discussions.  This research was supported in part by the ARO MURI under Grant No. W911NF-11-1-0268.
\end{acknowledgments}

\bibliographystyle{apsrev4-1}

redpajama_set_name: RedPajamaArXiv


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The pills are produced in bathroom-sized labs on the Myanmar side of the border, according to a Bangladeshi counter-narcotic official.
Nearly 700,000 Rohingya have entered Bangladesh since August 25, joining another 300,000 already living in camps along the border for many years.
"The number of people who were used as carriers has increased due to the influx. The internal carriers have increased. Some of them are desperate just for survival," Chowdhury said.
"It is a way to make easy money," said Chowdhury, who explained that a yaba pill bought on the Myanmar border for 20 US cents can be sold at US$3-4 dollars in the Bangladesh capital Dhaka.
Prime Minister Sheikh Hasina has called for "zero tolerance" on yaba smuggling from Myanmar and ordered an increased anti-narcotics drive along the border, the coastguard officer said, speaking on condition of anonymity.


Marea here is made with help of HDR photography techniques. To get the most colour out of every pixel! Really like the underwater like result.
This entry was posted on Wednesday, April 29th, 2009 at 22:13	and is filed under Photography. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


text:
Capture of Fiume, 3rd July 1813
Capture of Fiume
Part of : The Napoleonic Wars (1803 - 1815)
Previous action : Surrender of Guipanna and Mezzo 17.6.1813 - 22.7.1813
Next action : Reduction of San Sebastien 10.7.1813 - 8.9.1813
Milford (74) 1809-1846
John Duff MarklandBritish
Fleet Flagship
Eagle (74) 1804-1926
1918 Renamed "Eaglet"
Charles CowleyBritish
Elizabeth (74) 1807-1820
Bacchante (38) 1811-1858
William HosteBritish
Haughty (12) 1804-1816
Unrated Gun-brig
Notes on Action
DesctiprionTRN5
In 1813 Fiume was not a place of as much strength as it is now. It was nevertheless of much local importance. On July 3rd, Rear-Admiral Thomas Francis Fremantle, in the Milford, 74, Captain John Duff Markland, with the Elizabeth, 74, Captain Edward Leveson Gower, Eagle, 74, Captain Charles Rowley, Bacchante, 38, Captain William Hoste, and Haughty, 12, Lieutenant James Harvey, weighed from an anchorage about four miles from the port. Dropping the Haughty and a division of boats to storm the mole-head battery, the other ships proceeded to attack the sea-face batteries, which mounted 15 heavy guns. Owing to a shift of wind, the arrangements could not be entirely carried out as intended; but, after the Eagle had silenced one battery, Fremantle made the signal to storm. Rowley, leading in his gig, took the fort, which he had silenced; Hoste, with Marines from the Milford, occupied another battery, which had been evacuated after a cannonade by the Milford and Bacchante. Having turned the guns of his battery against the other works, Rowley dashed through the town, regardless of the fire from the windows, and from a field piece in the main street, and chased the defenders into a large house in the chief square. By that time a party from the Milford, under Markland, had begun to open with carronades upon the building, whereupon the French gave way and fled. Hoste and Rowley joining, the remaining batteries, with the field piece, stores, and shipping, were taken possession of. In all this fighting, only 1 of the attackers was killed, and but 6 were wounded. Of the 90 vessels captured, about half were handed back to their owners, 13 sent to Lissa, and the rest destroyed. Ere the place was abandoned the guns in the batteries were rendered useless.
TRN5The Royal Navy : a history from the earliest times to the present Vol VWilliam Laird Clowes, Digital Book


The best songs are probably the first two and "Every Breath You Take," although all of them are really awesome.
videos are immensly disturbing, so be careful.
be that melody that you fall asleep singing to.
Seeing them in concert is a life-changing experience because the passion that he pours through his music is intensely moving.


There's something lovely about Spring that makes me feel enthused and excited about being outside, planting things, enjoying foods from the garden and so on. What about you?
My name is still on the allotment waiting list, so for now I'm making do with planting sweet smelling herbs and flowers, and a few easy container vegetables in my tiny patio garden, and a few bits on windowsills.
First of all lets talk edibles – you know I'm always hungry! Last years Strawberry plant has somehow doubled in size over the winter despite very little attention from me, and it seems to have sprouted a few siblings as well – it's coming on nicely already, so I'm expecting a pretty good yield of strawberries in a few months. If the birds don't get to them first.
I'm bringing back the favourite tumbling toms in a couple of hanging baskets – these were delicious last year, and despite me not knowing about them preferring shade, I did well to have a constant yield right the way through from April to October. Fingers cross for the same this year!
In my herb "garden" (read: 2 massive pots) I've managed to sustain basic, corriander, parsley, dill, chives and lemon balm all year, plus a sage bush and a diddy bay tree thrive on the path too – so I'm all set for English garden spices. If only I could grow Cinnamon!
Finally, the last edible I'm feeling brave enough to grow from seed is a batch of chilli – hanabero and birds eye. As a novice last year I got them up to a decent size then popped them outside without protection. They were (obviously) dead within a week. Lesson learned, and this year they have lots of lovely windowsil space!
Now, on to scented florals and colourful shrubs – I want out patio to look & smell like something out of a wonderland come July!
I love having lavender and rosemary shrubs in the back yard, and will usually line the patio with them (doesn't take much) alternating them so wherever you turn, you get a sweet smell – and when the lavender starts to sprout, it's like a little purple paradise! So, with that in mind, I've uncovered the bushes this weekend to see what has survived, and have a few replacements to pick up.
This will mark the first year I'm attempting tulips and peonies. I know it's a bit early for peonies, but I got a bunch of them as a gift a few months ago and they were just so lovely that I have to have a whirl! The tulips are in, and I'm basically inspecting them every 30 minutes to see if there's any progress. What's that about a watched pot?
Finally, I was gifted some wildflower mix from Burts Bees at an event lately, so I've sewn a whole bunch of them into a few baskets, and a few pots. Hopefully they'll produce some lovely bee-friendly spreads for summertime and keep my buzzy wee friends happy.
Are you planting anything lovely this year? Tell me all about it!


text:
Q: Unable to mock httprequest for json response from HTTP trigger Azure function I want return json response hence I write this :
var nullPayloadResponse = req.CreateResponse(HttpStatusCode.BadRequest);
await nullPayloadResponse.WriteAsJsonAsync(new { Message = "Data not valid." });
But when I try to create mock call for this im getting value cannot be null error. I tried this :
var request = new Mock(context.Object);
request.Setup(r => r.Body).Returns(bodyStream);
request.Setup(r => r.Url).Returns(new System.Uri("http://localhost:7073/api/test"));
request.Setup(r => r.CreateResponse()).Returns(() =>
{
var response = new Mock(context.Object);
response.SetupProperty(r => r.Headers, new HttpHeadersCollection());
response.SetupProperty(r => r.StatusCode, httpStatusCode);
response.SetupProperty(r => r.Body, new MemoryStream());
// response.Setup(r => r.WriteAsJsonAsync(It.IsAny<byte[]>(), It.IsAny()));
return response.Object;
});
        return request;

Can someone please help how mock httprequest here ?

redpajama_set_name: RedPajamaStackExchange


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text:
Aviation in Ulster — 1909-1939
The history of powered, controlled flight in Ulster and Northern Ireland spans little more than 100 years. At no time during that period was the population of the region greater than around 2 million people yet the historical scope and importance of its aviation heritage bears very favourable comparison with other regions of Britain and Ireland. Selective germinal contributions during the first 30 years are attributable to a number of persons and organizations.
Harry Ferguson (left) — who later achieved world-wide fame as inventor of the three-point linkage and its application to the development of agricultural machinery and tractors — was the first person in Ireland and only the second native-born citizen of the United Kingdom to fly an aircraft designed and built to his own specification, at Hillsborough, County Down on
31 December 1909. Hard on his heels was Joseph Cordner, who was doing comparable things in the north-west of Ulster.
In one respect however, their achievements were eclipsed by Lilian Bland. Inspired by a postcard of Bleriot's monoplane sent from France by her uncle, Miss Bland first tested her aircraft, named Mayfly, as a glider at Carnmoney.
Then with an engine supplied by A.V Roe in the summer of 1910 at Randalstown, County Antrim, became the first woman in the world to fly an aircraft she had designed and built.
One hopes that her remarkable contribution has been largely ignored because of ignorance rather than for chauvinistic reasons.
Assembly-line production of aircraft in Ulster commenced in 1917, when Belfast shipbuilders Harland & Wolff were awarded a series of contracts for a combined total of 1,000 aircraft for the Royal Flying Corps and Royal Naval Air service. In response, during the period 1917-20, the company produced at least 600 aeroplanes, mostly DH.6 and Avro 504 types plus a comparatively small number of Handley-Page V1500 bombers.
To facilitate the test-flying of the V1500s, an airfield was constructed at Aldergrove in County Antrim and became No 16 Aircraft Acceptance Park. Whereas H & W stopped building aircraft in 1920, the experience gained was a factor which led to the creation of Short & Harland in 1936, the result of an amalgamation between the world's oldest aircraft manufacturers Short Brothers (Rochester & Bedford) Ltd and Harland & Wolff.
In 1947, following a decision by Short Brothers to cease their design and manufacturing work in England, Short & Harland took over the company which then became Short Brothers & Harland Ltd, then Short Brothers Ltd and then Short Brothers plc until taken over by Bombardier Inc in 1989.
by Ernie Cromie
Avro 504 serial C6001 manufactured in Belfast and pictured at the Harland & Wolff factory.
The fact that Short Brothers plc was considered worthy of incorporation into that huge conglomerate iis a tribute to the late Sir Philip Foreman, who was Managing Director and Chairman of Short Brothers plc at the time of his retirement in 1988 and the esteemed Patron of the Ulster Aviation Society for ten years, up until his death in 2013.
Harry Ferguson in a rather debonair pose with the 6th variant of his aircraft at Magilligan strand 1910.
Lillian Bland's Mayfly biplane