text: ==Background== The 1953 British Grand Prix was the fourth time that the British Grand Prix was held. Prior to the event, defending champion Alberto Ascari was leading the standings on 28 points, ahead of Mike Hawthorn, who had scored 14 points.<ref>[https://statsf1.com/en/1953/france/championnat.aspx ''STATS F1'' providing the championship standings heading into the 1953 British Grand Prix.] Retrieved 13 Oct '21</ref> The race, like others held in 1952 and 1953, would actually be run under Formula 2 regulations, to encourage a larger competitor field to attend events following the withdrawal of Alfa Romeo after the 1951 World Championship.<ref>[https://topgear-autoguide.com/category/formula-1/f1-world-cup-1952-and-1953-when-formula-2-became-formula-11607102762 ''Topgear-Autoguide'' detailing how the 1952 and 1953 World Championships were held under Formula 2 regulations.] Retrieved 13 Oct '21</ref> The event would experience more exposure than most Formula One races back then, as the BBC was interested in televising it. Prior to the Second World War, the BBC had broadcast many sports for the first time, including football, cricket, and rugby.<ref>[https://swlondoner.co.uk/sport/19032021-rugby-tv-debut-calcutta-cup-televised-83-years-on/#:~:text=On%20March%2019th%201938%2C%2083,Toft's%20English%20side%2021%2D16. ''SW Londoner'' listing some of the sports the BBC televised for the first time live prior to the Second World War.] Retrieved 13 Oct '21</ref> While Formula One did not exist until 1950, the BBC had a history of televising motor racing stretching back to 1937, when it televised an [[1937 International Imperial Trophy Race (lost footage of motor race; 1937)|International Imperial Trophy race on 9th October]].<ref>[https://genome.ch.bbc.co.uk/48df02d5f1aa49b2b55c6a12aa0a54f8 Issue 731 of ''Radio Times'' listing the International Imperial Trophy race, the first motor race to be televised live.] Retrieved 13 Oct '21</ref> BBC cameras were therefore situated at Silverstone,<ref>[https://motorsportmagazine.com/archive/article/august-1997/55/50-british-grands-prix ''Motorsport Magazine'' noting BBC cameras were present at Silverstone, capturing footage for the television broadcast.] Retrieved 13 Oct '21</ref> and commentary was provided by Raymond Baxter, a fighter pilot and prominent television and radio presenter.<ref>[https://books.google.co.uk/books?id=ec85DwAAQBAJ&pg=PA146&lpg=PA146&dq=1953+British+Grand+Prix+bbc&source=bl&ots=xNy0yszCyL&sig=ACfU3U3NPu_-j-tdqwHrERGhtW0Wy47q6Q&hl=en&sa=X&ved=2ahUKEwjUqfy4msjzAhU3AWMBHTEGCOU4ChDoAXoECBEQAw#v=onepage&q=1953%20British%20Grand%20Prix%20bbc&f=false ''Speed Read F1: The Technology, Rules, History and Concepts Key to the Sport'' crediting Raymond Baxter as the commentator for the race.] Retrieved 13 Oct '21</ref> Baxter would also be present for the Thursday practice session, in a television broadcast that explained some of the terminology and workings of motor racing, as well as introduce the audience to some of the racing personalities and cars that made motor racing unique.<ref>[https://genome.ch.bbc.co.uk/page/ff82e79220e04df88320ecfba16eba77?page=34 Issue 1,548 of ''Radio Times'', listing the Thursday broadcast as ''Background to Sport: Motor Racing''.] Retrieved 13 Oct '21</ref> The actual race would be broadcast on Saturday and would be expected to last from 11:30 to 2:30, complying with the international agreement where the Grand Prix must last three hours.<ref>[https://genome.ch.bbc.co.uk/page/ff82e79220e04df88320ecfba16eba77?page=43 Issue 1,548 of ''Radio Times'', detailing the expected race time start and end, as well as providing a ''Silverstone Special'' column from Raymond Baxter.] Retrieved 13 Oct '21</ref> The BBC would dedicate its Television Service to broadcasting the race for the entire expected duration plus a fifteen-minute overlap, with the broadcast being expected to end by 14:45. According to issue 1,548 of ''Radio Times'', multiple other events would also be held on Saturday, including 500cc., sports car and Formula Libre events. This explains further listings in the issue, including the 500cc. race expected to be held from 10-11, the sports car race from 15:30-15:45, and other events from 16:00-16:15 and 16:35-17:00. Further, a BBC recording would be broadcast showcasing highlights of the Silverstone races from 22:35-23:05.<ref>[https://genome.ch.bbc.co.uk/schedules/service_bbc_television_service/1953-07-18 BBC Genome archive of ''Radio Times'' issues listing the races that would be broadcast on the 18th, including the 1953 British Grand Prix.] Retrieved 13 Oct '21</ref><ref>[https://genome.ch.bbc.co.uk/page/ff82e79220e04df88320ecfba16eba77?page=46 Issue 1,548 of ''Radio Times'' listing all of the races BBC Television Service was expected to broadcast on the 18th.] Retrieved 13 Oct '21</ref> Despite the historic nature of the broadcast, the BBC would only occasionally broadcast Formula One races in the 1950s and 1960s, with regular coverage generally occurring from 1979 onwards.<ref>[https://motorsportbroadcasting.com/history/ ''Motorsport Broadcasting'' noting the BBC did not provide regular coverage of Formula One until 1979.] Retrieved 13 Oct '21</ref>


September 11, 2017 cota, Formula One, motorsports, usgp.
"We've got some time on the soccer front, and it only makes sense to wait until after the F1 race, which is obviously our biggest event of the year," COTA chairman and USL Austin majority owner Bobby Epstein told the Statesman on Saturday at the track.
Austin soccer blogger Jerome Cortinovis tweeted Monday that the name "Austin Bold FC" was trademarked by USL Austin official Roberto Pinto da Silva Jr.
Pinto da Silva Jr., special events coordinator at COTA, confirmed to the Statesman that he did obtain Austin Bold FC but cautioned not to read too much into it.
"This is one of about 40 names we want to consider," he said. "At least half of those names already were registered in some way.
Both Epstein and Pinto da Silva Jr. said more work will be put into the name selection late this fall.


Broadview Networks, founded in 1996 and headquartered in Rye Brook, New York, provide fully hosted cloud based Unified Communications (UC) services. Their services are available nationwide in the USA to all 50 states and they serve more than 250,000 people. Broadview also provides conferencing solutions, cloud computing services and network-based voice and data services.
Broadview offers a 100 percent cloud-based Unified Communications (UC) phone solution plan which they call OfficeSuite UC. Pricing starts at $23.95 per user per month but it is unclear whether this is based on a discount for a certain number of users. Also, make sure to ask about any term commitments when getting price quotes.
As a UC solution, features such as real time presence and company wide chat messaging are included in your monthly fees. Standard features such as auto attendant, call forward, call hold and transfer, call park, do not disturb, and music on hold are included also as well as many others. There are also optional features that can be added to your service for small additional monthly fees, such as call recording, HD conferencing, custom on-hold announcements, online fax, voicemail transcription and softphones. Various App integrations are available including Salesforce, Skype for Business, Google, Dentrix, various web-based CRMs (Hubspot etc), and Microsoft Office 365 and Outlook. Contact center call services are also available.
Broadview has created an intuitive and easy to use online portal that can be used by administrators and end-users to manage their services. The portal is called MyOfficeSuite, and they refer to it as the heart of their system. This portal provides control of all the features from one place and can be accessed by any device from anywhere. The portal can be used for launching meetings, sending faxes, accessing voicemails, real-time chat and presence for internal colleagues, as well as call control features like call forwarding. This really is a feather in their cap.
A great support section is available on their website, including many guides, videos and virtual tours to setting up phones and using their portal and features. A toll-free number is available to call 24x7, as well as a live chat option and a "Ask the Community" forum option through myOfficeSuite. A VoIP test is available as well as a list of routers that are recommended and ones that are not recommended.
Broadview phones (they have their own branded phones) and Polycom phones appear to be the main options available currently to new and existing customers. A selection of desk phones, cordless handsets, and conferencing solutions are available, as well as a few accessories like headsets and video cameras. A trade-in option is available and you can submit information on your older equipment to get a quote for how much they will pay you. As an example, a Polycom IP335 would get you $15 and an IP550 would get you $23.
Broadview takes security seriously, taking advantage of encryption from the handsets to their secure servers. Even stored voicemails and messages are encrypted. Their service is HIPAA compliant and they can sign Business Associate Agreements for businesses that need that. They are also SSAE16 compliant.
Terms and conditions mention a minimum 12 month term for OfficeSuite services. We always recommend contacting the provider to verify pricing and term commitments before signing up.
There are also mentions of cancellation fees if you leave your term early. Again we recommend verifying this before making a commitment.
Customers may need to sign up for their Internet services also as well as a minimum of 6 IP phones, according to the T&Cs. Verification is recommended here.
A re-stocking fee will be assessed on each OfficeSuite handset returned prior to expiration of the term.
OfficeSuite UC plan starting at $23.95 per user per month. Includes unlimited calling and various features such as auto-attendant and voicemail.
Unified Communications solution includes internal real-time presence and chat features.
Optional features such as softphones, online fax, contact center services, and call recording are available for additional fees.
Various integrations available, including Salesforce, Skype for Business, Microsoft 365, Outlook, and Google Apps, for additional fees.
Other services available, including video/web Conferencing, cloud computing and network voice and data services, for additional fees.
MyOfficeSuite online portal to manage and access all your service features from any device, anywhere at anytime.
Broadview Networks provides a Unified Communications (UC) solution that provides more than your standard VoIP service provider. Real-time presence and messaging features, along with a fully featured and user friendly online portal plus various integrations (Salesforce, Microsoft, Google) provides for a comprehensive and modern communication solution that can enhance productivity in any company. Additional services like conferencing, cloud computing and network services complement their VoIP solution. Make sure you verify items such as minimum terms, and any early cancellation fees that may apply to you. Also, with optional features, make sure you select the ones you really need as the monthly cost could add up quickly.
Broadview Networks reviews are listed in chronological order.
Their products do not work correctly, We installed their system in December and features do not work. They acknowledged the product does not work and keep telling us a new version is being rolled out this day then another day and so on. They are not truthful and their support is awful. Run.
We have used this system for just over a year and it is so much better than our old one. I love the fact that we can control everything from our desktop (you can change the hold music, change the extensions for ringing if someone is on vacation, can get your voicemail messages emailed to you when you miss that important client phone call, etc.). Repair tickets were also taken care of very quickly. Only 4 stars (would have given 4.5 stars but not an option on rating system) for customer service due to the fact that one representative couldn't help with routing a call and I wound up fixing it myself. Might have been a new employee and didn't know the ins and outs of the system yet. Other than that anytime we had a question, the customer service reps were able to help.
We are using Broadview as our primary VoIP and T1 Internet service. Our area only offers Comcast and the upload speeds for business are not available. We made the switch to 100/100 fiber line since we have large backups to the cloud. In the 5 years we have had VoIP and 2 years of T1 line we have not experienced any outages. We did have an issue with the router and technical support was at our location within 4 hours replacing and configuring the router. Our sales agent follows up frequently to ask us if we need anything and tell us about new products. We have 10 leased phones and overhead intercom.
Highly responsive service team. In the few situations where I needed assistance (which is infrequent due to the intuitive nature of the system), it was highly competent and accessible.
I've been a Customer of Broadview for 10 years. Through the years they have improved greatly with their products and Service. Most recent is OfficeSuite. You can't get any better than that. My Sales Rep was very knowledgeable and followed us through from placing the orders to the best follow through with the installation. He made sure we had a very competent Project Manager. Accolades to them on such a wonderful VoIP product.
Pros: Very customer oriented. Tickets/issues are dealt with quickly. Cons: Seems like there are some internal/corporate integration issues after a recent merger?
The sales representative was incredibly knowledgeable which made discussing the company's offerings all that much easier. Changing over from our current service to Broadview was also seamless with minimal issues. Configuring the system is straight forward with an elegant web based interface that makes management easy. When we have had an issue; creating a ticket is simplistic and getting a resolution is usually very quick. Overall the service is affordable and works VERY well!
The phone service is reliable and has high quality sound. When there has been a problem customer service is easy to reach, helpful and resolves the problem quickly.
Broadivew has a great service. The phones are always 100%, customer service is excellent and the system is super easy to use.
I would give customer service 10 stars if possible. Broadview is a very innovative company they always has new features coming out for its customers and they excel in providing great service.


Work continues at Villa Irene on the Greek Island of Evia, part of the Sporades Group located off the north-east coast of the mainland.
The new guesthouse, built with stone from the site, is now complete and works to form the terrace linking the old and new buildings, while also providing a small pool above and a large plant and utility room below are well advanced.
The project is due for completion in time for Greek Easter celebrations in April next year.


text:
Sun Go Nova is the third solo extended play (EP) by American rapper and producer Denmark Vessey. It was released on April 20, 2018 via Mello Music Group. The EP is supported by three singles: "Sun Go Nova", "Trustfall" and "Sellout".

Background
The instrumental for the track "Sellout" was originally produced for Vince Staples by Earl Sweatshirt in 2013. It was featured in the 2014 mini documentary Earl Sweatshirt and Vince Staples - Inside the Beat - Ep. 1.

Track listing

References

External links
 Sun Go Nova | Denmark Vessey on Bandcamp
 Sun Go Nova by Denmark Vessey on iTunes

2018 EPs
Mello Music Group albums
Albums produced by Knxwledge
Albums produced by Earl Sweatshirt

redpajama_set_name: RedPajamaWikipedia


text:
Hard-Fi to headline UK festival
NME Feb 21, 2006 12:29 pm GMT
Plus they extend their own tour
Hard-Fi are to headline this year's Orange Evolution Festival.
The band, who performed at the free bash in 2005, will close proceedings on May 29, Bank Holiday Monday.
Now in its fifth year, the North-East's biggest music festival takes place over ten days, culminating in a huge outdoor free music event on Bank Holiday Monday when Newcastle and Gateshead Quaysides will be transformed to accommodate three stages with over 20 bands performing to an audience expected to exceed 35,000 people.
Last year's performers also included Dizzee Rascal, Mylo, The Futureheads and Super Furry Animals.
Additionally Hard-Fi have added two extra dates to their UK tour. The band will now play Leeds Millennium Square on May 21 and the Brighton Centre May 26 as well as their sold out dates which include five nights at London's Brixton Academy.

redpajama_set_name: RedPajamaCommonCrawl


Prickly pear cacti (Optunia spp.) are easily recognized by their upside down pear shaped pads or segments. Their grey-green fleshy pads generate showy flowers Flagstaff that are big in the spring that change into red fruits later in the period. You'll find more than 150 types of prickly-pear cacti, which are hardy in Sunset's Environment Zones 1 2 through 24.These drought-tolerant crops make extremely low-maintenance house plants that include a bit of South-Western flair to the decor of a room. Prickly pear cacti are propagated through cuttings.
Put while operating together with the cactus, to protect your fingers from injuries. Have a cutting from a disease-free prickly-pear cactus. Harvest cuttings only if night-time temperatures are hotter or a constant 60 degrees Fahrenheit. Choose a pad, or section that is oval, without scars or blemish that's SIX MONTHS months old. Grasp the very top of the pad gently. Cut the pad off a knife being used by the parent plant Chico in the normal seam at its bottom.
Place the cutting on a flat area in sunlight that is filtered. Choose a area that's continuous temperatures of 60 degrees Fahrenheit or or more. Leave the cutting for seven to 10 times to to make tissue.
Mix one part perlite with one part compost to produce a well-draining developing medium to plant San Diego the cactus in. Fill a container with the combination, leaving the best one to two inches empty. Use a container.
Set the cactus cutting using the edge resting on the soil in the middle of the pot. Push one third to one half of the cutting to the soil combination. Tamp the soil down across the bottom of the cactus before the cutting is capable to stand up right on its own.
Water the can until the s Oil is moist. Place the pot in a warm area — 6 levels hotter or Fahrenheit — in sunlight. The cactus when the leading inch of s Oil starts to dry.
Once it begins creating new growth to begin acclimating it to brighter mild move the cactus to complete sunlight in doors. In the event that you fundamentally program to plant Flagstaff the cactus out doors in the ground Magnesium chloride snow melt Aurora Lake City move the cactus to total sunlight out doors after two weeks.
Grow the cactus in the container for a-T least one-year. Once s Oil temperatures achieve 6 levels Fahrenheit transplant the cactus plant Fresno in the spring. Plant and possesses nicely and the cactus in a area that receives complete sunlight -draining s Oil. Dig a hole as broad as the root ball of the plant Long Beach and equivalent comprehensive. Place the root ball in the guts of the hole. Backfill the hole with s Oil and t-Amp it down. Water the cactus just after after planting Redding.
When night-time temperatures are above 6-0 degrees Fahrenheit, water the planted cactus once-per week. When night-time temperatures fall below 6-0 levels Fahrenheit water the cactus every week or two.


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text:
Q: C++ pass class method as parameter I'm working with the ESP32. Currently I am migrating my library from an ESP8266 codebase.
I'm currently doing the following:
m_pubSubClient.setCallback( std::bind(&CL::callbackHandler, this, std::placeholders::_1, std::placeholders::_2, std::placeholders::_3));

But it seems like the author of the PubSubClient library has changed the signature.
// old
#define MQTT_CALLBACK_SIGNATURE std::function<void(char*, uint8_t*, unsigned int)> callback
//new
#define MQTT_CALLBACK_SIGNATURE void (*callback)(char*, uint8_t*, unsigned int)

I can't figure out, how to do that with the new signature.

A: As plain function pointers can not be bound to member functions, you basically just have three choices left: you can declare CL::callbackHandler as a static member function, make it a free function or use a lambda expression (without closures), like
m_pubSubClient.setCallback([](char* param1, uint8_t* param2, unsigned int param3) {
    // handling the event
});

redpajama_set_name: RedPajamaStackExchange


text:
Home/Switzerland/Everyday role plays on a different platform
Everyday role plays on a different platform
Switzerland NewsDesk November 11, 2018 19:47
November 11, 2018 at 20:23:
On Xbox One, you can also experience the day-to-day priming of an unfaithful photographer who moves into a mysterious place.
As part of Microsoft's X018, White Owls announced that "The Good Life" will now be released for the Xbox One in 2019. The PC version and PlayStation 4 have already been confirmed.
In The Good Life, actor takes on the role of photographer Naomi of New York, who moves with British Rainy Woods to pay off debts. In the newspaper he performs a job and makes a strange discovery: residents of the city at night have the form of dogs or cats. The next morning, people do not know what happened at night. Of course, Naomi will not be saved.
The title of Yukio Futatsugi, who is also responsible for "Panzer Dragoon" and "Phantom Dust", is a daily action-adventure game where you have to earn money and discover the secret, according to developers.
The first impression of "The Good Life" can be found in the PAX West Trailer from 2017.
Source: Gematsu
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Green tea is said to help fight the coronavirus
By Switzerland NewsDesk


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Volvo Launches New V40 Hatchback At A Price Of Rs 24.75 Lakh
| June 17 , 2015 , 14:40 IST
Volvo Auto India on Tuesday launched it's highly anticipated V40 hatchback. The car was first unveiled at the Geneva Motor Show in 2012, The V40 will compete with the likesof BMW 1 series and Mercedes A-Class. The V40 in looks is similar to the Cross Country. The V40 comes in two varients, The kinetic and R-Design, Both the varients come packed with a Drive-E, four-cylinder engine lineup, 2.0-litre D3 twin turbo diesel, same as the Cross Country. The diesel engine produces a maximum power of 150PS @ 350Nm of torque. The engine works on a 6-speed automatic transmission. The V40 comes fully loaded with security features with ABS, traction control, dual zone climate control, parking censors and airbags – front, side, curtain and pedestrian as standard. The R-Design comes with a blacked-out grille, air dam, alloys, panormic glass roof along with rain sensing wipers, adaptive digital display console, a climate control system, leather seats, powered front seats etc and keyless start. The Volvo V40 Kinetic will be available for Rs 24.75 lac, whereas the Volvo R-Design Rs 27.7 lac.


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Consistency is my mantra for success, reveals this IRS officer and Apeejay Pitampura alumni
Researchers study performance of India's hospitality sector in post-COVID times
Jalandhar student wins second prize at inter-college hammer throw competition
Scholar-Journalist of the Week: Antlo – The planet of super beings
'My child, who is in KG, has no stage fear and speaks confidently'
'My son is very happy at his school'
Having trouble managing your money? Learn it from the expert
"Digital-first communication will dominate the PR industry in the upcoming years"
Vasant Panchami Special: Sketch by Sonika Gaur, Apeejay School, Kharghar
'Republic Day': Poem by Noor Kohli, Apeejay School, Tanda Road
Centre open to views on NEP: Dharmendra Pradhan
Coverage of Apeejay College of Fine Arts, Jalandhar, in The Tribune
With pride, students and teachers celebrated 74th Republic Day
Republic Day 2023: Delhi school hoists tiranga, salutes freedom fighters
'Making India – the land of our dreams'
Student bags award under 'Pariksha Pe Charcha' program
Varun Singhal may have cleared the Civil Services in his fifth attempt, but his heart was set on ushering in change in society at the highest level as a policy maker
Aasheesh Sharma
Varun Singhal
Assistant Commissioner,
Indian Revenue Service
Varun Singhal is an assistant commissioner with the Indian Revenue Service posted in New Delhi at present. After completing his schooling from Apeejay School, Pitampura, Varun graduated in Commerce from Delhi University and subsequently did a Young India Fellowship at Ashoka University before embarking upon his quest to join the Civil Services in order to "bring change at the highest level through policy making," he tells us in an interview. He reveals how he gravitated towards a career in bureaucracy and also shares his memories of the time he spent at Apeejay Pitampura along with a few valuable tips for IAS aspirants. Edited excerpts:
Please take us through your professional journey so far
After completing my schooling from Apeejay School, Pitampura, I pursued B.Com (Honours) from the Sri Guru Gobind Singh College Of Commerce, Delhi University. I was the president of the entrepreneurship cell in my college. I enjoyed warm relations with my college principal and organised festivals. I even invited Delhi Chief Minister Arvind Kejriwal to one of the events for a public interaction. I think it is at this point that I began the process of questioning the kind of education that I was receiving. I was getting good marks because I knew how and what I had to study. At the same time I was completely engaged in work that impacted society. I was part of a social entrepreneurship platform called SIFE and launched a start-up called Sochbook. I was completely immersed into the world of business and start-ups till I stumbled upon the Young India Fellowship at Ashoka University. I read about it in the education supplement of a newspaper and it seemed like an experiment in education I would want to experience. Beyond just education, the fellowship helped you develop your overall life purpose. Without telling my friends or parents, I wrote a statement of purpose and submitted it. Finally, when I got an interview call, I went for the interview without telling anybody. Since I was dressed formally in a suit, my parents assumed I was going to attend an event in my college and I cleared the interview successfully.
Why do you call the Young India Fellowship a departure from convention?
The Young India fellowship (YIF) was one year of an entirely new revolutionary approach towards education and mindful exposure that you experienced with your peers. I was part of a group of 100 people from entirely different backgrounds. These included, among others, a painter, a car racer, a scholar, an entrepreneur and a filmmaker. We were staying under one roof and getting lessons in 24 subjects from the best professors one can hope to find in the world. Apart from that, we had guest lectures from amazing teachers such as economist Mihir Shah, sociologist Andre Beteille, Rudrangshu Mukherjee and historian Ramachandra Guha. These professors opened our minds like anything. We used to challenge the status quo and have day-long discussions. YIF helped open my mind and led me to questions such as what my purpose was and what I wanted to become.
How did you gravitate towards the Civil Services, coming from a business family?
Once you do the Young India Fellowship you can head in any direction. In a year we studied 24 diverse subjects which ranged from Statistics, Mathematics, history, painting, psychology, philosophy, you name it and we had it. Even Vinod Rai was one of the guest lecturers. These people open up your mind and encourage you to think out of the box and do something that might take you towards your life purpose. Basically, during the fellowship, I questioned my idea of life. I also went to my father's business for some time and just for the sake of getting out of business and even tried my hand at chartered and financial analyst (CFA) level one. But I couldn't imagine being a chartered and financial analyst for the next 30-40 years of my life. I met a guy at the library who told me about the Civil Services. Apart from that, celebrated economist Mihir Shah who taught us during the fellowship, used to tell us about how to bring change in society. At the policy level the highest means of change can be achieved, he said. If one makes it to the Civil Services or joins politics, you can bring policy level change that is the highest form of change. That helped me gravitate towards a career in bureaucracy.
What are your memories of the time you spent in school at Apeejay, Pitampura?
I studied for my entire school life at Apeejay Pitampura. I clearly remember my interaction with DK Bedi Sir, who was the principal at that time. He always used to ask us what our 'purpose' was. I remember a conversation I had with him in class 12. He would come and ask us: Why do you want to do what you want to do? For instance, if someone wanted to do chartered accountancy or an MBA, he would ask them for their purpose. One of my peers said he wanted to do a CA as well as an MBA. So, Dr Bedi said one shouldn't just pursue a subject just because it was the fad at that time and that no person in his right mind would do a CA as well as an MBA because they were two completely distinct profiles. CA is an accounting job and an MBA is a management profile. Think about what you would accomplish by doing both. These may appear as trivial , but we found answers to such small questions thanks to the personal touch with teachers and the attention they paid to every student. It helped students understand the purpose of their lives. Similarly I remember my interaction with Taruna Kalra and Sangeeta Ma'am, our class-teacher. Going beyond teaching their subjects, these teachers would treat us like their own sons and daughters. I remember Smita Ma'am teaching us history. She told us history is not just rote learning and you must understand the subject. Indu Malik Ma'am was my class teacher in class 11. I recollect a funny incident that featured her. I had opted for Commerce in class 11 and she was asking everyone why each of us had opted for their stream. I stood up and said my father is doing business and I have to go into business and that is why I took Commerce. My classmates laughed at my answer, but after everyone was done laughing, she said: What's wrong if he wants to go into business? He is being honest in saying that he chose Commerce for a reason. At least he has a reason to support his answer. I was encouraged by this. I remember these words even a decade later. Such words encourage you and help you move forward. One must be true to oneself even if others laugh at you.
I recently returned to my alma mater and met the current principal Dr Veena Goyal and other teachers and perceived the same feelings of warmth towards their alumni. It was a wonderful feeling to meet some of the teachers who taught me and are still around and revisiting memories of my school days.
Please elaborate upon your approach your preparation for the Civil Services.
I started preparing for the Civil Services in 2016. I started my preparation with a coaching centre and received some good knowledge and guidance there but left it soon to focus on self-study. I used to study with my friends in a group. Self-study and group study and trying to learn from each other is something that led me to my success I think. It was a long journey indeed and it was a slow journey full of disappointments as well, but later I got a taste of success.
Although law is not my background, fortunately I secured the highest all-India score in law and even taught law to UPSC and judiciary aspirants for some time. I cleared the exam and I got into the Indian Revenue Service, customs and indirect taxes and I had a year of theoretical training at the National Academy of Customs, Indirect Taxes and Narcotics (NACIN) in Faridabad and am now posted in Delhi as an assistant commissioner.
Did you clear the Civils in the first attempt itself?
No, in the first attempt I could not clear the prelims. In the second attempt I could not clear the mains and in the third attempt I could not clear the interview. In the fourth attempt, I could not clear the prelims again and I finally got through in the fifth attempt. Still, I was determined it was my calling and I wanted to bring in change at the highest form of policy making. Apart from this determination, when you take an exam and miss it by a mark or two, it gives you confidence that you would clear it the next time and makes you go on.
You come from a business family, did CFA and chose to be in the Civil Services and made it on the fifth attempt, how did you stay motivated?
If you have a clear purpose, it will definitely help you to stick to your plans. Even in the case of failures, your purpose will drive you over it. In my case I identified my purpose as bringing in change at the highest level through bureaucracy or politics, as advised by my Professor Mihir Shah. I found it to be a really meaningful purpose.
Now that you are in the IRS, what do your job responsibilities entail?
I am still new to the service. After completing my training, I am meeting my senior officers and getting introduced to various verticals in the Service. At different departments you have different responsibilities and my department primarily has three departments: GST, Narcotics and Customs. Apart from these, we have the directorate of revenue intelligence, the Directorate General of GST Intelligence and the Enforcement directorate. You work on various solutions and work closely with the customs, GST officials and Narcotics officers. You have responsibilities depending upon the department you are working in. Right now I am into GST. My work is to analyse GST returns, to apprehend the violators and trying to facilitate GST transactions and augmenting the government's revenue. Put simply: We are not revenue officers, we are law enforcement officers. Because at the end of the day, we also enforce the law as and when required. Normally all the bureaucrats work in tandem whether it is the Civil Services or the IRS or the IPS.
Please share some tips for those who want to follow your career trajectory?
I generally tell others that slow and steady wins the race. I say this because a lot of people clear Civil Services in the first attempt and a lot of others do not. And the majority of the aspirants do not. If you are true to your goal, slowly and steadily keep at it. You need not study 14 hours a day. You need to study six hours a day but do it every day without any breaks even on weekends or holidays. Once in a while you can skip it as well. Look at it as a marathon and not as a 100-metre sprint. Consistency and regularity always pay rewards and this is true for any profession.
Related Topics:Apeejay School PitampuraCivil ServicesFeaturedGSTIASIndian Revenue ServiceIRSSchools
Aasheesh Sharma is a seasoned journalist with an experience of more than 25 years spread over newspapers, news agencies, magazines and television. He has worked in leadership positions in media groups such as Hindustan Times, India Today, Times of India, NDTV, UNI and IANS. He is a published author and his essay on the longest train journey in India was included in an anthology of writings on the railways, brought out by Rupa Publications. As the Editor of Apeejay Newsroom, he is responsible for coverage of the latest news and developments in the Apeejay institutions. He can be reached at [email protected] He tweets @Aasheesh74
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The fair brought prospective workers and young professionals face to face with companies looking to expand their LGBT and ally workforce, including Walgreens, Boeing and NASA. Out For Work Operations Manager Alex Gant said it's important to create a career conference for the LGBT community and address the very specific issues of dealing with being open in the workplace.
Story length: 763 words; Access the full story here.
This particular news story highlights the complexities of difference in organizational communication and how they might influence everyday practices of organizing. Simply put, "difference is both a construct – a sensitizing device – through which the complexities of organizational communication processes can be examined and a constitutive feature of everyday organizing" (Mumby, 2011, p. ix). This past year I was fortunate enough to attend the national Out For Work conference in Chicago, highlighted here in the article, and so I offer three key takeaways as we engage in these conversations.
First, this conference begins the socialization process for many lesbian, gay, bisexual, and transgender (LGBT) students entering the workplace. The article points out the invisibility of the LGBT identity, in that unlike "typical" demographic characteristics, the LGBT identity marker is not apparent on the surface-level. In essence, a student could "pass" as straight at a career fair if they wanted to because it is not a defining physical attribute. Many LGBT students seeking jobs attempt to hide or mask this identity until they can discern whether or not the organization is a "safe space." However, this conference provides that "safe space" from the beginning and helps students move beyond differences towards organizational acceptance.
Second, the conference highlights that if an individual is able to be him/herself in the organization, they are much more likely to live up to their full potential as an employee. The article suggests that being "out" might not have anything to do with an individual's actual job duties or expectations, yet it does influence the individual's communication in the workplace. If we consider Petronio's (2002) Communication Privacy Management theory, "closested" employees may constantly be considering what types of information they should conceal or disclose. And if someone is managing their identity, they might have a more difficult time managing their communication and focusing on their job responsibilities in the workplace.
And finally, something else to question from this article is in relation to Kirby and Krone's (2002) journal article "The Policy Exists But You Can't Use It." While I am a strong advocate of career fairs and conferences such as these, the true test lies beyond the career fair in the lived experiences of organizational culture. Policies of non-discrimination and domestic partner benefits for LGBT populations (that these organizations were promoting at the conference) may exist on paper, but we do not know how they are enacted until we are immersed in the organizational culture.
With this I pose the following question: How do we as organizational scholars, practioners, and students enact and foster difference as a central feature in organizations?
Petronio, S. (2002). Boundaries of privacy: Dialectics of disclosure. New York: State University of New York Press.
Read more "Takeaways" from Tim and other scholars here.
Posted in Careers + Work, Diversity + Difference, Identity, Socializing/ Socialization, Work-LifeTagged employee benefits, GLBTQ, hiring/ recruitment, labor conditions, worker retention, workplace dignityBookmark the permalink.


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La Perla del Once fue un bar notable de Buenos Aires (Argentina), ubicado en la zona del Once, en el barrio de Balvanera, que cerró sus puertas en 2017. Fue conocido por ser lugar de reunión de intelectuales en la década de 1920, y más tarde, en la década de 1960, por ser el lugar donde se compuso La balsa, considerada la canción fundacional del rock argentino. Debido a ello, en 1994 fue declarado Sitio de Interés Cultural de la Ciudad Autónoma de Buenos Aires.

Historia

En la década de 1920 La Perla fue un lugar de encuentro de jóvenes intelectuales, entre ellos Jorge Luis Borges, que acudían al local a escuchar al filósofo y escritor Macedonio Fernández, quien realizaba largas charlas y debates sobre metafísica y filosofía.

En 1957, algunos miembros de la Unión Nacionalista de Estudiantes Secundarios (UNES) entre los que se encontraba Alberto Ezcurra,  decidieron crear el Movimiento Nacionalista Tacuara durante una reunión celebrada en La Perla. Diez años más tarde, en 1967, Rodolfo Galimberti decidió abandonar Tacuara para formar su propia organización, las Juventudes Argentinas para la Emancipación Nacional (JAEN), cuyo inicio también se gestó en La Perla, y de la que participaron, en su primera época, importantes nombres del rock argentino, como Luis Alberto Spinetta y Emilio del Guercio.

Durante la década de 1960, junto con La Cueva, fue uno de los locales en los que se desarrollaron los inicios del rock argentino. Fue en los baños de La Perla donde, en mayo de 1967, Tanguito y Lito Nebbia compusieron el tema La balsa, que interpretado por Los Gatos se convirtió en un éxito y está considerado como el tema emblemático y fundacional del rock argentino.

El local, que fue declarado Sitio de Interés Cultural de la ciudad en 1994, cerró sus puertas el 14 de enero de 2017. Un mes más tarde reabrió como parte de una cadena de pizzerías, tras ser enteramente reformado, y conservando únicamente del histórico local la placa en la puerta de los baños conmemorativa del origen de La balsa.

Galería

Véase también
 Bares de Buenos Aires
 Bares Notables
 Lista de bares notables de Buenos Aires
 Bienes de Interés Cultural de la Ciudad de Buenos Aires

Referencias

Bibliografía

Enlaces externos

Bares desaparecidos de la ciudad de Buenos Aires
Bares notables de la ciudad de Buenos Aires
La Perla
Rock_nacional_(Argentina)
Balvanera (Buenos Aires)
Sitios de Interés Cultural de la Ciudad de Buenos Aires


Your October update of competitions has just been uploaded to the Simply Prizes website! You can get started on your entries right now, and this month we've got 425 brand new comps for you, including your chance to win £20,000 cash, the chance to switch on the Blackpool Illuminations, a tour of Vietnam and Cambodia, a Fiat 500 and much, much more. So what are you waiting for!
Your magazine will be with you on Saturday, and with that I have some rather exciting Simply Prizes news. From November we are going completely online, which means this will be your last paper issue of the magazine! With the world of comping moving almost entirely online, by introducing a new weekly service instead of a printed magazine, we'll be able to deliver an even better comping experience for you. And although we will miss bringing you a monthly paper issue, this new format will not only give you lots more competitions, but it'll also make entering them easier and spread the fun of your hobby out over the month.
The new weekly format isn't the only thing we have up our sleeves; we'll be bringing you a weekly exclusive subscriber-only in-house competition, alongside lots of regular blog and video updates from me. As the service develops there will be lots more exciting benefits for all of our customers too. When your magazine arrives, head on over to page 34 where you can read more about this change straight from our publisher, Dave Gibson.
I hope you enjoy this new format; it'll mean extra competitions, and extra content for you, as well as helping to make Simply Prizes the best comping community on the web.
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text:
All categories City of San Luis Economic Development Home Press Release Public Notice
City of San Luis
CITY OF SAN LUIS TO HONOR CESAR CHAVEZ
San Luis, Arizona — Back in January 2019, the Mayor and City Council of the City of San Luis, Arizona passed Resolution No. 2062; a resolution of the Mayor and City Council adding Cesar Chavez Day as an official holiday of the City of San Luis. This would only be the beginning of what was to come with the celebration of the life and legacy of Cesar Chavez.
The Cesar Chavez Life and Legacy Committee, a local committee consisting of San Luis Elected Officials, community members, and different organizations, has been working since mid-2018 in putting together a month-long celebration of various events and activities for the community to work together on and celebrate Cesar Chavez hard work.
Cesar Chavez was born on March 31, 1927, in North Gila Valley, near Yuma, Arizona. At just the age of 10, Cesar Chavez knew the struggles farm workers faced, as he was one himself. At a young age, Cesar Chavez started dedicating his life to improving the treatment, pay, and working conditions of farmworkers; founding the United Farm Works (UFW) in 1962.
"Cesar Chavez is a hero to many and an inspiration to all, who made history by speaking up for the voiceless and advocating for the rights and dignity of farmworkers," stated Mayor Gerardo Sanchez. "This is why the City of San Luis has approved to observe his birthday as a city holiday and will be working alongside The Cesar Chavez Life and Legacy Committee on celebrating his legacy the entire month of March," he added. "Let's continue to celebrate his history and hear from the pioneers of the movement that are still in San Luis, Arizona; by recognizing Cesar Chavez Day and honoring his legacy we are returning the favor to the pioneers and their families. We are keeping the life and legacy of Cesar Chavez fresh in the memory of all our residents," Mayor Gerardo Sanchez concluded.
The calendar of events for the Cesar Chavez Life and Legacy Celebration is as follows:
Cesar Chavez Day Proclamation
San Luis Council Chambers
Cesar Chavez Life & Legacy Parade
Juan Sanchez Blvd from 8th Ave to Joe Orduño Park
Cesar Chavez Life & Legacy Celebration
Joe Orduño Park
Harvest Dinner
Cesar Chavez Cultural Center
Cesar Chavez Birthday Celebration
For more information about this topic, or to schedule a meeting with the City of San Luis Public Affairs Office, please call us (928) 341-8520 or email us at publicrelations@cityofsanluis.org.
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text:
NL Rookie of the Year Heats Up
Your City Sports-Cleveland
You have to go back to 2003 to find the last National League Rookie of the Year Award that was handed out to a pitcher. Jaime Garcia is doing everything in his power to end the six-year drought. His Sunday outing, a complete-game 9-0 victory against the Giants, brought his name back into the RoY discussion.
2010 was supposed to be the year of Steven Strasburg and Aroldis Chapman. Yet these two prodigies have combined for just twelve Major League starts and two trips to the Disabled List (all statistics accumulated by Strasburg alone).
In effect, the RoY predictions of baseball insiders were looking towards the wrong horizon for the next great National League pitcher. The incredible start of a 24 year-old Cardinal caught the entire Major Leagues off-guard--including the St. Louis organization.
Unfortunately for Garcia, there was one position player on everyone's radar in Spring Training that has panned out. The Braves' Jason Heyward burst on the scene with a home run in his first Major League at-bat and has not looked back since.
Heyward and Garcia do have something in common. They both left Spring Training with a spot on the Opening Day roster for their Big League clubs. They joined Ian Desmond (Nationals), Alcides Escobar (Brewers), Mike Leake (Reds), David Freese (Cardinals), Jon Niese (Mets), Tyler Colvin (Cubs), and Gaby Sanchez (Marlins) as the only National League rookies to do so. The rookie spotlight was there's alone; a nice head start over the rest of the field.
Unfortunately, Heyward was given a media boost to push his name out there even further. The Opening Day homer did nothing to hurt the hype. So even though Garcia has current statistics that are far superior to Heyward, scouts and analysts christened the Braves' outfielder as the Rookie of the Year before the season even started. He parlayed all of this publicity into an unworthy All-Star starting position. It would not surprise anyone if he rode the hype train straight to the Rookie of the Year Award.
This is not a knock on Heyward. He is a franchise-changing player, one that will probably have a better career than anyone else in this rookie class. The Braves should lock him into their right field spot for the next 20 years. But he has missed a few games with an injury, and is only hitting .265 for the season.
The award is not given out to the player that exudes the most potential for future greatness. Heyward would win that, no arguments. It is for the best statistical season by a rookie, one year sample size.
It probably will not happen, but Cubs' shortstop Starlin Castro and Florida's Gaby Sanchez should each receive more votes than Heyward. They are frankly having better seasons. And even they are not at the head of the rookie class.
Garcia's ERA is now seventh in the National League, not just among rookies. He is one of the best left-handed pitchers in the game right now, regardless of Major League tenure. That is the mark of a Rookie of the Year--someone who blends in so seamlessly to a lineup or rotation that their first-year status can quickly slip people's minds. A winner of the award sure does not look like a rookie by midseason.
Every year, however, there is one player that bursts on the scene late and never cools down. This year's player is San Francisco's Buster Posey. The catcher/first baseman exceeded the Giants' organization expectations so rapidly that he left manager, Bruce Bochy, scrambling to find an everyday spot for him. Eventually, Posey forced out the incumbent backstop, Bengie Molina.
If the writers want to reward pure hitting, they will vote for Posey. Whereas Heyward is currently sitting eighth in batting average among NL rookies (minimum 250 at-bats), Buster Posey leads that category with a .341 average. This is nearly 90 points higher than Heyward.
Posey also leads in slugging and on-base percentages, which means he is also tops in the most unnecessarily talked about statistic in baseball--OPS (on-base plus slugging).
It is just another fluff category that Posey can say he leads. Pitchers could throw two categories together and make up a new stat too. Jaime Garcia leads all rookies in WPSO (wins plus strikeouts). This amalgamation of indirectly-related figures makes about as much sense as OPS. There is no narrative to substantiate its value. It simply has John Madden logic that great players will have great numbers.
People need to be more conscientious about what they put under a microscope. Data has to speak to something you wish to prove, something that cannot be put into words. News flash: a great power hitter will have both a high on-base percentage and a high slugging percentage. It might be a better tool than batting average, but it is not a pure offensive value. Why summate two known numbers if it is not going to tell you anything different?
If the writers seek to reward a freakishly talented power hitter, Posey is not even close to the best candidate. Gaby Sanchez, of the Florida Marlins, has 48 extra base hits. His teammate, Mike Stanton, has 14 home runs in just 63 games.
Voters cannot sleep on Ike Davis' 15 homers, playing his home games in the caverns of Citi Field. The leader among all rookies in the stat is Tyler Colvin, with 18 home runs.
So there is a pair of C's in Chicago: Castro and Colvin. There is also an alliteration of candidates in Florida: Stanton and Sanchez. Sprinkle in a Posey in San Fran, a Davis in the Big Apple, and a Heyward in Hotlanta and that is the list. The Cardinal faithful can hope that the "same-team vote split," the one that infamously plagued Wainwright and Carpenter in last year's postseason awards, can take down the two Cubs and two Marlins.
This canceling out would only act like the 50/50 in "Who Wants to Be a Millionaire?" The game show staple never took away anything contestants actually wanted; only those they already knew had no shot at being the answer. The two favorites, the leaders in the proverbial clubhouse, would still be in the race.
If each hitter statistically negates one another, there is only one pitcher in the National League that would emerge. This first-year player leads the entire Major League rookie class (not just the NL) in wins, ERA, innings pitched, strikeouts, and now tied in shutouts.
Garcia has also been with a pennant-contending team for the entirety of the season. Buster Posey did not see his first Major League action until Memorial Day. By that time, Jaime already had a 5-2 record and a 1.32 ERA.
What the RoY cannot turn into is a team-MVP contest. Heyward is already one of the best players on his club; same with Posey. You could make an argument that Garcia is the sixth or seventh best player on the Cardinals. This is hardly his fault, nor what the award is about.
Sadly, past votes would appear to carry that line of thinking. In 2003, Angel Berroa was the best thing going for a lousy Royals team. It definitely hindered Hideki Matsui that he had the protection of a Yankees lineup.
Garcia will make a maximum of eight more starts. A realistic 5-2 record would push him to 16-8 overall, and most likely sustain his sub-3.00 ERA. This would really put the pressure on the voters to take notice of someone not named Heyward or Posey.
Regrettably, the preseason brainwashing might be too much for the Cardinals' lefty to overcome.


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text:
Are problems in Quantum Information Theory (un)decidable?
Michael M. Wolf, Toby S. Cubitt, David Perez-Garcia Department of Mathematics, Technische Universität München, Email: Departamento de Analisis Matematico, U. Complutense de MadridDepartamento de Analisis Matematico, IMI, U. Complutense de Madrid
This note is intended to foster a discussion about the extent to which typical problems arising in quantum information theory are algorithmically decidable (in principle rather than in practice). Various problems in the context of entanglement theory and quantum channels turn out to be decidable via quantifier elimination as long as they admit a compact formulation without quantification over integers. For many asymptotically defined properties which have to hold for all or for one n∈N, however, effective procedures seem to be difficult if not impossible to find. We review some of the main tools for (dis)proving decidability and apply them to problems in quantum information theory. We find that questions like "can we overcome fidelity 1/2 w.r.t. a two-qubit singlet state?" easily become undecidable. A closer look at such questions might rule out some of the "single-letter" formulas sought in quantum information theory.
\titlecontents
subparagraph[5.25em]\contentslabel4.6em\titlerule*[0.6pc].\contentspage
2 Decidable problems
2.1 A general tool from Tarski and Seidenberg
2.2 Entanglement
2.3 n-distillability
2.4 Existence of an (n,m)-hidden variable model
2.5 Existence of a d−dimensional quantum representation
2.6 Birkhoff property
2.7 n−shot zero-error capacity
2.8 Additive minimal output entropy
3 Undecided problems
3.1 Distillability:
3.2 LHV models:
3.3 Quantum representation for correlations:
3.4 Birkhoff property:
3.5 Zero error-capacities:
3.6 (Non-)additivity of other capacities:
3.7 More problems
4 Undecidable problems
4.1 Tools for proving undecidability
The halting problem
Post's correspondence problem
Diophantine equations.
4.2 Reachability of fidelity thresholds is undecidable
The elementary objects of quantum information theory are often rather simple small dimensional matrices—for instance representing density matrices, quantum channels or observables. In spite of their innocent looking mathematical description, their properties can be difficult to compute quantitatively or even to decide qualitatively: Is a given quantum state entangled? Is it distillable? Does it admit a local hidden variable description? Is a quantum channel a mixture of unitaries? Does it have non-zero quantum capacity? None of these questions seems to have a simple answer.
From an abstract point of view, the difficulty in deciding such questions arises from quantification over infinite sets. That is, answering these questions requires taking for instance all decompositions, protocols, models, codings, etc. into account. Moreover, in particular information theoretic questions often involve asymptotic limits regarding the numbers of copies, rounds within a protocol, measurements, etc..
In the present work we analyse problems like the ones mentioned above from the point of view of algorithmic decidability. This means we don't ask (at least not on the following pages) whether there are resource-efficient algorithms, we rather want to know whether there is any effective algorithm, irrespective of the scaling of resources.
One of our motivations---although the following work is only a minor step in this direction---is to address the (im)possibility of ''single-letter'' formulas/criteria, especially for asymptotically defined quantities. Clearly, those arise not only in information theoretic contexts but rather naturally also in statistical physics, where phases and their properties are defined in the thermodynamic limit. In the present paper we will, however, focus exclusively on problems occurring in quantum information theory.111Undecidablility of properties of quantum spin chains will be the content of a separate article [CPGW] . Moreover, we will restrict ourselves to decidability problems and, for the moment, leave closely related questions of (quantitative) computability aside.222Note that a quantity, e.g. some capacity, might be computable although it is undecidable whether or not it is different from zero.
We will go through three stages: decidable problems (Sec.2), undecided problems (Sec.3), and undecidable problems (Sec.4). The notion of decidability which we have in mind is the one of recursive function theory rather than, for instance, the one of real computation (as in [BCSS97] ).
Others have followed similar lines: decidability problems in quantum automata were for instance studied in [BJKP05, DJK05, Hir08] and results on quantum implications and interpretations of the undecidable 'matrix mortality' problem are found in [JE] .
As already mentioned in the introduction, deciding properties of quantum information theory often involves taking for instance all decompositions, protocols, models, codings, etc. into account. Mathematically this can be expressed in terms of quantification over the respective sets. In this section we discuss a general tool for proving decidability via elimination of all these quantifiers.
Let us introduce the general idea via a simple example: if a1 and a2 are real numbers, then the equation a1x2>a2 has a real solution iff a2 is negative or a1 is positive. More formally,
∃x a1x2>a2, (1)
⇔ a2<0∨a1>0. (2)
That is, in Eq.(2) we have a quantifier-free expression which has the same truth value as the initial formula in Eq.(1).
Tarski [Tar51] and Seidenberg [Sei54] have shown that such an elimination of both existential ∃ and universal ∀ quantifiers is possible in a more general context. Sufficient requirements are that the initial formula (i) contains real variables and parameters together with the objects and relations 1,0,+,⋅,>,= of the ordered real field, and (ii) is expressed in first-order logic. The latter means that in addition to quantifiers over individual variables the formula is allowed to involve Boolean operations ∨,∧,¬ and brackets for unambiguous readability. Something like ∀n∈N… or quantification over sets, however, is not allowed. A more precise formulation of the underlying theorem is (cf. [Mar08] ):
Theorem 1 (Tarski-Seidenberg, quantifier elimination)
Let R be a real closed field. Given a finite set {fi(x,a)}ki=1 of polynomial equalities and inequalities with variables (x,a)∈Rn×Rm and coefficients in Q. Let ϕ(x,a) be a Boolean combination of the f′is (using ∨,∧,¬) and
Ψ(a):=(Q1x1…Qnxn:ϕ(x,a)), Qj∈{∃,∀}. (3)
Then there exists a formula ψ(a) which is (i) a quantifier-free Boolean combination of finitely many polynomial (in-) equalities with rational coefficients, and (ii) equivalent in the sense
∀a: (ψ(a)⇔Ψ(a)). (4)
Moreover, there exists an effective algorithm which constructs the quantifier-free equivalent ψ of any such formula Ψ.
A real closed field is an ordered field which satisfies the intermediate value theorem for polynomials. Examples are the field R of real numbers, the field of computable real numbers and the field RA of algebraic real numbers. Algorithms for quantifier elimination are effective but not generally efficient [BPR94] . For a comparison of some of the implemented algorithms see [DSW98] .
Suppose now we want to decide whether or not some Ψ(a) is true for given a. Then we can use quantifier elimination in order to obtain a simpler, quantifier-free expression ψ(a), so that it remains to check a set of (in-)equalities for polynomials in the ai's, like in Eq.(2) of our example. In order for this to be feasible we need a computable ordered field in which the basic operations ⋅,+ as well as the relations =,> are computable. The latter is not the case for the fields of real or computable real numbers—being able to compute an arbitrary number of decimals for a1 is not sufficient for an effective procedure for deciding a1>0.
In the following we restrict the parameters a to any computable ordered field Rc which is real closed and contains the field of algebraic numbers Ralg. The latter forms a computable ordered and real closed field on its own [LM70] so that we may simply set Rc=Ralg. With some care, however, including (computable) transcendental numbers is possible: one can for instance start with non-algebraic extension fields of Q as in [O.L09] and then use the fact that any real closure of a computable ordered field is again a computable ordered field [Mad70] . By Cc:=Rc(√−1) we will denote the algebraic closure of Rc. If for instance Rc=Ralg, then Cc is the field of complex algebraic numbers.
In spite of all this, we will apply Thm.1 to R=R, since the problems we will discuss involve quantification over the reals.
Problems arising in quantum information theory often involve quantification over matrices or linear maps, such as unitaries, positive semidefinite matrices, positive or completely positive maps, matrices with bounded rank or bounded norm, etc. Moreover, relevant inequalities are often w.r.t. the semidefinite partial ordering of Hermitian matrices.
Before discussing specific problems from quantum information theory, we will convince ourselves that the Tarski-Seidenberg trick works as well if quantification is over any of the sets just mentioned. We will denote by Md(G) the set of d×d matrices with entries in G.
Let x∈R2d2 be a vector containing the real and imaginary parts of a matrix X∈Md(C). The membership relation X∈S is equivalent to a Boolean combination of polynomial (in-)equalities in x with integral coefficients, if S⊂Md(C) is one of the following sets:
the set positive semidefinite matrices,
the set of density matrices ,
the set of unitaries,
the set of Hermitian matrices with bounded norm ||X||p≤1 for any p∈N∪{∞},
the set of matrices with constrained rank rank(X)=r for any r≤d.
Proof. 1. follows from the fact that X≥0 holds iff X=X† and all the principal minors of X are non-negative.
2. follows from 1. and the linear equation tr[X]=1.
3. follows from the quadratic equation X†X=1.
4. since ||X||∞≤1 iff (1−X†X)≥0 we can use 1. For p∈N we obtain a polynomial inequality from tr[Xp]≤1.
5. follows from the fact that the rank is the size of the largest non-vanishing minor.
In general, sets which can be characterized by means of polynomial (in)-equalities in a finite-dimensional real vector space are called semialgebraic.
We will call a relation F on Md1(C)×…×MdK(C) a polynomial matrix equation with rational coefficients iff it can be expressed as ^F(M1,…,MK)⊳0, where ^F is a matrix (possibly one-dimensional) whose entries are polynomials in the entries of the Mi's with rational coefficients and ⊳∈{>,≥,=}. The inequalities refer to the partial order induced by the cone of positive semidefinite matrices.
The above Lemma together with the Tarski-Seidenberg theorem now leads to the following useful observation:
Corollary 1
Let A be a tuple of matrices with entries in Cc and let {Fi(X,A)}ki=1 be a finite set of polynomial matrix equations with rational coefficients and variables X∈Md1(C)×…×Mdn(C). Given a Boolean combination ϕ(X,A) of the Fi's and
Ψ(A):=(Q1(X1∈S1)…Qn(Xn∈Sn):ϕ(X,A)), (5)
where Qj∈{∃,∀} and the Si's are semialgebraic sets (e.g. those specified in Lemma 1). Then there exists an effective algorithm which decides Ψ(A).
This corollary uses Thm.1 after we have expressed everything in terms of the real and imaginary parts of the matrix entries, used Lemma 1 and brought the equation into prenex normal form. After the quantifier elimination we then exploit that A is specified over a computable ordered field which finally admits an effective algorithm.
Let us apply this observation to some standard problems in the context of entanglement and quantum channels. All the problems discussed in the remainder of this section are such that they involve quantification (∃,∀) over semialgebraic sets, like the ones in Lemma 1, and a semialgebraic equation to be checked. Consequently, Cor.1 applies to all of them.
A density matrix ρ∈Md(C)⊗n is called classically correlated or unentangled [Wer89] iff there exist sets of density matrices {ρ(α)i∈Md(C)}d2ni=1 for α=1,…,n and a probability vector λ∈Rd2n such that333The bound on the range of i, which is not part of the definition, follows from Caratheodory's theorem.
ρ=∑iλiρ(1)i⊗⋯⊗ρ(n)i. (6)
If the entries of ρ are taken from Cc, then the question whether or not ρ is entangled is of the form in Cor.1 and can thus be effectively decided.
A density matrix ρ∈Md(Cc)⊗2 is n−distillable iff there exists a rank-2 matrix Y∈Mdn(C) such that
⟨y|(ρ⊗n)T1|y⟩<0, (7)
where |y⟩=∑k,lYk,l|k,l⟩ and T1 denotes the partial transposition (both understood w.r.t. the bipartite rather than the n-partite partitioning). For fixed n∈N this is again such that Cor.1 implies an effective decision procedure.
Let P(i,j|k,l) be a conditional probability distribution so that ∑mi,j=1P(i,j|k,l)=1 for all k,l=1,…,n. P∈Rm2n2 admits a local hidden variable (LHV) model [WW01] iff there exists a non-negative matrix Λ∈Mmn(R) satisfying ∑a,bΛa,b=1 such that for all i,j,k,l it holds that
P(i,j|k,l)=∑a,bΛa,bδa(k),iδb(l),j, (8)
where a,b∈{1,…,m}n. We say that a bipartite density matrix ρ∈Md(Cc)⊗2 admits an (n,m)-hidden variable description iff the distribution
P(i,j|k,l):=tr[ρ(Q(k)i⊗P(l)j)] (9)
admits a LHV model for all sets of POVMs {Q(k)i∈Md(C)} and {P(l)j∈Md(C)}, where i,j label the m effect operators of each of the 2n POVMs. Since the latter are again characterized by semialgebraic means, the question whether or not a quantum state admits an (n,m)-hidden variable description is effectively decidable by Cor.1. Note that the quantifier structure of this problem involves also universal quantifiers as it is of the form ∀Q∀P∃Λ….
Conversely, we may ask whether a probability distribution P∈Rm2n2c admits a d−dimensional quantum representation in the sense that there exists a density matrix ρ∈Md(C)⊗2 and sets of POVMs {Q(k)i∈Md(C)} and {P(l)j∈Md(C)} such that Eq.(9) holds for all i,j,k,l. Again Cor.1 applies.
Let us turn to quantum channels T:Md(C)→Md(C), i.e., completely positive, trace-preserving linear maps. For a unital quantum channel, i.e., one which satisfies T(1)=1 in addition, there may exist an n∈N, unitaries {Ui∈Mdn(C)} and a probability vector λ such that
T⊗n(ρ)=∑iλiUiρU†ifor% all ρ. (10)
For d=2 this holds in fact for all unital quantum channels (even for n=1). However, it fails to be generally true if d≥3 [HM11] . Since we can bound the range of the index by Caratheodory's convex hull theorem to i={1,…,d4n} the problem of deciding whether a unital quantum channel can be represented as in Eq.(10) again falls into the field of activity of Cor.1 if n is fixed and the channel's Choi matrix or its Kraus operators are over Cc.
A quantum channel T:Md(C)→Md(C) has a non-vanishing n−shot classical zero-error capacity, iff there exist density matrices ρ,σ∈Mdn(C) such that
tr[T⊗n(ρ)T⊗n(σ)]=0. (11)
One more time, since the quantification is over the set of density matrices and all equations are (semi-)algebraic, Cor.1 applies for any fixed n∈N.
More generally, the n−shot classical zero-error capacity is greater than logm/n iff there exist m density matrices ρi∈Mdn(C) (i=1…m) such that
tr[T⊗n(ρi)T⊗n(ρj)]=0 (12)
for all i≠j and Cor.1 applies once again for any fixed n,m∈N.
For any p∈N∪{∞} define a functional νp on the set of quantum channels by
νp(T):=sup{||T(ρ)||p∣∣ρ≥0∧tr[ρ]=1}, (13)
and consider the decision problem whether or not
νp(T⊗T′)=νp(T)νp(T′), (14)
holds for all quantum channels T′:Md(C)→Md(C) of a fixed dimension. Since the l.h.s. in Eq.(14) is always lower bounded by the r.h.s., this problem can be phrased using quantifiers (over the respective sets) in the following way:
∃ρ1∀T′∃ρ2∀ρ12:||(T⊗T′)(ρ12)||p≤||T(ρ1)||p||T′(ρ2)||p. (15)
Note that all sets and equations are semialgebraic. Hence, for fixed p and d, Cor.1 implies the existence of an effective algorithm which, upon input of a finite-dimensional channel T which is defined over Cc, determines the truth-value of Eq.(15).
Consider the problems discussed in 2.2 to 2.8 of the last section again. Apart from 2.2, all problems share the appearance of one or more integers whose quantification leads to a more fundamental problem. However, adding such a "∃n∈N" or "∀n∈N" disables the Tarski-Seidenberg tool and leaves us with problems for which, to the best of our knowledge, no effective procedure is currently known. The catch is, that for every n∈N we have an effective procedure, but as long as there is no upper bound on the "largest relevant" n, these cannot be combined to a universal effective algorithm. Of course, for some of the discussed problems quantification over the remaining integer(s) may eventually again lead to a semialgebraic set for which membership can be decided using Tarski-Seidenberg.
Let us have a closer look, problem by problem.
A bipartite quantum state is called distillable iff there exists an n∈N such that it is n−distillable. In this case, it was shown in [Wat04] that there is indeed no dimension-dependent upper bound: for any n there exist a density matrix acting on C9⊗C9 such that the state is distillable but not n−distillable. At present it is not known whether or not being not distillable coincides with the property of the density matrix having a positive partial transpose. If this would be true, then distillability would be decidable since, as we've seen in Lemma1, positive semi-definiteness is a semialgebraic relation. Conversely, this means that undecidability of distillability would imply the existence of undistillable states whose partial transpose is not positive—with all its puzzling consequences [SST01, VW02] .
Regarding the existence of a universal hidden variable model (one which holds for all n,m∈N) for a given density matrix, not much seems to be known. There are quantum states which are entangled and nevertheless admit such a universal LHV model [Wer89, Bar02] . However, it is also clear that there are states which admit a LHV description for n=m=2 but fail to do so for larger n or m.
The relevant integral parameter in the problem raised in 2.5 is d∈N, so that the underlying question is: given a probability distribution P(i,j|k,l) is there any d∈N for which there exists a quantum representation. To the best of our knowledge, this question is still open. There is some evidence [PV10] that already for n=3,m=2 the dimension d might be generally unbounded or even infinite. The recent connection between Tsirelson's problem and the long standing Connes' embedding problem [JNP+11, Fri] also points in this direction. This could explain why attempts to find a characterization of the set of quantum correlations, like the hierarchy of semidefinite relaxations given in [NPA07] , lead in the worst case scenario to infinite running times.
For d=3 there are unital quantum channels known for which Eq.(10) does not hold for n=1 but becomes true for n=2 [MW09] . It is also known that the set of channels for which there exists such an n∈N is a subset of the so-called factorizable channels [HM11] .
Already in the case of the zero-error capacity of classical channels, it is known that there is no upper bound on the block-length n required to achive the capacity (see [KO98] and references therein). Since classical channels are a special case of quantum channels, this carries over immediately to the classical zero-error capacity of quantum channels.
In the important case of deciding whether the zero-error capacity vanishes, classical channels are no longer any use, since it is trivial to see that the n−shot zero-error capacity of a classical channel vanishes iff it vanishes for n=1. Thanks to entanglement, this is not true for quantum channels. A simple implication of the superactivation results of [CCH, Dua] is that there exist quantum channels for which the 1−shot capacity vanishes but the 2−shot capacity does not, but it is not known at present whether this extends to arbitrary n.
The maximal output p−norm is known not to be multiplicative in general [HW02, WH08, Has09] . Channels for which multiplicativity of the form νp(T⊗2)=νp(T)2 has been proven, are often such that νp(T⊗T′)=νp(T)νp(T′) holds for any T′ irrespective of its dimension.
The failure of additivity of various single-shot quantities (such as the minimal output entropy corresponding to νp for p→1) leads to the fact that we do currently neither have a single-letter formula for the classical capacity of quantum channels nor for its quantum capacity. The latter is known to be itself non-additive in the strong sense that 0+0>0 [SY08] . Questions like "is the quantum capacity zero?" or, if it is, "can it be activated by another zero-capacity channel" also seem undecided.
The above list can easily be extended as there are numerous problems which involve quantification over one or several integers. Further examples would be: is a quantum channel implementable by means of LOCC (for which there is no a priori bound on the number of required rounds)? Can a source or a channel asymptotically simulate another one at a pre-given rate and under given constraints? Etc.
Before we briefly review some tools for proving that a problem is algorithmically undecidable, let us recall what this means and implies. The reader familiar with the basic notions of computability theory may skip this subsection. The reader not satisfied by it may consult [Cut80, Bar77] .
Informally, an algorithm is an effective procedure based on finitely many instructions which can be carried out in a stepwise fashion. Crucial for this notion is that there has to be a finite description. Any formal definition has to specify who is carrying out what kind of instructions. The two historically first frameworks which made this precise were Turing machines and recursive function theory. However, any other reasonable framework people came up with turned out to be no more powerful than those. That this will continue to hold in the future is the content of the Church-Turing thesis.
For a decision problem to be algorithmically undecidable, an infinite domain is necessary. Suppose there were only finitely many, say N, distinct inputs for which we want a yes-no question to be decided. Then we may write one algorithm for each of the 2N truth-value assignments and since this exhausts all possibilities one of the algorithms will return the correct answer for all the inputs.444We tacitly assume the tertium non datur. Finite problems are thus decidable which, for the problems we have discussed, implies that introducing "ϵ-balls" will make them algorithmically decidable (although we still wouldn't know the algorithm). An infinite domain, on the other hand, should be countable infinite, since otherwise the inputs have no finite description.
Requiring algorithms to have a finite description and considering decision problems with countable infinite domains already implies that there are undecidable problems (viewed as subsets of N) since algorithms are countable but decision problems aren't, i.e., |N|<|2N|.
is the mother of all undecidable problems. This is not only true historically, but also technically: most problems which are known to be undecidable are proven to be so by showing that the halting problem can (directly or indirectly) be reduced to them. The halting problem asks whether a Turing machine halts upon a given input. If we identify the inputs with natural numbers, then the subset of N upon which a fixed Turing machine halts is called recursively enumerable—a notion we will come back to later. Clearly, membership in a recursively enumerable set cannot generally be decidable since otherwise the halting problem would be decidable.
(PCP) is an example of an undecidable problem to which the halting problem can be reduced. It is frequently used as an intermediate step in undecidability proofs by reducing it in turn to the problem under consideration.
Let A be a finite alphabet for which we consider the set of words A∗ as a monoid with respect to which X,Y:K→A∗ are two homomorphisms from K={1,…,k}. Post's correspondence problem is then to decide whether or not there is a non-empty word w∈A∗ such that X(w)=Y(w). An equivalent, possibly more intuitive, depiction of the problem is to think about k types of dominos and ask whether they admit a finite chain in which the upper row coincides with the lower row.
While PCP is decidable if k=2, it's known to be undecidable for k≥7 [MS05] . In fact, it is undecidable whether there is a so-called Claus instance solution of the form 1w7 with w∈{2,…,6}∗ (cf. [Har09] ).
The main reason why we emphasize PCP, is that it can be reduced to problems involving products of matrices—something we will make use of in the next subsection. The main tool in this context is a monoid morphism due to Paterson [Pat70] : take A={1,…,m} and define an injection σ:A∗→N as the m−adic representation of words, i.e., σ(w)=∑|w|j=1wjm|w|−j where |w| denotes the length of the word and σ maps the empty word onto 0. Note that σ(uw)=m|w|σ(u)+σ(w). The map γ:A∗×A∗→M3(N0)
γ(u,w):=⎛⎜⎝m|u|000m|w|0σ(u)σ(w)1⎞⎟⎠, (16)
is then a monoid monomorphism, i.e., it holds that γ(uu′,ww′)=γ(u,w)γ(u′,w′). Defining vectors |x⟩:=(1,−1,0)T and ⟨y|:=(0,0,1) this implies that u=w, as required by a PCP solution, holds iff ⟨y|γ(u,w)|x⟩=0. If one wants to have a non-negative expression of this type (as in the following subsection), one can take the square, express it in terms of γ⊗γ and restrict to the anti-symmetric subspace—Lemma 3 is based on such a representation (cf. [BC01, Hir07] for more details).
One of the problems to which PCP can be reduced is the matrix mortality problem. This asks whether a set of matrices M:={Mi∈Md(Z)}ki=1 admits a finite string of products for which Mi1⋯Min=0. M is then called 'mortal'. In [JE] this has been applied to quantum information problems by constructing a set M′⊂Md′(Q) which is mortal iff M is, and which in addition can be interpreted as a set of 'Kraus operators' for which ∑jM′jM′†j=1. Note that by allowing algebraic entries, one can simply achieve this by setting M′i:=X−1MiX/√λ using a positive eigenvector ∑iMiX2M†i=λX2 which is always guaranteed to exist.555If there is no positive definite eigenvector X2, one can replace the Mi's by a direct sum of their irreducible blocks, for which the corresponding positive map then has a positive definite eigenvector. The advantage of using Ralg instead of Q is that the number of matrices and their size doesn't change when going from M to M′. In this way the argumentation in [JE] then holds for (d,k)=(3,7) rather than only for (15,9).
A subset D∈N is called diophantine iff for some n∈N there is a polynomial p:Nn+1→N with integer coefficients, such that
D={x|∃y∈Nn:p(x,y)=0}. (17)
W.l.o.g. one can bound the degree of the polynomial by 4 and, at the same time, the number of variables n≤56. A remarkable theorem [DMR76] , which eventually proved the undecidability of Hilbert's 10th problem, states that a set is diophantine if and only if it is recursively enumerable. Since membership is generally undecidable for the latter, the same has to hold for the former. This result should be compared with the earlier mentioned Tarski-Seidenberg theorem. While Thm.1 implies that the existence of roots of a system of polynomial equations with integer coefficients can be decided over R, the same problem becomes undecidable over N. Whether it is decidable over Q is still an open problem.
So much for general undecidability in a nutshell. Let's have a look at a specific problem and apply some of the mentioned results.
As an example which naturally occurs in the context of quantum information theory, we will discuss the problem of deciding whether a small set of "noisy gates" allows to create a state which overcomes some pre-given fidelity threshold w.r.t. a given target state. The latter might for instance be a two-qubit singlet state and the aim, motivated by the requirements of entanglement distillation protocols, to achieve an overlap strictly larger than one half.
In the following we will for simplicity let R and C denote the fields of real and complex algebraic numbers respectively.
Let (k,d) be a pair of integers which is either (2,5) or (5,3) (or pointwise larger than either) and take any real number 0<λ<1 and normalized vector |ϕ⟩∈Cd. Then, there is no algorithm which upon input of a density matrix ρ∈Md(C) and of a set of quantum channels {Ti:Md(C)→Md(C)}ki=1 decides whether or not there is an integer n≥1 and a sequence (i1,…,in)∈{1,…,k}n for which
⟨ϕ|Ti1⋯Tin(ρ)|ϕ⟩>λ. (18)
The proof of this theorem is decomposed into two steps: (i) we establish a mapping from unconstrained matrices to quantum channels in such a way that concatenations of channels are closely linked to products of matrices, and (ii) we exploit known undecidability results for problems which are expressed in terms of products of matrices.
Let {Hi∈Md(C)}d2i=1 be any Hermitian operator basis which is orthonormal in the sense that tr[HiHj]=δi,j and satisfies H1=1/√d. Note that the latter implies tr[Hi]=0 for all i>1.
We represent any linear map T:Md(C)→Md(C) in terms of a d2×d2 matrix with entries ^Ti,j:=tr[HiT(Hj)]. T is Hermiticity preserving iff ^T is real, and trace-preserving iff the first row of ^T is (1,0,…,0).
Let d≥2 and H2:=(1−dψ)/(√d2−d) for any one-dimensional projector ψ=|ψ⟩⟨ψ|∈Md(C). For every M∈Md2−2(R) and every ν∈(−√d−1,1/√d−1) there exists an ϵ>0 such that
^T:=(10ν0)⊕ϵM (19)
represents a completely positive, trace-preserving linear map.
Proof. By construction ^T represents a trace-preserving and Hermiticity preserving linear map. First note that for ϵ=0 the matrix ^T corresponds to the map
T(X)=tr[X]d(1+ν√d−1(1−dψ)). (20)
The Choi matrix of this map is positive definite iff
1+ν√d−1(1−dψ)>0, (21)
which holds in turn iff ν∈(−√d−1,1/√d−1). Within this range, full support of the Choi matrix then allows us to add a sufficiently small multiple of any other Hermiticity preserving map without violating complete positivity.
Let d≥2, λ∈(0,1) and let ϕ=|ϕ⟩⟨ϕ| be a one-dimensional projector acting on Cd. For any x,y∈Rd2−2 and any set of matrices {Mi∈Md2−2(R)}ki=1, one can construct a set of quantum channels {Ti:Md(C)→Md(C)}ki=1, a density matrix ρ∈Md(C) and positive numbers ϵ,δ>0 such that for all natural numbers n≥1
tr[ϕTi1⋯Tin(ρ)]=λ+δϵn⟨x|Mi1⋯Min|y⟩ (22)
holds for all sequences (i1…in)∈{1,…,k}n.
Proof. First of all, we exploit the freedom in the choice of the Hermitian and orthonormal operator basis {Hi}. As indicated above, we fix H1:=1/√d,H2:=(1−dψ)/(√d2−d), with ψ≠ϕ to be chosen later, and we choose H3,…,Hd2 such that up to some factor δ1>0 we have tr[ϕHi+2]=δ1xi for all i=1,…d2−2.
We define ρ∈Md(C) in this basis as being represented by the vector (1/√d,0,δ2y). In this way, ρ=ρ†, tr[ρ]=1 and we can choose δ2>0 sufficiently small, so that ρ≥0.
To every matrix Mi we assign a quantum channel Ti of the form in Eq.(19) where a common ϵ>0 is chosen such all the Ti's are completely positive. In this way, we obtain
tr[ϕTi1⋯Tin(ρ)] = 1d(1+ν1−d|⟨ψ|ϕ⟩|2√d−1) (24)
+δϵn⟨x|Mi1⋯Min|y⟩,
where we have set δ:=δ1δ2. By choosing appropriate ν∈(−√d−1,1/√d−1) and ψ≠ϕ, the r.h.s. of Eq.(24) can be set to any value in the open interval (0,1) and Lemma 2 guarantees that the Ti's are completely positive and trace-preserving as requested.
The following Lemma is distilled from [Hir07] where it has been proven by reduction from PCP with Claus instances:
Let (k,m) be either (5,6) or (2,21). There is no algorithm which upon input of vectors x,y∈Zm and of matrices {Mi∈Mm(Z)}ki=1 decides whether or not there is an integer n≥1 and a sequence (i1,…,in)∈{1,…,k}n so that
⟨x|Mi1⋯Min|y⟩>0. (25)
Together with the previous proposition this Lemma now proves Thm.2. Inserting a different classical result [BC01, Hir07] into Prop.1 we obtain an analogous result in which > is replaced by ≥ in Eq.(18). Variation on the theme of Prop.1 also allows to use targets other than the overlap with a given pure state such as the expectation value of some observable or the overlap with a given quantum channel.
Loosely speaking: while semi-algebraic problems are decidable, word problems are typically not. We have discussed to what extent this fact separates quantum information problems into decidable and undecidable ones. Many of the interesting problems, however, remain undecided.
In the direction of proving decidability of some of them, one approach might be to look into tools which extend Tarski-Seidenberg in various directions. In [DMM94] it has for instance been shown that quantifier elimination is possible for the real field augmented by exponentiation and all restricted analytic functions. Needless to say, there are also limitations on the possibility of quantifier elimination, which are discussed for instance in [GAB07] . Also tools for deciding specific second-order theories might be worthwhile looking at [Rab69] .
In the other direction, most of the known undecidable problems are, in one way or the other, word problems. It would therefore be interesting to see a word problem structure identified in one of the undecided problems we have discussed. Is there an undecidable word problem where strings are formed by tensor products rather than by ordinary products?
Acknowledgements: We acknowledge financial support from the European projects COQUIT and QUEVADIS, the CHIST-ERA/BMBF project CQC, the Alfried Krupp von Bohlen und Halbach-Stiftung and the Spanish grants I-MATH, MTM2008-01366, S2009/ESP-159 and QUITEMAD.
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