Looking for deals in Wengen?
Hotel Schönegg is a 3-star superior hotel located right in the center of car-free Wengen, which is only reachable by train. We stayed here for part of our honeymoon. The room was amazing, with beautiful views overlooking Wengen. The staff are very friendly and extremely helpful in providing suggestions for activities and sightseeing. The breakfast buffet was excellent. Wengen is a great location and great access to all surrounding towns and hiking trails. We would absolutely stay in this hotel again. Loved it!
Built in 1881, the family-run Hotel Alpenrose is the oldest hotel in the car-free Wengen, which is only reachable by train, and offers you completely renovated rooms in a traditional Alpine style and... The staff are extremly helpful. The hotel is actually located under a very big slope and if you call the owner, they would pick you up and send you to the train station. The food here is delicious. They also serve dinner with reasonable price. Not to mettion the scenery here.
The Arenas Resort & Spa Victoria-Lauberhorn, built in 1897, is centrally located next to the train and cable car station of the car-free Wengen and features up-to-date spa facilities including a... Perfect location , very good staff.
Hotel Schönegg is a 3-star superior hotel located right in the center of car-free Wengen, which is only reachable by train. We stayed here for part of our honeymoon.
The Beausite Park Hotel in the car-free village of Wengen is a little, sunny paradise at the foothills of the Eiger, Mönch and Jungfrau mountain range offering superb views, fine cuisine and great spa... Comfortable rooms, great staff, great views, and food was outstanding.
Which hotels in Wengen are good for families?
Many families visiting Wengen loved staying at Hotel Schönegg, Hotel Edelweiss, and Beausite Park Hotel.
How much does it cost to stay in a hotel in Wengen?
On average, 3-star hotels in Wengen cost $282 per night, and 4-star hotels in Wengen are $391 per night. If you're looking for something really special, a 5-star hotel in Wengen can be found for $435 per night, on average (based on Booking.com prices).
Which hotels in Wengen are good for couples?
These hotels in Wengen are highly rated by couples: Hotel Edelweiss, Hotel Alpenrose Wengen, and Hotel Schönegg.
Which hotels in Wengen offer an especially good breakfast?
For hotels in Wengen that serve highly-rated breakfasts, try Hotel Schönegg, Beausite Park Hotel, and Hotel Bären.
Which hotels are the best ones to stay at in Wengen?
Hotel Schönegg, Hotel Edelweiss, and Hotel Alpenrose Wengen are some of the popular hotels in Wengen.
Which hotels in Wengen have nice views?
Hotel Schönegg, Hotel Maya Caprice, and Beausite Park Hotel got great room view-related reviews from travelers in Wengen.
We cannot say enough good things about the Berghaus in Wengen. This town is a wonderful center point for hiking in the summer. Snow still affects where you can walk until mid-June, but the masses of crocus covering ground where the snow has just melted are a thrill.
This town was an amazing experience in the Jungfrau Ski Region of the Swiss Alps. The people, the mountain views were spectacular. The restaurant options and shopping were also wonderful. The train, ski lift, and the gondola were all perfect.
Wengen is a quieter town and the perfect, central location to partake in all that the region has to offer. There is a Coop grocery store across the street from the train station, but note the hours - it is not open late.
Beautiful! If you like to hike then this area has many options to offer and easy access to trailheads either out the front door or the many forms of transport. Jungfrau pass works well and seems expensive but if you use various transportation it will cover the cost quickly. If you have a Swiss half card or rail pass you get a discount on the Jungfrau pass. Have been in this area four times now over the past 11 years and I will return again.
Beautiful area, great hotel with spectacular views.
Beautiful area, great hotel with spectacular views. Lots of hiking opportunities and excursions. We absolutely loved this city and the hotel and stay was the highlight of our trip. You won't be disappointed!
Absolutely beautiful and close to a bunch of great hikes.
Absolutely beautiful and close to a bunch of great hikes. Switzerland is obviously expensive and the trains to get around from here were even more pricy than we expected. It is much more beautiful than interlaken and feels less touristy, even though the town is mostly hotels.
Warm environment and kind staffs. Excellent breakfast and dinner. Dinner has four courses with only 20 francs to the hotel guests. We stayed for three nights and never ate same courses and looked forward to the dinner on our way back to hotel everyday. Glad to see the distinguished history make the hotel progress continuously. Wengen is a small town with no outside cars. The hotel's small shuttle can take guest to and from the railway station.
Undoubtedly one of the best experience of amazing Wengen,Switzerland . Staying in the iconic chalet with it's beautiful architecture was a great feeling. Stunning view of the the snow caped mountain and the green lawns was just superb! Mr Michel and his wife were very accommodating & courteous host. All in the family including my daughter had a super time. Keep up the great work. Looking forward to visiting again.
I had two great nights in this hotel. I love hear. And the view is amazing. I had book only one night on Booking.com. Then we decided to stay one more night. They agreed immediately. The room is quite clean and we got anything we need. I will recommend my friends to choose this hotel if they want to travel in Switzerland. Btw, we got a pick-up service when we arrived the train station at 9:30.
The most beautiful hotel i've stayed:) I stayed in 'lauberhorn suite' for 3 days. The room was very clean and antique stylish. This hotel has astonishing view that you can see Jungfrau in the room or in the balcony of this hotel. The location is also good that you can reach the hotel from Wengen station in 3 minutes. Thanks to helpful staffs i could enjoy this town much easily.
Best stay ive had in a hotel by far. Cant believe its only a 3 star hotel. Very service oriented hotel. The staff were pleasant and helpful. I had booked 3 individual rooms but the rooms were surprisingly bigger than expected. The toilet was super clean. The view from the room spectacular. And very easy to get around in wengen. I hope to be back someday soon!
The views were absolutely breathtaking, the staff were amazing 10/10 and willing to help in every way they could to make our stay memorable, the hotel is a short walk from lovely restaurants and the train station aswell, would highly recommend this hotel if you are staying in Wengen. We would definitely stay again. Thank you.
Nice views of mountains. Decent bathroom. Nice front desk staff helped us buy tickets (don't go to Top of Europe unless weather is clear). Amazing breakfast in hotel!!!
Best views in Wengen. Best team, cannot ask for more than what the management will offer.
Quiet location. Near wengen train station. Close to coop. Hotel admin staff is nice.
The bed was comfortable and breakfast was a good variety. The view from the balcony was great.
Everything, the view, the staffs, cleanliness, breakfast, i can't even tell.


text:
Reading Transparency Report #13: underwater is breathtaking
Next Rethinking the way we travel and wander
Transparency Report #13: underwater is breathtaking
While we were standing still and burning rubber in our previous report, we now find ourselves with our heads underwater. While we paddle and explore the "beauty" underneath, we also run with less oxygen, which comes with all kinds of costs and breakthroughs.
By pixelgrade in Along our way
The last few months brought on a pace we never had in almost a decade. We're slower, more exhausted, more scattered. It's a reality we could sense on many levels within Pixelgrade.
The most direct effect is reflected in our cash flow because the revenue got thinner and more erratic. The not-so-obvious, yet important ripples are across the team. In the first half of 2021, some of our colleagues were sick, had faced unpleasant events, so their energy and drive slipped accordingly.
A small crew of only eight people is a fragile system that absorbs vibrations more intensely when someone is out of duty.
Swimming in the sea comes with both serenity and burden. The water is calming, the horizon line brings hope and tranquillity, the sun heats our bodies and takes the pressure off our shoulders. But there's also uncertainty and tiredness, physical and emotional. It's hard to keep the tempo, be safe, maintain a certain amount of control.
As if by a reverse-miracle, some unfathomable hand decided we didn't have enough and we should swim with our heads underwater, for added urgency and drama.
As with most mammals out there, we couldn't live underwater for too long. We have limited resources in our bodies. They are finite; it's that simple. As much as we find meaning in embracing conundrums and complex challenges, there's a point where we need to ask ourselves how much we can carry on.
In this report, we try to depict an honest image of our becoming in this specific moment in time and space. As we've always done, we are committed to be transparent and surface the whole landscape (yes, the cracks, too). In the end, stormy seas (and unfathomable hands) are part of any organization's path, so why try to hide them?
Quick navigation in this article:
Overview of the deck
Revenue and Expenses
Team momentum changes
An updated version of the employee handbook
The first book signed by Upstairs Community is here
We have a new business partner, captain!
The next big wave
"Thought is the wind, knowledge the sail, and mankind the vessel."
August Hare
A quick yet frank summary of the first half of 2021 comes with a sharp message: we made little progress. There are two aspects that weighed heavily on us, and we think it's essential to provide a broader context for each.
First of all, the blunt truth is that we've built various pieces of a giant ship at once. We're not done with any of them, so that's one of the main reasons why we stood silent and avoided making a big fuss about it. We don't like gathering a big crowd, cutting the ceremonial ribbon, and show up with only a few parts of a working product. It would be messy and frustrating for both us and our potential customers.
Second of all, despite this conscious decision to keep things under the radar, there's no doubt that all the events we've been through, both personally and collectively, left marks on our focus and performance. From there, the domino effect rapidly occurred.
It's not at all comfortable to acknowledge and live with any of the above, but it's part of our shared reality. We showed up one day after the other, doing our best to put our ducks in a row. Sometimes we succeeded, but unfortunately, we faced plenty of setbacks. For us, it's crystal clear that fatigue, along with less oxygen and weak bodies, brought us here today. It's a spot with mixed feelings, but it's not the end of the journey.
Six months ago, we had one of the smoothest and most predictable sailings of all time. While not unexpected, now it seems like we are taking a look at the seabed. Hopefully, it's just another journey of discovery which we will swim through successfully without emptying the oxygen tanks. It won't be a quick bounce to the surface, but we will do our best.
When it comes to actual numbers, the last six months look like this:
$29,187 total monthly average revenue (–6.5% down from $31,217)
↳ $13,359 from our shop at Pixelgrade.com (-7.3% down from $14,414)
↳ $12,557 from the WordPress.com marketplace (-3% down from $12,943)
↳ $3,157 from the Envato marketplace (–10% down from $3,510)
Pixelgrade monthly revenue evolution of the first six months of 2021
Murky waters and all, we thought we had been alone underwater, but maybe we weren't. Speaking with other people from the WordPress community, it looks like it's an overall drop in the field of traditional WordPress themes, sales being affected regardless of whether the products are available through a marketplace or independent shops (like ours).
One likely reason might be the impact of shifting to the new Gutenberg editor and the confusion created by the way the future of WordPress themes might look like, taking some prospective customers aback. Some just sit-and-wait for things to become clearer, while others look to other solutions, hopefully only temporarily. I think this attitude will change once the trend of creating block-based sites in WordPress will get more traction.
Our average monthly expenses increased slightly to $27,489 (+6% from $25,934), broken down into:
→ 76% of salaries
→ 19% maintenance (rent, software, accounting fees, suppliers)
→ 5% occasional expenses (office remodeling, hardware)
The numbers and percentages are in line with the previous months with a slight increase in salaries and maintenance costs.
Happenings within the crew
1. Team momentum changes
Regardless of how eerie or murky the waters are, life has its flow and rhythm. We cannot oppose it. Whatever each of our teammates experienced in the last year or so twisted the perception on big topics, such as life, death, love, impact, and so on. All of them leaned and weighed on Pixelgrade's ship.
We've been eight people at Pixelgrade for almost three years now. Mădălin (front-end developer) and Alex (customer support) are the last comrades to come along and have stuck with us ever since. Besides them, we did not bring anyone new on board, which is comforting and delicate at the same time.
On the one hand, we're aligned with our culture and core values; we have a common understanding of the organizational intricacies; we know how to handle most of the situations we face without feeling awkward or insecure. It saves time and mental bandwidth, which is a-m-a-z-i-n-g.
On the other hand, the current team size, along with the fact that we're not students anymore and our lives got more complex with time, makes tackling big projects less sustainable. Wherever someone takes the paddles off, the entire ship shakes.
In a way, it's ironic.
Four or five years back, we were younger and with fewer dilemmas, which meant that we could have pushed the boundaries with impunity. In a way, we did that, barely. Now, with more significant issues on our table due to adulthood, we don't have the same drive and stamina to maintain a certain velocity. It feels like we missed something on both ends.
Regardless of how cynical it might sound, a company does not expect anyone to take time for its needs. Pixelgrade, like any organization out there, requires constant fuel and dedication. Once we stop, the damage is very broad, especially in today's WordPress ecosystem. We don't have a solution nor a magic recipe, but it's definitely something we need to sleep on in the next few months.
2. An updated version of the employee handbook
During the first months of 2021, Oana reopened the conversation about the perks we offer. The need emerged from the fact that some of the advantages we've been providing have brought little to no value, while others felt de facto.
We did not capitalize on valuable perks we were giving at that moment, and others became irrelevant.
The updated version of the house book it's a comprehensive resource that functions as a standard reference for everyone at Pixelgrade. It showcases the current moment we are as a team and business, alongside our particular needs: new benefits anchored in today's reality and a few rituals to create stronger bonds.
Here are some specific decisions we took:
during August we work four days per week
we have three Fridays off during the year (long weekends)
we offer four days for remote work per month
we make team huddles every few months (two of our teammates live in other cities)
once a year, we sign out and spend time together, preferably in nature.
Oana documented part of her process in an in-depth article where she shares the urge to rethink such perks within our company and get a frank overview.
3. The first book signed by Upstairs Community is here
The highlight of this celebration is the publication of our first physical book, with all the 27 narratives published so far. It was a challenge from top to toe, but we're beyond grateful for walking through the process and making it happen.
It's such a special feeling to see it, to touch it, to smell it, to skim through it. At least for us, there is something absolutely mesmerizing in the materiality of things, the capability of physical creations to activate so many of our senses.
We work in the digital world, and most of the time our work does not get to see the light of the actual Sun, which is a blessing and a curse. It remains neatly tucked away in folders, files, Basecamp, GitHub, Paper, Dropbox, Drive, and other similar apps and tools.
Therefore, we lack tactility, and we pay a cost for that. For instance, it's easier to miss some progress that we've made, nor how much effort we've put into a project because we can't see the pile of papers on our desks. Publishing this book reminds us about the beauty that lies within human contact because it will reach 70 members of our community: real people who already sent us real photos with the book. Hooray!
This tiny object beautifully encompasses the Upstairs Community's spirit of generosity towards everything this adventure represents and stands for.
4. We have a new business partner, captain!
Răzvan has been with us for more than nine years, while Pixelgrade celebrates ten this fall. Without further ado, he's a pioneer and a witness to all of our cycles of transformation and growth.
As a front-end developer who proved so many times that he could come up with witty solutions for all kinds of problems, and a teammate who gets his hands dirty on various organizational endeavors, it felt right to make this jump and bring him closer to the partners' table.
It took several internal discussions to grasp his intentions and specific expectations better, but we made it happen in late spring. Our handshake confirms that we share similar goals regarding how we imagine Pixelgrade both in the short and long run and relief that the boat has another set of paddles.
Three years back, Oana joined forces to bring her know-how in nurturing relationships on a team level and reinforcing the culture consistently. In 2021, Răzvan got closer to bridge the gap between George and Vlad by putting his soft and hard skills at work.
We're happy about this milestone, and we genuinely believe it happened at the right time for both him and us. Congrats!
Swimming with our heads underwater managed to open our eyes in ways we never experienced before.
One of our core values is excellence, and we put it at the forefront of everything we do. Sometimes, this means looking at the opposite side of the spectrum most Valley-entrepreneurs are focused on.
When they shout out that the secret is to create MVPs and launch them fast, we take a step back and ship when we think it's good enough. We've always been embracing this attitude in the last decade, and we like to believe it's a substantial part of what brought us here today, with all the goods and the bad.
The project we've been crafting is a massive, deep-sea creature that has the power to make memorable ripples within the WordPress ecosystem and bring us closer to our mission. It's also one of the most challenging in terms of motivation across the team. The road is full of ups and downs, we iterate a lot, there's plenty of back-and-forths, and we don't release stuff loudly, which definitely impacts the squad's mojo and makes their wheels spin slower.
We're looking forward to shipping it in a couple of months and celebrate accordingly. Big time! And hey, once it is there, it's your turn to tell us if you feel safe enough to scuba dive with us and discover the fascinating beauty underneath in a way that will stick with you forever. Can't wait to enjoy such adventures together!
What gives you fuel to keep navigating whenever you encounter murky water?
Conversations 10 comments
Ionutsays:
Relevant commenter background or experience:chief failure officer
Hey! Most important for me in the past was the fact that I was not alone, as long as the other people involved wanna keep going, I would too + fear of failure. We see the same trends in terms of revenue and I see it as a continuation of 2017-2019 trend.
Oanasays:
Relevant commenter background or experience:Storyteller & marketing gal
Hi, Ionuț! 👋 Thanks for sharing your thoughts on this one.
Feeling less alone helps us relate, which definitely helps. As for the trends, I'm afraid I became quite a skeptic since we often got caught off guard. That's one of the reasons we've been investing so hard in making our brand rock-solid and offering an outstanding customer experience.
It's a neverending tango, but one thing is crystal clear: trends or not, the future is challenging and full of unknown. Best of luck, and keep it up!
Sim Filipsays:
Relevant commenter background or experience:All at sea these days
Heya! Just wanna say this is a very poetic transparency report and I love how you used the sea/water methapor throughout the article. 🌊 Plain sailing from now on!
Don't we all need more poetry in our lives? Thank you so much! Stay serene. 🤗
Chris McCoysays:
Relevant commenter background or experience:WordPress Geek
been a pixelgrade customer for years, love your products, looking at the chart did you lose money this month after paying expenses?
George Olarusays:
Relevant commenter background or experience:Co-founder at Pixelgrade
Thanks for staying with us, Chris! It's nice to see you around. Yeah, the cash flow turned negative last month.
Paul Laceysays:
Relevant commenter background or experience:Podcaster, Designer
Hi Oana, Thanks for this transparency and honesty. I'm sorry it's been a struggle. Aniut some of the numbers you have experienced, I saw this recently and it looks like your are experiencing perhaps whatever it is that caused the same trends. https://poststatus.com/active-install-growth-of-wordpress-plugins-declines/
On August 3, 2021 Reply
Thanks for your kind words, Paul! It means a lot to me. While we care about how these trends impact us, I think it's wise to focus on making our ship stronger, too. The WordPress ecosystem is a blessing and a curse at once, and that's okay. But what we need to do is to keep our promise to the customers and reach our mission. If we need a different route than what we experienced in the last decade, so be it. We're up for the challenge! 💪
Joe Robisonsays:
Relevant commenter background or experience:Marketer
Thanks for sharing this – I think you guys are too 5% in the marketplace – hope things start trending upwards soon!
Andreisays:
Relevant commenter background or experience:Marketer at Pixelgrade
Thanks for the kind words and support. 🤗 We'll do our best to change this trend for the better.
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redpajama_set_name: RedPajamaCommonCrawl


text:
What Kinda Boy You Looking For (Girl)
Hot Chocolate	Buy This Song
Hot Chocolate was a chart-topping British pop band popular during the 1970s and 1980s, formed by Errol Brown. The act had at least one hit every year between 1970 and 1984 inclusive, and their song "You Sexy Thing" made the Top 10 in three decades. more »
The easy, fast & fun way to learn how to sing: 30DaySinger.com
hey say I'm not your kind  they say that I should take you from my mind.  But that is hard to do  girl  when everything I care about is you.  What kinda boy you're lookin' for  girl?  Must he be just like your favourite movie star?  What kinda boy you're lookin' for  girl?  Could you give your heart to someone  Who is crazy about you  longing for you  loving you each day?  They say I'm far too plain  they say you're much too vain  To want this love of mine. But what else can I do  But let you know my love waits here for you?  What kinda boy you're lookin' for  girl?  Must he be just like your favourite movie star?  What kinda boy you're lookin' for  girl? Are you sure I'm not the one  Who you should give your love to? Darling  I would never break your heart@  Girl  I love you  and I want you to know  Whenever you need a love that's true  My love's waiting here for you  Girl  it's true - love's waiting here.  What kinda boy  You're looking for  girl?   Must he be just like  Your favourite movie star?   What kinda boy   You're lookin' for  girl?  Must he be just like  Your favourite movie star?   What kinda boy  You're lookin' for girl? ...
Sheet Music Playlist
Written by: ERROL AINSWORTH GLENSTOR BROWN
Discuss the What Kinda Boy You Looking For (Girl) Lyrics with the community:
"What Kinda Boy You Looking For (Girl) Lyrics." Lyrics.com. STANDS4 LLC, 2021. Web. 16 Jan. 2021. <https://www.lyrics.com/lyric/28106866/Hot+Chocolate>.
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text:
Australian Researchers Develop Tech to Turn Regular Glasses into Night-Vision Devices
Researchers in Australia have developed a new technology that uses infrared imaging to allow people to see clearly in the dark. These ultra-thin films could one day turn regular spectacles into night-vision goggles and also revolutionise several sectors, such as autonomous vehicle navigation, optical tomography, and food quality control. The film is thinner than human hair and is made of nanoscale crystals. Researchers from the Australian National University (ANU) say the prototype tech could replace the bulky night-vision goggles used by military and law-enforcement officials with a lightweight alternative, and could also make it safer to drive at night or walk home after dark.
The technology, which works at room temperature, converts infrared light into light visible to the human eye. Researchers say it's cheap and easy to produce at a mass scale, making it a preferred choice over the current high-end infrared imaging tech that requires cryogenic freezing and is costly to produce.
The first-of-its-kind thin film has been described in a new report published in the journal Advanced Photonics. It could be very useful to security personnel who currently use the heavy helmet-mounted night vision goggles that they say have led to neck pain.
"We have made the invisible visible," lead researcher Dr. Rocio Camacho Morales said in a report published on the ANU website. "Our technology is able to transform infrared light, normally invisible to the human eye, and turn this into images people can clearly see — even at distance."
Dragomir Neshev, a professor in physics at the university, said this was the first time that infrared light has been successfully transformed into visible images in an ultra-thin screen, and called it a "really exciting development".
The scientists, who had received help from European and British researchers on this project, are now working to advance the technology further. It's not yet clear when the technology would be available for commercial use.
For the latest tech news and reviews, follow Gadgets 360 on Twitter, Facebook, and Google News. For the latest videos on gadgets and tech, subscribe to our YouTube channel.
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\section*{\MakeUppercase{#1}}  \raisebox{2em}[0pt][0pt]{\rule{\textwidth}{.5pt}}}
\def\ensuremath\mathcal{\ensuremath\mathcal}
\def\ensuremath\mathbf{\ensuremath\mathbf}

\begin{document}
\title{Improved device-independent randomness expansion rates from tight bounds on the two sided randomness using CHSH tests}
\author{Rutvij Bhavsar}
\email{rutvij.bhavsar@york.ac.uk}
\affiliation{Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom}
\author{Sammy Ragy}
\email{sammyragy@gmail.com}
\affiliation{Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom}
\author{Roger Colbeck}
\email{roger.colbeck@york.ac.uk}
\affiliation{Department of Mathematics, University of York, Heslington, York, YO10 5DD, United Kingdom}
\date{$23^{\text{rd}}$ May 2021}

\begin{abstract}
A device-independent randomness expansion protocol aims to take an initial random string and generate a longer one, where the security of the protocol does not rely on knowing the inner workings of the devices used to run it.  In order to do so, the protocol tests that the devices violate a Bell inequality and one then needs to bound the amount of extractable randomness in terms of the observed violation. The entropy accumulation theorem gives a bound in terms of the single-round von Neumann entropy of any strategy achieving the observed score. Tight bounds on this are known for the one-sided randomness when using the Clauser-Horne-Shimony-Holt (CHSH) game. Here we find the minimum von Neumann entropies for a given CHSH score relevant for one and two sided randomness that can be applied to various protocols. In particular, we show the gain that can be made by using the two-sided randomness and by using a protocol without spot-checking where the input randomness is recycled.  We also discuss protocols that fully close the locality loophole while expanding randomness. Although our bounds are mostly numerical, we conjecture analytic formulae for the curves in two cases.
\end{abstract}

\maketitle
\section{Introduction}
Random numbers have a wide variety of uses. In some applications, only the distribution of the random numbers matters, while in others it is important that the generated numbers are also private, for instance when used for cryptography.  According to our current understanding of physics, generating fundamentally random numbers requires quantum processes. While it is easy to come up with a quantum process that can in theory generate random numbers, given a candidate quantum random number generator, verifying that it is indeed generating random numbers and at what rate is a difficult task that requires detailed understanding of the physical device. Any mismatch between the model of the device used in the security proof and the real device could in principle be exploited by an adversary.

Device-independent protocols aim to circumvent the mismatch problem by designing the protocol to abort unless the devices used are performing sufficiently well, without needing to know the internal mechanism by which they operate.  In the context of device-independent randomness expansion (DIRE) the main idea is that the ability to violate a Bell inequality implies that the devices doing so must be generating randomness~\cite{ColbeckThesis,CK2}. Thus, in a sense, the protocol self-tests~\cite{MayersYao} the devices during its operation, leading to enhanced security.  Although challenging to accomplish, recently the first experimental demonstrations of DIRE were performed~\cite{LZL&,Shalm_rand,LLR&}, following earlier experiments considering randomness generation~\cite{PAMBMMOHLMM,BKGZM&,LZL&gen}. On the theoretical side, an increasingly sophisticated series of proofs~\cite{VV,MS1,MS2} led to the current state of the art~\cite{ZFK,ARV}. In this work we consider the proofs based on the entropy accumulation theorem~\cite{DFR,DF}, which establishes a lower bound on the amount of randomness after many Bell tests in terms of the von Neumann entropy of a single round achieving the observed score (see Appendix~\ref{app:EAT} for a precise statement). In this work we focus on generating bounds on the von Neumann entropy, before combining these bounds with the entropy accumulation theorem to illustrate the improvement.

In a DIRE protocol, randomly chosen inputs are made to two separated devices in such a way that each device cannot learn the input chosen by the other. We use $X$ and $Y$ to label the inputs and $A$ and $B$ to label the outputs, and consider an adversary with side information $E$. This side information could be quantum, i.e., the general strategy has an adversary hold the $E$ part of a state $\rho_{A'B'E}$ where the $A'$ and $B'$ systems are held by the devices. Each input $X$ to the first device corresponds to a measurement on $A'$ giving outcome $A$ and, analogously, for each input $Y$ to the other device there is a corresponding measurement on $B'$ with outcome $B$. There are two quantities of interest, both of which depend on the post-measurement state: the first is the score in some non-local game, which is a function of the conditional distribution $p_{AB|XY}$; the second is the von Neumann entropy of either one or both of the outputs.  More precisely, we want to express the minimum von Neumann entropy as a function of the score. In this work we study six such von Neumann entropies: $H(AB|X=0,Y=0,E)$, $H(AB|XYE)$, $H(AB|E)$, $H(A|X=0,Y=0,E)$, $H(A|XYE)$ and $H(A|E)$\footnote{In the one sided cases, conditioning on $Y$ is irrelevant, but we keep this for notational symmetry.}.

The difficulty in bounding these stems from the fact that a priori there is no upper bound on the dimension of the systems $A'$, $B'$ and $E$ that need to be considered. For instance, for some Bell inequalities it is known that the maximum quantum violation cannot be achieved if $A'$ and $B'$ are finite dimensional~\cite{Slofstra}, and there is evidence that this is true even in the case where $X$ and $Y$ are binary and where $A$ and $B$ have three possible values~\cite{PV}.  However, in the case where $A$, $B$, $X$ and $Y$ are all binary, Jordan's lemma~\cite{Jordan} implies that there is no loss in generality in considering a convex combination of strategies in which $A'$ and $B'$ are two-dimensional. This observation was used to give a tight analytic lower bound on $H(A|X=0,Y=0,E)$ in terms of the CHSH score~\cite{PABGMS} and is also crucial for the present work. This analytic bound was also extended to a family of CHSH-like inequalities in~\cite{WAP}. In cases where such simplifications do not hold, we are limited to lower bounding the single-round von Neumann entropy instead.  For instance, it can be bounded using the single-round min entropy, which can be optimized at a particular level of the semi-definite hierarchy~\cite{NPA2}. Although this method can generate numerically certified lower bounds for arbitrary protocols~\cite{BRC}, there is a significant loss in tightness when moving to the single-round min entropy. More recent methods give tighter computational bounds on the von Neumann entropy~\cite{SBVTRS,BFF}. Forming more direct and tighter bounds is a key open problem in the field of device-independence and would go along way towards making device independent quantum key distribution (DIQKD) experimentally implementable.

Which of the six entropic quantities is of most interest depends on the situation. The one-sided entropies are most-relevant in the context of DIQKD, for instance, while using both outputs is useful for DIRE. The quantities that are conditioned on $X=0$ and $Y=0$ are useful for spot-checking protocols in which particular fixed measurements (taken here to be the $X=0$ and $Y=0$ measurements) are used to generate randomness/key, and where the other measurements are used only rarely to check that the devices are behaving honestly. The quantities conditioned on $X$ and $Y$ are useful for analysing protocols in which the generation rounds involve multiple settings, where the input randomness is recycled, and can also be used to close the locality loophole in randomness generation. The remaining quantities, $H(AB|E)$ and $H(A|E)$, tell us something about the fundamental randomness based on the Bell violation, and the second of these can be useful for DIQKD protocols in which the measurement settings in the generation rounds are chosen randomly and to which an adversary, Eve, does not have access.

In this paper we give numerically generated curves for each of these six quantities when using the CHSH game as the Bell test. For $H(A|XYE)$ and $H(AB|XYE)$ we additionally give conjectured analytic forms for the curves. We then use these together with the EAT to show the improvement based on finite statistics.

\section{The significance of various entropic quantities}\label{sec:signif}
In this section we discuss the significance of the six entropic quantities given above in the context of DIRE, noting that the one-sided quantities are also useful for DIQKD. To do so we first describe the general structure of the raw randomness generation part of a spot-checking and non-spot-checking DIRE protocol. A more complete description of the protocols is in Section~\ref{sec:protocols}.

In a protocol without spot-checking, there are two untrusted devices, and in every round their inputs $X_i$ and $Y_i$ are generated according to some distribution $p_{XY}$. Often two independent random number generators are used for this, so that $p_{XY}=p_Xp_Y$.  The generated numbers are used as inputs to the devices, which return two outputs $A_i$ and $B_i$ respectively. This is repeated for $n$ rounds generating the raw randomness ${\bf A}$, ${\bf B}$, where the bold font denotes the concatenation of all the outputs.

In a spot-checking protocol there is an additional step in which each round is declared to be either a test round ($T_i=1$) or a generation round ($T_i=0$), where test rounds occur with probability $\gamma$, which is typically small. On test rounds $X_i$ and $Y_i$ are generated according to some distributions $p_{XY}$. On generation rounds, $X_i$ and $Y_i$ are set according to some other distribution -- in this work we use the deterministic distribution $X_i=Y_i=0$. These are used as inputs to the devices, which return two outputs $A_i$ and $B_i$ respectively. The rationale behind using a spot-checking protocol is that randomness is required to perform a Bell test and it is desirable to be able to run the protocol with a smaller requirement on the amount of input randomness required. Choosing whether to test or not requires roughly $H_\mathrm{bin}(\gamma)$ bits of randomness per round\footnote{Here $H_\mathrm{bin}$ denotes the binary entropy.}, so choosing $\gamma$ small enough leads to an overall saving.  Furthermore, protocols often discard the input randomness, in which case for many Bell tests spot-checking is necessary in order to achieve expansion. In the CHSH game, for instance, if $p_{XY}$ is chosen uniformly, each test round requires $2$ bits of randomness, but the amount of two-sided randomness output by the quantum strategy with the highest possible winning probability is only $1+H_\mathrm{bin}(\frac{1}{2}(1+\frac{1}{\sqrt{2}}))\approx1.60$ bits. However, as we discuss later, the input randomness need not be discarded.

In the case of small $\gamma$, almost every round is a generation round and hence an eavesdropper wishes to guess the outputs for the inputs $X=0$ and $Y=0$. The entropy $H(AB|X=0,Y=0,E)$ is thus the relevant quantity for spot-checking DIRE protocols.  The one-sided quantity $H(A|X=0,Y=0,E)$ has often been used instead because of the existing analytic bound for this~\cite{PABGMS,WAP}, but, because this ignores one of the outputs, it is wasteful as an estimate of the generated randomness. For DIQKD protocols, on the other hand, the one-sided entropy is the relevant quantity, since to make key Alice's and Bob's strings should match. We also remark that these quantities can be useful bounds for protocols without spot-checking if the distribution $p_{XY}$ is heavily biased towards $X=Y=0$.

The quantities $H(AB|XYE)$ and $H(A|XYE)$ are useful for protocols without spot checking. One might imagine, for example, using a source of public randomness, such as a randomness beacon to choose the inputs to the protocol, in which case $X$ and $Y$ become known to the adversary (but are not known before the devices are prepared).  In this case, rather than being interested in randomness expansion, the task is to turn public randomness into private randomness in a device-independent way. One can also use $H(AB|XYE)$ and $H(A|XYE)$ in protocols when the input randomness is recycled. In this case we are really interested in $H(ABXY|E)$, but, because $X$ and $Y$ are chosen independently of $E$, this can be expanded as $H(XY)+H(AB|XYE)$. Hence $H(AB|XYE)$ is the relevant quantity in this case as well. The one-sided quantity $H(A|XYE)$ could also be used for DIQKD without spot-checking.

In addition we consider the quantities $H(AB|E)$ and $H(A|E)$.  The second of these could be useful for QKD protocols in which the key generation rounds do not have a fixed input and where Alice and Bob do not publicly reveal their measurement choices during the protocol.  For instance, the sharing of these choices could be encrypted using an initial key, analogously to a suggested defence against memory attacks~\cite{bckone}\footnote{Note that such protocols would only be useful if more key is generated than is required, so the protocol we are thinking of here is really quantum key expansion. Furthermore, the results presented in Figure~\ref{fig:rates} show that the use of $H(A|E)$ only gives a minor advantage over $H(A|XYE)$.}. $H(AB|E)$ would be a useful quantity for randomness generation in a protocol without spot-checking and in which $X$ and $Y$ are kept private after running the protocol and not used in the overall output.  When such protocols are based on the CHSH inequality, they cannot allow expansion.  These quantities can also be thought of as quantifying the fundamental amount of randomness obtainable from a given Bell violation. Although we have computed the graphs for $H(AB|E)$ and $H(A|E)$, existing versions of the EAT cannot be directly applied to them --- see Appendix~\ref{app:EAT}.

\section{Rates for CHSH-based protocols}
We calculate various one sided and two sided rates for protocols based on the CHSH game, which involves trying to violate the CHSH Bell inequality~\cite{CHSH}. In this game, each party makes a binary input and receives a binary output and the game is won if $A\oplus B=XY$. We define the CHSH score by
$$\frac{1}{4}\left(\sum_ap_{AB|00}(a,a)+\sum_ap_{AB|01}(a,a)+\sum_ap_{AB|10}(a,a)+\sum_ap_{AB|11}(a,a\oplus1)\right)\,,$$
which is the probability of winning the CHSH game when the inputs are chosen at random\footnote{Note that even if nonuniform distributions of inputs are used when running protocols, in this work the CHSH score is always defined as here.}.
Classical strategies can win this game with probability at most $3/4$, while quantum strategies can get as high as $\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right)\approx0.85$. For a fixed CHSH game score, $\omega$, we wish to compute the minimum von Neumann entropy over all strategies achieving that score. In this context a strategy comprises three Hilbert spaces $\mathcal{H}_{A'}$, $\mathcal{H}_{B'}$ and $\mathcal{H}_E$, POVMs $\{M_{a|x}\}_a$ on $\mathcal{H}_{A'}$ for both $x=0$ and $x=1$, POVMs $\{N_{b|y}\}_b$ on $\mathcal{H}_{B'}$ for both $y=0$ and $y=1$, and a state $\rho_{A'B'E}$ on $\mathcal{H}_{A'}\otimes\mathcal{H}_{B'}\otimes\mathcal{H}_E$.  Given a strategy, and a distribution $p_{XY}$ there is an associated channel $\mathcal{N}$ that acts on $A'B'$ and takes the state to the post-measurement state, i.e.,
$$\tau_{ABXYE}=(\mathcal{N}\otimes\mathcal{I}_E)(\rho_{A'B'E})=\sum_{abxy}p_{XY}(x,y)\proj{a}_A\otimes\proj{b}_B\otimes\proj{x}_X\otimes\proj{y}_Y\otimes\mathrm{tr}_{A'B'}\left((M_{a|x}\otimes N_{b|y}\otimes \openone_E)\rho_{A'B'E}\right)\,,$$
where $\mathcal{I}_E$ is the identity channel on $E$.
The entropic quantities we consider all pertain to this state\footnote{In the cases where we condition on $X=0$ and $Y=0$, we can project this state onto $\proj{0}_X\otimes\proj{0}_Y$ and renormalize -- see Appendix~\ref{app:simp} for more detail.}. Note also that the score is a function of $\tau$, which we denote $S(\tau)$ -- in particular, $p_{AB|xy}(a,b)=\mathrm{tr}\left((M_{a|x}\otimes N_{b|y}\otimes \openone_E)\rho_{A'B'E}\right)$.

For each of the six entropic quantities previously discussed we consider the infimum over all strategies that achieve a given score.  We use this to define a set of curves.  We write $F_{AB|XYE}(\omega,p_{XY})=\inf H(AB|XYE)_{\tau}$, where the infimum is over all strategies for which $S(\tau)=\omega$. In the same way we define $F_{AB|E}(\omega,p_{XY})$, $F_{A|XYE}(\omega,p_{XY})$ and $F_{A|E}(\omega,p_{XY})$, replacing the objective function by the corresponding entropy. We also define $F_{AB|00E}(\omega)$ and $F_{A|00E}(\omega)$ analogously, noting that these are independent of $p_{XY}$. Furthermore, if we write $F_{AB|XYE}(\omega)$ etc.\ (i.e., leaving out the $p_{XY}$), we refer to the case where $p_{XY}$ is uniform over $X$ and $Y$. For a more precise writing of these optimizations, see~\eqref{opt_1}.

We also consider a related set of functions $G_{AB|XYE}(\omega,p_{XY})$, $G_{AB|E}(\omega,p_{XY})$ etc.\ that are defined analogously, but while optimizing over a smaller set of allowed strategies. More precisely, the $G$ functions are defined by restricting $\mathcal{H}_{A'}$ and $\mathcal{H}_{B'}$ to be two dimensional and $\mathcal{H}_E$ to have dimension 4, taking $\rho_{A'B'E}$ to be pure with $\rho_{A'B'}$ diagonal in the Bell basis, and taking the POVMs to be projective measurements onto states of the form $\cos(\alpha)\ket{0}+\sin(\alpha)\ket{1}$ (see~\eqref{opt_simp}).

It turns out that $F_{AB|00E}(\omega)=G_{AB|00E}(\omega)$, $F_{A|00E}(\omega)=G_{A|00E}(\omega)$, and that in each of the other four cases $F$ is formed from $G$ by taking the convex lower bound. The underlying reason for this is that Jordan's lemma~\cite{Jordan} implies that in the case of Bell inequalities with two inputs and two outputs, any strategy is equivalent to a convex combination of strategies in which $A'$ and $B'$ are qubit systems. This means that if we solve the qubit case, the general case follows by taking the convex lower bound\footnote{The convex lower bound is not needed in the cases where we condition on $X=0$ and $Y=0$ because $G$ is already convex in these cases.}. Such an argument was made in~\cite{PABGMS} and we give the details in Appendix~\ref{app:Proof1}.

\section{Numerical analysis for rates}
The optimizations that generate the $G$ functions can be expressed in terms of $7$ real parameters ($3$ to specify the state and $4$ to choose the measurements). They are hence amenable to numerical optimizations. We note also that except in the cases $G_{AB|E}(\omega,p_{XY})$ and $G_{A|E}(\omega,p_{XY})$ we can remove an additional parameter. We discuss simple ways to write the entropic expressions in Appendix~\ref{app:simp}.

Our program for computing $G$ runs in $N$ iterations. In each iteration the program starts by making a random guess for the parameters from within the valid range. It then uses sequential quadratic programming to minimize $\bar{H}((\mathcal{N}\otimes\mathcal{I}_E)(\rho_{A'B'E}))$ subject to the CHSH score being fixed.
On each iteration, the program arrives at a candidate for the minimum value, and we run $N\approx10^4$ iterations to arrive at the conjectured minimum value for $\bar{H}((\mathcal{N}\otimes\mathcal{I}_E)(\rho_{A'B'E}))$. The numerical optimization is performed in Python using the SLSQP solver in SciPy.  Since these optimizations are not guaranteed to converge, strictly speaking the generated curves are upper bounds on the infima. For the purposes of applying the EAT, we assume they are tight.

$G_{AB|00E}$ and $G_{A|00E}$ are convex functions, and hence $F=G$ for these. For the other cases we generate the graphs in the case where $p_{XY}$ is uniform, observing that each of the $G$ curves starts with a concave part and switches to convex for larger CHSH scores. Since the minimum entropy is always zero for classical scores, each of the $G$ curves approach $0$ as $\omega$ approaches $3/4$. Each of the $F$ curves can be found from $G$ by finding the tangent to $G$ that passes through $(3/4,0)$.  We call the score at which this tangent is taken $\omega^*$, defined by $(\omega^*-3/4)G'(\omega^*)=G(\omega^*)$. We then have
\begin{equation}
    F(\omega) =     \begin{cases}
    G'(\omega^*) (\omega -\frac{3}{4}) & \text{if}\ \omega \leq \omega^* \\
    G(\omega) & \text{otherwise} \\
    \end{cases}\,.
\end{equation}
We give estimates for $\omega^*$ for each of the cases below. In essence, what this means is that for $\omega<\omega^*$ the optimal strategy for Eve is to either use a deterministic classical strategy with score $3/4$ or a strategy that achieves score $\omega^*$, mixing these such that the average score is $\omega$. Eve can remember which strategy she used, and hence the entropy from her perspective is also the convex mixture of the entropies of the endpoints.

Figure~\ref{fig:rates} shows the curves we obtained for the functions $F$ in each of the six cases. Note that, except in the cases where we condition on $X=0$ and $Y=0$, the graphs all have linear sections as a result of taking the convex lower bound. In Appendix~\ref{app:graphs} we show the graphs for $G$ together with those for $F$. The approximate coordinate of the top of the linear segment for $F_{AB|E}$ is $(0.8523,1.8735)$ and for $F_{A|E}$ it is $(0.8505,0.967)$. Note also that $1+H_\mathrm{bin}\left(\frac{1}{2}+\frac{1}{\sqrt{32}}\right)\approx1.908$ is the maximum value on the graph $F_{AB|E}(\omega)$.

\begin{figure}
\includegraphics[width=\textwidth]{Finalrates1.png}\\
\phantom{m}(a)\hspace{26em}(b)
\caption{Graphs of the rates for (a) the one-sided and (b) the two-sided randomness with uniformly chosen inputs. Each of these curves has a non-linear part and the blue curves do not have a linear part.}
\label{fig:rates}
\end{figure}

By examining the parameters that come out of the numerical optimizations we have the following.

\begin{lemma}\label{lem:ABgXYE}
  Consider the curve $g_1(\omega)=1+H_\mathrm{bin}(\omega)-2H_\mathrm{bin}(\frac{1}{2}+\frac{2\omega-1}{\sqrt{2}})$. $F_{AB|XYE}(\omega)$ is upper bounded by
\begin{equation}
F_{AB|XYE}(\omega)\leq\begin{cases}g_1(\omega)&\omega_{AB|XYE}^*\leq\omega\leq\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right) \\
g_1'(\omega_{AB|XYE}^*)(\omega-3/4)&3/4\leq\omega\leq\omega_{AB|XYE}^*\end{cases}\,.
\end{equation}
where $\omega_{AB|XYE}^*\approx0.84403$ is the solution to $g'_1(\omega)(\omega-3/4)=g_1(\omega)$. Note that $g_1(\omega^*)\approx1.4186$ and the maximum value reached is $1+H_\mathrm{bin}(1/2+1/(2\sqrt{2}))\approx1.601$.
\end{lemma}
\begin{proof}
In Appendix~\ref{app:simp} we give a parameterization of a two-qubit state (with parameters $R$, $\theta$ and $\delta$) and measurements (with parameters $\alpha_0$, $\alpha_1$, $\beta_0$ and $\beta_1$) before computing an expression for $H(AB|XYE)$ in terms of these (cf.~\eqref{eq:HABgXYE}).  We also obtained an expression for the CHSH score (cf.~\eqref{eq:score}).  Choosing $R=\sqrt{2}(2\omega-1)$, $\theta=0$, $\delta=0$, $\alpha_0=0$, $\alpha_1=\pi/2$, $\beta_0=\pi/8$, $\beta_1=-\pi/8$ we find a score $\omega$, and calculating $H(AB|XYE)$ we obtain $H(AB|XYE)=g_1(\omega)$ and hence $G_{AB|XYE}(\omega)\leq g_1(\omega)$. Since, $G_{AB|XYE}(3/4)=0$, and $F_{AB|XYE}$ is formed from $G_{AB|XYE}$ by taking the convex lower bound, we establish the claim.
\end{proof}

\begin{lemma}\label{lem:AgXYE}
Consider the curve $g_2(\omega)=1-H_\mathrm{bin}\left(\frac{1}{2}+\frac{2\omega-1}{\sqrt{2}}\right)$.
$F_{A|XYE}(\omega)$ is upper bounded by the convex lower bound of $g_2(\omega)$. In other words,
\begin{equation}
F_{A|XYE}(\omega)\leq\begin{cases}g_2(\omega)&\omega_{A|XYE}^*\leq\omega\leq\frac{1}{2}\left(1+\frac{1}{\sqrt{2}}\right) \\
g_2'(\omega_{A|XYE}^*)(\omega-3/4)&3/4\leq\omega\leq\omega_{A|XYE}^*\end{cases}\,.
\end{equation}
where $\omega_{A|XYE}^*\approx0.84698$ is the solution to $g'_2(\omega)(\omega-3/4)=g_2(\omega)$. Note that $g_2(\omega^*)\approx0.92394$.
\end{lemma}
\begin{proof}
The proof is the same as for Lemma~\ref{lem:ABgXYE}, except that $g_1$ is replaced by $g_2$ --- the choice of state and measurements remains the same.
\end{proof}

\begin{conjecture}
The upper bounds in Lemmas~\ref{lem:ABgXYE} and~\ref{lem:AgXYE} are tight.
\end{conjecture}

\section{Protocols for randomness expansion}\label{sec:protocols}
In this section we discuss CHSH-based protocols for DIRE of both the spot-checking and non spot-checking types. We pick specific protocols for concreteness, but there are many possible variations. For instance, the protocols we discuss condense the observed statistics to a single score, but this is not necessary, and in some cases and for some sets of experimental conditions it can be advantageous to use multiple scores~\cite{BRC,TSGPL}.

Before getting to the protocols, we first describe the setup, assumptions and security definition. Although DIRE requires no assumptions on how the devices used operate, the setup for DIRE involves a user who performs the protocol within a secure laboratory, from which information cannot leak. Individual devices can also be isolated within their own sub-laboratory and the user can ensure that these devices only learn the information necessary for the protocol (in particular, they cannot learn any inputs given to other devices).  The user has access to a trusted classical computer and an initial source (or sources) of trusted randomness.

The quantum devices used for the protocol are only limited by the laws of quantum theory and may share arbitrary entanglement with each other and with an adversary. However, they cannot communicate with each other, or to the adversary after the protocol starts. Furthermore, we assume they are kept isolated after the protocol (cf.\ the discussion in Appendix~\ref{app:compos}).

For security of the protocols, we use a composable security definition. Consider a protocol with output $Z$ and use $\Omega$ to denote the event that it does not abort. The protocol is $(\epsilon_S,\epsilon_C)$-secure if
\begin{enumerate}
\item $\frac{1}{2}p_{\Omega} || \rho_{ZE|\Omega} - \frac{1}{d_Z}\openone_Z \otimes \rho_{E|\Omega} ||_1 \leq \epsilon_S$,
  where $E$ represents all the systems held by an adversary and $d_Z$ is the dimension of system $Z$; and
  \item There exists a quantum strategy such that $p_{\Omega}\geq1-\epsilon_C$.
\end{enumerate}
Here $\epsilon_S$ is called the soundness error, and $\epsilon_C$ is the completeness error.

\subsection{CHSH-based spot-checking protocol for randomness expansion}
We now describe a spot-checking protocol for randomness expansion. It uses a central biased random number generator $R_T$ and two other random number generators, $R_A$ and $R_B$ that are near each of the devices used to run the protocol.

\begin{protocol}\label{prot:spotcheck}{\bf (Spot-checking protocol)}
\phantom{a}\\

\noindent{\bf Parameters:}\\
$n$ -- number of rounds\\
$\gamma$ -- test probability\\
$\omega_{\exp}$ -- expected CHSH score\\
$\delta$ -- confidence width for the score
\begin{enumerate}
    \item Set $i=1$ for the first round, or increase $i$ by 1.
    \item Use $R_T$ to choose $T_i\in\{0,1\}$ where $T_i=1$ occurs with probability $\gamma$.
    \item If $T_i=1$ (test round), $R_A$ is used to choose $X_i$ uniformly, which is input to one device giving output $A_i$. Likewise $R_B$ is used to choose $Y_i$ uniformly, which is input to the other device giving output $B_i$. Set $U_i=1$ if $A_i\oplus B_i=X_iY_i$ and $U_i=0$ otherwise.
    \item If $T_i=0$ (generation round), the devices are given inputs $X_i=Y_i=0$, and return the outputs $A_i$ and $B_i$. Set $U_i=\bot$.
    \item Return to Step~1 unless $i=n$.
    \item Calculate the number of rounds in which $U_i=0$ occurred, and abort the protocol if this is larger than $n\gamma(1-\omega_{\exp}+\delta)$.
    \item \label{st:7} Process the concatenation of all the outputs with a quantum-proof strong extractor $\mathrm{Ext}$ to yield $\mathrm{Ext}({\bf AB},{\bf R})$, where ${\bf R}$ is a random seed for the extractor. Since a strong extractor is used, the final outcome can be taken to be the concatenation of ${\bf R}$ and $\mathrm{Ext}({\bf AB},{\bf R})$.
\end{enumerate}
\end{protocol}

There are a few important points to take into account when running the protocol. Firstly, it is crucial that each device only learns its own input and not the value of the other input, or of $T_i$. If this is not satisfied it is easy for devices to pass the protocol without generating randomness. Secondly, for implementations in which devices can fail to record outcomes when they should, it is important to close the detection loophole, which can be done by assigning an outcome, say $0$, when a device fails to make a detection.

In order to run the protocol, some initial randomness is needed to choose which rounds are test rounds, to choose the inputs in the test rounds and to seed the extractor. Since the extractor randomness forms part of the final output, it is not consumed in the protocol, so for considering the rate at which the protocol consumes randomness we can work out the amount of uniform randomness needed to supply the inputs.  Using the rounded interval algorithm~\cite{HaoHoshi} to make the biased random number generator, $n(H_\mathrm{bin}(\gamma)+2\gamma)+3$ is the expected amount of input randomness required. To achieve expansion the number of output bits must be greater than this. We use the entropy accumulation theorem (EAT) to lower bound the amount of output randomness. Asymptotically the relevant quantity is $H(AB|X=0,Y=0,E)$. The quantity $H(A|X=0,Y=0,E)$ acts as a lower bound for this, and can be used in its place if convenient, for instance in analyses that are more straightforward with an analytic curve.

\subsection{CHSH-based protocols without spot-checking}
In this section we discuss two such protocols.  Protocol~\ref{prot:biased} uses two biased local random number generators to choose the inputs on each round.  Protocol~\ref{prot:nonspotcheck} eliminates the bias, but also recycles the input randomness.  Recycling the input randomness is necessary when unbiased random number generators are used, since otherwise more randomness is required to run the protocol than is generated. Protocol~\ref{prot:nonspotcheck} gives the highest randomness generation rate of all the protocols we discuss.

\begin{protocol}\label{prot:biased} {\bf (Protocol with biased local random number generators)}

\phantom{a}\\

\noindent\textbf{Parameters}:\\
$n$ -- number of rounds \\
$\zeta^A$ -- probability of 1 for random number generator $R_A$ (taken to be below $1/2$)\\
$\zeta^B$ -- probability of 1 for random number generator $R_B$ (taken to be below $1/2$)\\
$\omega_{\text{exp}}$ -- expected CHSH score. \\
$\delta$ -- confidence widths for each score.
\begin{enumerate}
    \item Set $i=1$ for the first round, or increase $i$ by 1.
    \item Use $R_A$ to choose $X_i\in\{0,1\}$, which is input to one of the devices giving output $A_i\in\{0,1\}$. Likewise use $R_B$ to generate $Y_i\in\{0,1\}$, which is input to the other device giving output $B_i\in\{0,1\}$. Here $X_i=1$ occurs with probability $\zeta^A$ and $Y_i=1$ occurs with probability $\zeta^B$. Set $U_i=(X_i,Y_i,1)$ if $A_i\oplus B_i=X_iY_i$ and $U_i=(X_i,Y_i,0)$ otherwise.
    \item Return to Step~1 unless $i=n$.
    \item \label{st:omega} Compute the value
    \begin{eqnarray}\label{eq:omega}
    \omega=\frac{1}{4}\sum_{x,y} \frac{|\{i:U_i=(x,y,1)\}|}{n p_{X}(x) p_Y(y)}
    \end{eqnarray}
    and abort the protocol if $\omega<\omega_{\exp}-\delta$.
    Here $p_{X}(1)=\zeta^A$, $p_{X}(0)=1-\zeta^A$, $p_{Y}(1)=\zeta^{B}$ and $p_{Y}(0)=1-\zeta^{B}$.
    \item Process the concatenation of all the outputs with a quantum-proof strong extractor $\mathrm{Ext}$ to yield $\mathrm{Ext}({\bf AB},{\bf R})$, where ${\bf R}$ is a random seed for the extractor. Since a strong extractor is used, the final outcome can be taken to be the concatenation of ${\bf R}$ and $\mathrm{Ext}({\bf AB},{\bf R})$.
\end{enumerate}
\end{protocol}

Note that the quantity $|\{i:U_i=(x,y,1)\}|/(n p_{X}(x) p_Y(y))$ in~\eqref{eq:omega} is an estimate of the probability of winning the CHSH game for inputs $X=x$ and $Y=y$, and hence the $\omega$ computed in Step~\ref{st:omega} is an estimate of the CHSH value that would be observed if the same setup was used but with $X$ and $Y$ chosen uniformly.

The input randomness required per round in this protocol is roughly $H_\mathrm{bin}(\zeta^A)+H_\mathrm{bin}(\zeta^B)$.
To quantify the amount of output randomness (before randomness extraction is performed), in the asymptotic limit similar to the spot checking protocol, the relevant operational quantity is the von Neumann entropy $H(AB|XYE)$. Expansion hence cannot be achieved if $H(AB|XYE)-H_\mathrm{bin}(\zeta^A)-H_\mathrm{bin}(\zeta^B)<0$, which places constraints on the pairs of possible $(\zeta^A,\zeta^B)$. For $\zeta^A$ and $\zeta^B$ smaller than $1/2$, the quantity $H(AB|XYE)-H_\mathrm{bin}(\zeta^A)-H_\mathrm{bin}(\zeta^B)$ increases as $\zeta^A$ and $\zeta^B$ decrease, and hence we want to take these to be small. They only need to be large enough to ensure that $X=1,Y=1$ occurs often enough to give a good estimate of the empirical score.

Since
\begin{eqnarray}
H(AB|XYE) &=&  \sum_{xy} p_{XY}(x,y)H(AB|XYE) \nonumber\\
&\geq&\min_{x,y}H(AB|X=x,Y=y,E)   \label{Bakchodi}
\end{eqnarray}
we can use the bounds formed for $H(AB|X=0,Y=0,E)$ instead\footnote{There is nothing special about the choice $X=0$ and $Y=0$ when computing the bounds for $H(AB|X=0,Y=0,E)$.}, albeit with a loss of entropy (this loss of entropy is small if $\zeta_A$ and $\zeta_B$ are small).

One reason for using Protocol~\ref{prot:biased} rather than Protocol~\ref{prot:spotcheck} is that the former enables the locality loophole to be closed while expanding randomness.  In order to perform the Bell tests as part of a device-independent protocol we need to make inputs to two devices in such a way that neither device knows the input of the other. One way to ensure this is by using independent random number generators on each side of the experiment, and ensuring the the outcome of each device is given at space-like separation from the production of the random input to the other. Although space-like separation can provide a guarantee (within the laws of physics) that each device does not know the input of the other\footnote{Provided we have a reasonable way to give a time before which the output of $R_A$ and $R_B$ did not exist.}, in a cryptographic setting it is necessary to assume a secure laboratory to prevents any unwanted information leaking from inside the lab to an eavesdropper. The same mechanism by which the lab is shielded from the outside world can be used to shield devices in the lab from one another and hence can prevent communication between the two devices during the protocol.  However, although unnecessary for cryptographic purposes, it is interesting to consider closing the locality loophole while expanding randomness.

This is not possible in a typical spot-checking protocol, where a central random number generator is used to decide whether a round is a test round or not. Considering Protocol~\ref{prot:spotcheck}, the locality loophole can be readily closed during the test rounds, but the use of the central random number generator means that, if one is worried that hidden communication channels are being exploited, there is a loophole that the devices could behave differently on test rounds and generation rounds. For instance, measurement devices that know whether a round is a test or generation round could supply pre-programmed outputs in generation rounds, while behaving honestly in test rounds. Thus, spot-checking protocols do not enable fully closing the locality loophole while expanding randomness.

When using Protocol~\ref{prot:biased} with $\zeta^A=\zeta^B-\zeta$, the main difference to Protocol~\ref{prot:spotcheck} is that the distribution of $X$ and $Y$ is $((1-\zeta)^2,\zeta(1-\zeta),\zeta(1-\zeta),\zeta^2)$ rather than $(1-3\gamma/4,\gamma/4,\gamma/4,\gamma/4)$. In the analysis this manifests itself in the statistics, and the much lower probability of $X=1,Y=1$ requires an adjustment of $\delta$ to achieve the same error parameters for the protocol. A comparison between the output rates for Protocols~\ref{prot:spotcheck} and~\ref{prot:biased} is shown in Figures~\ref{fig:EAT_n} and~\ref{fig:EAT_score}.\bigskip

\begin{protocol}\label{prot:nonspotcheck} {\bf (Protocol with recycled input randomness)}
\phantom{a}\\

\noindent\textbf{Parameters}:\\
$n$ -- number of rounds \\
$\omega_{\exp}$ -- expected CHSH score. \\
$\delta$ -- confidence width.
\begin{enumerate}
    \item Set $i=1$ for the first round, or increase $i$ by 1.
    \item Use $R_A$ to choose $X_i\in\{0,1\}$ uniformly, serving as the input to one of the devices giving output $A_i\in\{0,1\}$. Likewise use $R_B$ to generate $Y_i\in\{0,1\}$ uniformly, which is input to the other device giving output $B_i\in\{0,1\}$. Set $U_i=1$ if $A_i\oplus B_i=X_iY_i$ and $U_i=0$ otherwise.
    \item Return to Step~1 unless $i=n$.
    \item Count the number of rounds for which $U_i=0$ occurred and abort the protocol if this is above $n(1-\omega_{\exp}+\delta)$.
    \item \label{st:5} Process the concatenation of all the inputs and outputs with a quantum-proof strong extractor $\mathrm{Ext}$ to yield $\mathrm{Ext}({\bf ABXY},{\bf R})$, where ${\bf R}$ is a random seed for the extractor. Since a strong extractor is used, the final outcome can be taken to be the concatenation of ${\bf R}$ and $\mathrm{Ext}({\bf ABXY},{\bf R})$.
\end{enumerate}
\end{protocol}

An important difference in this protocol compared to Protocols~\ref{prot:spotcheck} and~\ref{prot:biased} and is in the extraction step, which now extracts randomness from the input strings ${\bf X}$ and ${\bf Y}$ as well as the outputs. Without recycling the inputs expansion would not be possible in Protocol~\ref{prot:nonspotcheck}. With the modification the relevant quantity to decide the length of the output is $H(ABXY|E)$, and so $H(AB|XYE)=H(ABXY|E)-H(XY)=H(ABXY|E)-2$ is the relevant quantity for calculating the rate of expansion. Note that in order to reuse the input in a composable way, it also needs to be run through an extractor~\cite{CK2} (for a discussion of why it is important to do so and a few more composability-related issues, see Appendix~\ref{app:compos}).

We could also consider an adaptation of Protocol~\ref{prot:spotcheck} in which the input randomness is recycled, forming Protocol~\ref*{prot:spotcheck}$'$ from Protocol~\ref{prot:spotcheck} by replacing Step~\ref{st:7} by
\begin{enumerate}
\item[\ref*{st:7}$'$.] Process the concatenation of all the inputs and outputs with a quantum-proof strong extractor $\mathrm{Ext}$ to yield $\mathrm{Ext}({\bf ABXY},{\bf R})$, where ${\bf R}$ is a random seed for the extractor. Since a strong extractor is used, the final outcome can be taken to be the concatenation of ${\bf R}$ and $\mathrm{Ext}({\bf ABXY},{\bf R})$.
\end{enumerate}
In this case, as the number of rounds, $n$, increases the advantage gained by this modification decreases, becoming negligible asymptotically. This is because as $n$ increases, the value of $\gamma$ required to give the same overall security tends to zero, and hence the amount of input randomness required becomes negligible.
Note that recycling the input randomness in Protocol~\ref{prot:biased} in the case where $\zeta^A=\zeta^B=1/2$ is equivalent to Protocol~\ref{prot:nonspotcheck}.

Like Protocol~\ref{prot:biased}, Protocol~\ref{prot:nonspotcheck} also allows the locality loophole to be closed if on each round $i$, the random choice $X_i$ is space-like separated from the output $B_i$ and the random choice $Y_i$ is space-like separated from the output $A_i$.

\begin{figure}
\includegraphics[width=\textwidth]{EATvsN-final_4.png}\\
\phantom{m}(a)\hspace{16em}(b)\hspace{16em}(c)
\caption{Graphs of the net amount of certifiable randomness according to the EAT for (a) the spot checking protocol (Protocol~\ref{prot:spotcheck}), (b) Protocol~\ref{prot:biased}, and (c) the protocol with recycled input randomness (Protocol~\ref{prot:nonspotcheck}), showing the variation with the number of rounds for three different scores, $\omega$.  The error parameters used were $\epsilon_S=3.09\times10^{-12}$ and $\epsilon_C=10^{-6}$. For each point on the curve (a) an optimization over $\gamma$ was performed to maximize the randomness; similarly, the values of $\zeta^A=\zeta^B$ were optimized over to generate the curves in (b).}
\label{fig:EAT_n}
\end{figure}

\begin{figure}
\includegraphics[width=\textwidth]{EAT-vs-omega-final_1.png} \\
\phantom{m}(a)\hspace{16em}(b)\hspace{16em}(c)
\caption{Graphs of the net amount of certifiable randomness according to the EAT for (a) the spot checking protocol (Protocol~\ref{prot:spotcheck}), (b) Protocol~\ref{prot:biased}, and (c) the protocol with recycled input randomness (Protocol~\ref{prot:nonspotcheck}), showing the variation with the CHSH score $\omega$. The round numbers, $n$, are indicated in the legend. The error parameters used were $\epsilon_S=3.09\times10^{-12}$ and $\epsilon_C=10^{-6}$. As in Figure~\ref{fig:EAT_n}, the values of $\gamma$ and $\zeta^A=\zeta^B$ were optimized over for each point.}
\label{fig:EAT_score}
\end{figure}

In each of the protocols, the parameter $\delta$ should be chosen depending on the desired completeness error. For the spot-checking protocol, the relation between the two is discussed in~\cite[Supplementary Information I~D]{LLR&}.  The analysis there can be applied to the protocol with recycled input randomness by setting $\gamma=1$ and the protocol with biased local random number generators is discussed in Appendix~\ref{app:comp}. The chosen soundness error affects the length of the extractor output, and, if chosen too small the output length becomes zero (see Appendix~\ref{app:EAT} for further discussion).

Figures~\ref{fig:EAT_n} and~\ref{fig:EAT_score} show how the amount of certifiable randomness varies with the score, $\omega$, and round number, $n$. Note that in the cases where the rate curves are linear, they are linear for most of their ranges.  Extending the linear part to the full range of quantum scores makes it easier to use the EAT while only resulting in a small drop in rate for scores close to the maximum quantum value. Note that, as mentioned above, strictly the numerical curves we gave for the von Neumann entropy are upper bounds; the curves in Figures~\ref{fig:EAT_n} and~\ref{fig:EAT_score} are generated under the assumption that these upper bounds are tight.

To demonstrate the increased practicality of the two-sided curves, we use the parameters from a recent experiment~\cite{LLR&} with Protocol~\ref{prot:spotcheck}. There a score of just over $0.752$ was obtained, for which it would require about $9\times10^{10}$ rounds to achieve expansion using Protocol~\ref{prot:spotcheck} with $\gamma=3.383\times10^{-4}$, $\epsilon_S=3.09\times10^{-12}$, $\epsilon_C=10^{-6}$ and taking the one-sided randomness~\cite{LLR&}.  Using Protocol~\ref{prot:nonspotcheck} instead, and taking the two-sided randomness for the same score and error parameters allows expansion for $n\gtrsim 8\times10^7$, significantly increasing the practicality. For instance, the main experiment of~\cite{LLR&} was based on a spot-checking protocol and took $19.2$ hours; the use of Protocol~\ref{prot:nonspotcheck} instead would allow the same amount of expansion in about $60$ seconds (this time holds under the assumption that the same repetition rate of the experiment can be met in the non-spot checking protocol\footnote{In some experiments, the rate at which we can switch between the two measurements is relatively slow, and hence when using Protocol~\ref{prot:nonspotcheck}, where switching is required on most rounds, the switching rate dominates, slightly increasing the time.}).
Protocol~\ref{prot:biased}, however, produces lower randomness rates compared to the spot-checking protocol. This is partly because more input randomness is required ($2H_{\text{bin}}(\sqrt{\gamma})\gg H_{\text{bin}}(\gamma)+2\gamma$ for small $\gamma$), and also because the completeness error has a worse behaviour. Protocol~\ref{prot:biased} is hence useful when inputs are not recycled and when closing the locality loophole is desirable.

When discussing randomness expansion we have considered the figure of merit to be the amount of expansion per entangled pair shared.  An alternative figure of merit is the ratio of the final randomness to the initial randomness, i.e., here we are considering how much randomness we can get from a given amount of initial randomness.  For the latter figure of merit, Protocol~\ref{prot:nonspotcheck} is no longer optimal, since the amount of expansion cannot exceed the amount of input randomness.  For the other two protocols the ratio of output randomness to input randomness can be made much higher by taking either $\gamma$ or $\zeta^A\zeta^B$ to be small.

\section{Discussion}
In this paper, we have given numerical and analytic bounds on various conditional von Neumann entropies that are relevant for CHSH-based device-independent protocols and discussed when each can be applied.  We have applied these results to explicit protocols, comparing the finite statics rates using the EAT. We discussed variations of the usual spot-checking protocol, the first removes the spot checking to allow expansion while closing the locality loophole, and the second recycles the input randomness, so allowing expansion when performing a CHSH test on every round.

It remains an open question to find an analytic form for $F_{AB|X=0,Y=0,E}$, $F_{A|E}$ and $F_{AB|E}$.  Since the curves $F_{A|E}$ and $F_{AB|E}$ are linear for all but the very highest (experimentally least achievable) scores, in these cases not much is lost by extending the line to all scores forming a lower bound that tightly covers all of the experimentally relevant cases.  On the other hand $F_{AB|X=0,Y=0,E}$ is a convex curve throughout and hence a tight analytic form would be particularly useful in this case. Our initial analysis suggests that form of the parameters achieving the optima for these is sufficiently complicated that any analytic expression would not be compact. A reasonably tight analytic lower bound for $F_{AB|X=0,Y=0,E}$ could also be useful for theoretical analysis. Note also that the bound $F_{A|E}\geq F_{A|XYE}$ is fairly tight (see Figure~\ref{fig:rates}(a)) so the analytic form for $F_{A|XYE}$ can be used to bound $F_{A|E}$ with little loss.  Another open problem is to find a concrete scenario in which $F_{AB|E}$ is directly useful.

The use of Jordan's lemma in this work prevents the techniques used being extended to general protocols, and finding improved ways to bound the conditional von Neumann entropy numerically in general cases remains of interest, in particular to ease the experimental burden required to demonstrate DIQKD in the lab.\bigskip

\noindent{\bf Additional note:} While completing this work a new paper appeared~\cite{TSBSRSL}, which also found numerical curves for $H(A|XYE)$ and used them with some new protocols to generate improved rates for DIQKD.

\acknowledgements
We thank Erik Woodhead for useful comments. This work was supported by the Quantum Communications Hub of the UK Engineering and Physical Sciences Research Council (EPSRC) (grant nos.\ EP/M013472/1 and EP/T001011/1).

redpajama_set_name: RedPajamaArXiv


text:
The peppermint breath test: a benchmarking protocol for breath sampling and analysis using GC-MS
Laura Di Francesco, Denise Biagini, Tommaso Lomonaco, Francesca G. Bellagambi, Sven Schuchardt, Olaf Holz, Katie Hamshere, Iain R White, Maxim Wilkinson, Stephen J. Fowler, 2020, original scientific article
Abstract: Exhaled breath contains hundreds of volatile organic compounds (VOCs) which offers the potential for diagnosing and monitoring a wide range of diseases. As the breath research field has grown, sampling and analytical practices have become highly varied between groups. Standardisation would allow meta-analyses of data from multiple studies and greater confidence in published results. The Peppermint Consortium has been formed to address this task of standardisation. In the current study we aimed to generate initial benchmark values for thermal desorption-gas chromatography-mass spectrometry (TD-GC-MS) analysis of breath samples containing peppermint-derived VOCs. Headspace analysis of peppermint oil capsules was performed to determine compounds of interest. Ten healthy participants were recruited by three groups. Each participant provided a baseline breath sample prior to taking a peppermint capsule, with further samples collected at 60, 90, 165, 285 and 360 min following ingestion. Sampling and analytical protocols were different for each institution, in line with their usual practice. Samples were analysed by TD-GC-MS and benchmarking values determined for the time taken for detected peppermint VOCs to return to baseline values. Sixteen compounds were identified in the capsule headspace. Additionally, 2,3-dehydro-1,8-cineole was uniquely found in the breath samples, with a washout profile that suggested it was a product of peppermint metabolism. Five compounds (α-pinene, β-pinene, eucalyptol, menthol and menthone) were quantified by all three groups. Differences in recovery were observed between the groups, particularly for menthone and menthol. The average time taken for VOCs to return to baseline was selected as the benchmark and were 441, 648, 1736, 643 and 375 min for α-pinene, β-pinene, eucalyptol, menthone and menthol respectively. An initial set of easy-to-measure benchmarking values for assessing the performance of TD-GC-MS systems for the analysis of VOCs in breath is presented. These values will be updated when more groups provide additional data.
Summary of found: ...Exhaled breath contains hundreds of volatile organic compounds (VOCs) which offers the potential...
Keywords: Volatile organic compounds, breath, diagnostics, standardisation
Published: 11.12.2020; Views: 95; Downloads: 0
Detection and quantification of exhaled volatile organic compounds in mechanically ventilated patients–comparison of two sampling methods
Iain R. White, Pouline M. van Oort, Waqar Ahmed, Craig Johnson, Jonathan Bannard-Smith, Timothy Felton, Lieuwe D. Bos, Royston Goodacre, Paul Dark, Stephen J. Fowler, 2020, original scientific article
Abstract: Exhaled breath analysis is a promising new diagnostic tool, but currently no standardised method for sampling is available in mechanically ventilated patients. We compared two breath sampling methods, first using an artificial ventilator circuit, then in "real life" in mechanically ventilated patients on the intensive care unit. In the laboratory circuit, a 24-component synthetic-breath volatile organic compound (VOC) mixture was injected into the system as air was sampled: (A) through a port on the exhalation limb of the circuit and (B) through a closed endo-bronchial suction catheter. Sorbent tubes were used to collect samples for analysis by thermal desorption-gas chromatography-mass spectrometry. Realistic mechanical ventilation rates and breath pressure–volume loops were established and method detection limits (MDLs) were calculated for all VOCs. Higher yields of VOCs were retrieved using the closed suction catheter; however, for several VOCs MDLs were compromised due to the background signal associated with plastic and rubber components in the catheters. Different brands of suction catheter were compared. Exhaled VOC data from 40 patient samples collected at two sites were then used to calculate the proportion of data analysed above the MDL. The relative performance of the two methods differed depending on the VOC under study and both methods showed sensitivity towards different exhaled VOCs. Furthermore, method performance differed depending on recruitment site, as the centres were equipped with different brands of respiratory equipment, an important consideration for the design of multicentre studies investigating exhaled VOCs in mechanically ventilated patients.
Keywords: Volatile organic compounds, infection, breath, ventilator associated pneumonia
Published: 10.12.2020; Views: 101; Downloads: 0
Increased sensitivity in proton transfer reaction mass spectrometry by incorporation of a radio frequency ion funnel
Andrew M Ellis, Stephen Mullock, Fraser Reich, Paul S Monks, Iain R White, Robert S Blake, Shane Barber, 2012, original scientific article
Abstract: A drift tube capable of simultaneously functioning as an ion funnel is demonstrated in proton transfer reaction mass spectrometry (PTR-MS) for the first time. The ion funnel enables a much higher proportion of ions to exit the drift tube and enter the mass spectrometer than would otherwise be the case. An increase in the detection sensitivity for volatile organic compounds of between 1 and 2 orders of magnitude is delivered, as demonstrated using several compounds. Other aspects of analytical performance explored in this study include the effective E/N (ratio of electric field to number density of the gas) and dynamic range over which the drift tube is operated. The dual-purpose drift tube/ion funnel can be coupled to various types of mass spectrometers to increase the detection sensitivity and may therefore offer considerable benefits in PTR-MS work.
Summary of found: ...An increase in the detection sensitivity for volatile organic compounds of between 1 and 2...
Keywords: Analytical performance, Detection sensitivity, Drift tube, Dynamic range, Ion funnels, Proton-transfer reaction mass spectrometry, Volatile organic compounds
Fulltext (799,47 KB)
Real-time multi-marker measurement of organic compounds in human breath: Towards fingerprinting breath
Iain R White, Kerry A Willis, Christopher Whyte, Rebecca Cordell, Robert S Blake, Andrew J Wardlaw, 2013, original scientific article
Abstract: The prospects for exploiting proton transfer reaction-time of flight-mass spectrometry (PTR-ToF-MS) in medical diagnostics are illustrated through a series of case studies. Measurements of acetone levels in the breath of 68 healthy people are presented along with a longitudinal study of a single person over a period of 1 month. The median acetone concentration across the population was 484 ppbV with a geometric standard deviation (GSD) of 1.6, whilst the average GSD during the single subject longtitudinal study was 1.5. An additional case study is presented which highlights the potential of PTR-ToF-MS in pharmacokinetic studies, based upon the analysis of online breath samples of a person following the consumption of ethanol. PTR-ToF-MS comes into its own when information across a wide mass range is required, particularly when such information must be gathered in a short time during a breathing cycle. To illustrate this property, multicomponent breath analysis in a small study of cystic fibrosis patients is detailed, which provides tentative evidence that online PTR-ToF-MS analysis of tidal breath can distinguish between active infection and non-infected patients.
Summary of found: ... Volatile Organic Compounds, breath, proton transfer reaction mass...
Keywords: Volatile Organic Compounds, breath, proton transfer reaction mass spectrometry, Cystic Fibrosis
Headspace volatile organic compounds from bacteria implicated in ventilator-associated pneumonia analysed by TD-GC/MS
Stephen J Fowler, Roy Goodacre, Iain R White, Tamara M E Nijsen, Waqar M Ahmed, Howbeer Muhamadali, Oluwasola Lawal, 2018, original scientific article
Abstract: Ventilator-associated pneumonia (VAP) is a healthcare-acquired infection arising from the invasion of the lower respiratory tract by opportunistic pathogens in ventilated patients. The current method of diagnosis requires the culture of an airway sample such as bronchoalveolar lavage, which is invasive to obtain and may take up to seven days to identify a causal pathogen, or indeed rule out infection. While awaiting results, patients are administered empirical antibiotics; risks of this approach include lack of effect on the causal pathogen, contribution to the development of antibiotic resistance and downstream effects such as increased length of intensive care stay, cost, morbidity and mortality. Specific biomarkers which could identify causal pathogens in a timely manner are needed as they would allow judicious use of the most appropriate antimicrobial therapy. Volatile organic compound (VOC) analysis in exhaled breath is proposed as an alternative due to its non-invasive nature and its potential to provide rapid diagnosis at the patient's bedside. VOCs in exhaled breath originate from exogenous, endogenous, as well as microbial sources. To identify potential markers, VAP-associated pathogens Escherichia coli, Klebsiella pneumoniae, Pseudomonas aeruginosa, and Staphylococcus aureus were cultured in both artificial sputum medium and nutrient broth, and their headspaces were sampled and analysed for VOCs. Previously reported volatile markers were identified in this study, including indole and 1-undecene, alongside compounds that are novel to this investigation, cyclopentanone and 1-hexanol. We further investigated media components (substrates) to identify those that are essential for indole and cyclopentanone production, with potential implications for understanding microbial metabolism in the lung.
Keywords: bacteria, exhaled breath, infection, ventilator-associated pneumonia, volatile organic compounds
TD/GC–MS analysis of volatile markers emitted from mono- and co-cultures of Enterobacter cloacae and Pseudomonas aeruginosa in artificial sputum
Craig Portsmouth, Pedro Povoa, Jan H Leopold, Pouline M P van Oort, Emili Diaz, Gemma Goma, Timothy Felton, Paul Dark, Alan Davie, Luis Coelho, Lieuwe D Bos, Marta Camprubi, Antonio Artigas, Jonathan Barnard-Smith, Waqar M Ahmed, Stephen J Fowler, Tamara M E Nijsen, Royston Goodacre, Weda Hans, Hugo knobel, Oluwasola Lawal, Iain R White, 2018, original scientific article
Abstract: Introduction: Infections such as ventilator-associated pneumonia (VAP) can be caused by one or more pathogens. Current methods for identifying these pathogenic microbes often require invasive sampling, and can be time consuming, due to the requirement for prolonged cultural enrichment along with selective and differential plating steps. This results in delays in diagnosis which in such critically ill patients can have potentially life-threatening consequences. Therefore, a non-invasive and timely diagnostic method is required. Detection of microbial volatile organic compounds (VOCs) in exhaled breath is proposed as an alternative method for identifying these pathogens and may distinguish between mono- and poly-microbial infections. Objectives: To investigate volatile metabolites that discriminate between bacterial mono- and co-cultures. Methods: VAP-associated pathogens Enterobacter cloacae and Pseudomonas aeruginosa were cultured individually and together in artificial sputum medium for 24 h and their headspace was analysed for potential discriminatory VOCs by thermal desorption gas chromatography–mass spectrometry. Results: Of the 70 VOCs putatively identified, 23 were found to significantly increase during bacterial culture (i.e. likely to be released during metabolism) and 13 decreased (i.e. likely consumed during metabolism). The other VOCs showed no transformation (similar concentrations observed as in the medium). Bacteria-specific VOCs including 2-methyl-1-propanol, 2-phenylethanol, and 3-methyl-1-butanol were observed in the headspace of axenic cultures of E. cloacae, and methyl 2-ethylhexanoate in the headspace of P. aeruginosa cultures which is novel to this investigation. Previously reported VOCs 1-undecene and pyrrole were also detected. The metabolites 2-methylbutyl acetate and methyl 2-methylbutyrate, which are reported to exhibit antimicrobial activity, were elevated in co-culture only. Conclusion: The observed VOCs were able to differentiate axenic and co-cultures. Validation of these markers in exhaled breath specimens could prove useful for timely pathogen identification and infection type diagnosis.
Summary of found: ...method is required. Detection of microbial volatile organic compounds (VOCs) in exhaled breath is proposed...
Keywords: Bacteria, Enterobacter cloacae, Gas Chromatography-Mass Spectrometry, Infection, Pseudomonas aeruginosa, Volatile organic compounds
Development of an adaptable headspace sampling method for metabolic profiling of the fungal volatome
Stephen J Fowler, Nicholas D Read, Royston Goodacre, Michael J Bromley, Tamara M Nijsen, Oluwasola Lawal, Iain R White, Pavlos Geranios, Waqar M Ahmed, 2018, original scientific article
Abstract: Pulmonary aspergillosis can cause serious complications in people with a suppressed immune system. Volatile metabolites emitted by Aspergillus spp. have shown promise for early detection of pathogenicity. However, volatile profiles require further research, as effective headspace analysis methods are required for extended chemical coverage of the volatome; in terms of both very volatile and semi-volatile compounds. In this study, we describe a novel adaptable sampling method in which fungal headspace samples can be sampled continuously throughout a defined time period using both active (pumped) and passive (diffusive) methods, with the capability for samples to be stored for later off-line analysis. For this method we utilise thermal desorption-gas chromatography-mass spectrometry to generate volatile metabolic profiles using Aspergillus fumigatus as the model organism. Several known fungal-specific volatiles associated with secondary metabolite biosynthesis (including α-pinene, camphene, limonene, and several sesquiterpenes) were identified. A comparison between the wild-type A. fumigatus with a phosphopantetheinyl transferase null mutant strain (ΔpptA) that is compromised in secondary metabolite synthesis, revealed reduced production of sesquiterpenes. We also showed the lack of terpene compounds production during the early growth phase, whilst pyrazines were identified in both early and late growth phases. We have demonstrated that the fungal volatome is dynamic and it is therefore critically necessary to sample the headspace across several time periods using a combination of active and passive sampling techniques to analyse and understand this dynamism.
Summary of found: ...in people with a suppressed immune system. Volatile metabolites emitted by Aspergillus spp. have shown... ...Volatile Organic Compounds, Fungi, Mycelial growth... ...terms of both very volatile and semi-volatile compounds. In this study, we describe a novel...
Keywords: Volatile Organic Compounds, Fungi, Mycelial growth
A microbiome and metabolomic signature of phases of cutaneous healing identified by profiling sequential acute wounds of human skin: An exploratory study
Mohammed Ashrafi, Yun Xu, Howbeer Muhamadali, Iain R White, Maxim Wilkinson, Mohamed Baguneid, Roy Goodacre, Ardeshir Bayat, 2020, original scientific article
Abstract: Profiling skin microbiome and metabolome has been utilised to gain further insight into wound healing processes. The aims of this multi-part temporal study in 11 volunteers were to analytically profile the dynamic wound tissue and headspace metabolome and sequence microbial communities in acute wound healing at days 0, 7, 14, 21 and 28, and to investigate their relationship to wound healing, using non-invasive quantitative devices. Metabolites were obtained using tissue extraction, sorbent and polydimethylsiloxane patches and analysed using GCMS. PCA of wound tissue metabolome clearly separated time points with 10 metabolites of 346 being involved in separation. Analysis of variance-simultaneous component analysis identified a statistical difference between the wound headspace metabolome, sites (P = 0.0024) and time points (P<0.0001), with 10 out of the 129 metabolites measured involved with this separation between sites and time points. A reciprocal relationship between Staphylococcus spp. and Propionibacterium spp. was observed at day 21 (P<0.05) with a statistical correlation between collagen and Propionibacterium (r = 0.417; P = 0.038) and Staphylococcus (r = -0.434; P = 0.03). Procrustes analysis showed a statistically significant similarity between wound headspace and tissue metabolome with non-invasive wound devices. This exploratory study demonstrates the temporal and dynamic nature of acute wound metabolome and microbiome presenting a novel class of biomarkers that correspond to wound healing, with further confirmatory studies now necessary.
Summary of found: ...metabolomics, skin, volatile organic compounds, VOCs, wound healing...
Keywords: metabolomics, skin, volatile organic compounds, VOCs, wound healing