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o hare airport to best western grant park
Services depart every 10 minutes, and operate every day. I'm also going to Lolla, woohoo! Find all the transport options for your trip from O'Hare to Best Western Grant Park Hotel, Chicago right here. We are also just one mile from McCormick Place Convention Center and DePaul University and within a few miles from two major airports, O'Hare and Midway. Rome2rio also offers online bookings for selected operators, making reservations easy and straightforward. As of 2009, Millennium Park trailed only Navy Pier as a Chicago tourist attraction and by 2017 it had become the number one tourist attraction in the Midwestern United States. Rome2rio displays up to date schedules, route maps, journey times and estimated fares from relevant transport operators, ensuring you can make an informed decision about which option will suit you best. Surrounded by award-winning restaurants, shops, and a vibrant nightlife, the Best Western Grant Park Hotel is easily accessible via the CTA, L trains, and city bus. The area was previously occupied by parkland, Illinois Central rail yards, and parking lots. O'Hare → Best Western Grant Park Hotel, Chicago, O'Hare to Best Western Grant Park Hotel, Chicago, Take the subway from Cumberland to LaSalle, Take the bus from East River Rd & Ainslie to Jefferson Park Transit Center, Take the subway from Jefferson Park to LaSalle, Take the line 56 bus from Jefferson Park Transit Center to Washington & Wabash, Take a taxi from O'Hare to Best Western Grant Park Hotel, Chicago, Drive from O'Hare to Best Western Grant Park Hotel, Chicago. Rome2rio also offers online bookings for selected operators, making reservations easy and straightforward. Visitor Pass Vending Machine available 24 hours per day. O'Hare to Best Western Grant Park Hotel, Chicago train services, operated by Chicago Transit (CTA), arrive at LaSalle station. The price estimate for this ride is $ 39.11.. From: O'Hare Airport To: Best Western Grant Park. View Hotel. Millennium Park is a public park located in the Loop community area of Chicago in Illinois, US, and originally intended to celebrate the third millennium. ... Chicago O'Hare International Airport 16.2 miles * All distances are measured in straight lines. BEST WESTERN Grant Park Hotel Shuttle Service. Get driving directions Prices start at Rp1265000 per night. This estimate was last updated on 01 January 2019, over 1 year ago. Find all the transport options for your trip from O'Hare to Best Western Grant Park Hotel, Chicago right here. Rome2rio makes travelling from O'Hare to Best Western Grant Park Hotel, Chicago easy. is situated in the lower level concourse, which connects Airline Terminals 1, 2 and 3, and is conveniently accessed from Terminal 5 by free, frequent airport shuttle trains. It takes approximately 23 min to drive from Best Western Grant Park Hotel, Chicago to Chicago O'Hare Airport (ORD). Airport (ORD) - 32 min drive; Midway Intl. If you need an ORD airport ride to or from Best Western at O'Hare we can help you find and book quality ground transportation options from trusted partners. The hotel provides complimentary 24-hour shuttle service to the airport and the airport train station. Last updated: 30 Nov 2020 Tickets cost Rp35000 - Rp70000 and the journey takes 44 min. Visit the Navy Pier for sightseeing by boat, and indulge in the dizzying flavours of the city's unique deep-dish pizza. O'Hare to Best Western Grant Park Hotel, Chicago bus services, operated by Chicago Transit (CTA), depart from East River Rd & Ainslie station. Domestic travel is not restricted, but some conditions may apply. The journey takes approximately 44 min. The best way to get from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago without a car is to bus which takes 57 min and costs . The fully-accessible O'Hare station. Rome2rio also offers online bookings for selected operators, making reservations easy and straightforward. It measures 33 by, and weighs 110 ST. United Center is an indoor sports arena located in the Near West Side neighborhood of Chicago, Illinois. Chicago O'Hare Airport (ORD) → Best Western Grant Park Hotel, Chicago, Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago, by bus, subway, taxi, car, shuttle or towncar, Take the bus from Chicago, IL - O'Hare International Airport to Chicago Union Station, Take the subway from O'Hare to Jackson-Blue, Take a taxi from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago, Drive from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago, Take a shuttle bus from Chicago O'Hare Airport to Best Western Grant Park Hotel, Chicago, Take a town car from Chicago O'Hare Airport to Best Western Grant Park Hotel, Chicago. The road distance is 31 km. What is the national COVID-19 helpline number in Best Western Grant Park Hotel, Chicago? The walk to the BW Grant Park from the LaSalle St Station of the Blue Line -- which is at LaSalle & Congress Parkway -- is about 3/4 of a mile. O'Hare to Best Western Grant Park Hotel, Chicago train services, operated by Chicago Transit (CTA), depart from Cumberland station. The ATS is a free, 24-hour rail system that operates between the three domestic terminals (1, 2 and 3), the international terminal (5), all parking facilities, the Kiss 'n' Fly drop-off point, and the PACE Bus stop. It takes approximately 20 min to drive from O'Hare to Best Western Grant Park Hotel, Chicago. We're working around the clock to bring you the latest COVID-19 travel updates.This information is compiled from official sources. The Art Institute of Chicago, founded in 1879 and located in Chicago's Grant Park, is one of the oldest and largest art museums in the United States. Book now. Am I allowed to travel from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago? Save this link to stay updated on COVID-19 restrictions, Domestic border crossings may be subject to approval, testing and quarantineObserve COVID-19 safety rules, If you need help, visit the national COVID-19 website or call the COVID-19 Helpline 800-232-4636. O'Hare Airport Blue. Its permanent collection of nearly 300,000 works of art is augmented by more than 30 special exhibitions mounted yearly that illuminate aspects of the collection and present cutting-edge curatorial and scientific research. View all hotels. Do I have to wear a face mask on public transport in Best Western Grant Park Hotel, Chicago? Best Western Grant Park Hotel is 19.1 mi (32-min drive) away from the O'Hare Intl. That easy, just a little walk. The area was previously occupied by parkland, Illinois Central rail yards, and parking lots. Find all the transport options for your trip from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago right here. Best Western Inn & Suites - Midway Airport 8220 S Cicero Ave, Burbank, IL 60459. Yes, the driving distance between O'Hare to Best Western Grant Park Hotel, Chicago is 17 miles. Make yourself known to an official member of staff and/or call the national coronavirus helpline number on 800-232-4636. Make yourself known to an official member of staff and/or call the national coronavirus helpline number on 800-232-4636. The journey, including transfers, takes approximately 1h 53m. It is a prominent civic center near the city's Lake Michigan shoreline that covers a 24.5 acre section of northwestern Grant Park. Get driving directions A Downtown Chicago Hotel in Grant Park Discover a unique urban experience at our Grant Park hotel in the heart of downtown Chicago. With thoughtfully chosen amenities and a genuine focus on the best standards of hospitality, it offers a comfortable stay that suits the needs of every type of modern traveler. The hotel offers a 24-hour shuttle to O'Hare airport. Yes, travel within United States is currently allowed. Best Western Grant Park Hotel offers genuine hospitality and thoughtful amenities, from our outdoor terrace with sweeping city views to complimentary Wi-Fi, a business center, fitness center, and a 24-hour front desk to accommodate your schedule. This taxi cost estimation from O'Hare Airport to Best Western Grant Park might be a bit outdated. The cheapest way to get from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago is to drive which costs Rp26000 - Rp39000 and takes 23 min. Its permanent collection of nearly 300,000 works of art is augmented by more than 30 special exhibitions mounted yearly that illuminate aspects of the collection and present cutting-edge curatorial and scientific research. Location. There are 618+ hotels available in Best Western Grant Park Hotel, Chicago. Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago bus services, operated by Vangalder Bus, arrive at Chicago Union Station. Am I allowed to travel from O'Hare to Best Western Grant Park Hotel, Chicago? What should I do if I have COVID-19 symptoms when I arrive in Best Western Grant Park Hotel, Chicago? Wearing a face mask on public transport in Best Western Grant Park Hotel, Chicago is subject to regional advice. The bus from Chicago, IL - O'Hare International Airport to Chicago Union Station takes 30 min including transfers and departs every four hours. Wearing a face mask on public transport in Best Western Grant Park Hotel, Chicago is subject to regional advice. Exceptions may apply, for full details: Centers for Disease control and prevention (CDC). What are the travel restrictions in Best Western Grant Park Hotel, Chicago? Read our range of informative guides on popular transport routes and companies - including Italian Lakes: Getting around by train, 10 amazing places to get away from it all in Australia and Thanksgiving 2018: How to travel stress-free - to help you get the most out of your next trip. Best Western Grant Park Hotel The Windy City, the third-largest in the US, is located alongside majestic Lake Michigan. The social distance requirement in Best Western Grant Park Hotel, Chicago is 2 metres. It takes approximately 23 min to drive from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago. Made up of 168 stainless steel plates welded together, its highly polished exterior has no visible seams. Cloud Gate is a public sculpture by Indian-born British artist Sir Anish Kapoor, that is the centerpiece of AT&T Plaza at Millennium Park in the Loop community area of Chicago, Illinois. The journey takes approximately 38 min. In addition to O'Hare hotel parking and off-airport parking, you can also park on airport property. Best Western Grant Park Hotel does not have a pool. To the best of our knowledge, it is correct as of the last update.Visit Rome2rio travel advice for general help. The sculpture and AT&T Plaza are located on top of Park Grill, between the Chase Promenade and McCormick Tribune Plaza & Ice Rink. This Best Western is 4 blocks from Allstate Arena and 1mile from O'Hare International Airport. Find all the transport options for your trip from Chicago O'Hare Airport (ORD) to Best Western Grant Park Hotel, Chicago right here. It's $2.25 one way and so easy! Constructed between 2004 and 2006, the sculpture is nicknamed The Bean because of its shape. The journey takes approximately 30 min. Rome2rio displays up to date schedules, route maps, journey times and estimated fares from relevant transport operators, ensuring you can make an informed decision about which option will suit you best. Yes, travel within United States is currently allowed. The distance between O'Hare and Best Western Grant Park Hotel, Chicago is 13 miles. It was really easy to travel from the O'hare airport from the L train to Grant Park Best western. Alternatively, you can take a bus from O'Hare to Best Western Grant Park Hotel, Chicago via Jefferson Park Transit Center, Washington & Wabash, Michigan & Washington, and Michigan & 11th Street in around 1h 53m. The road distance is 17 miles. You'll be provided with a choice of transfer companies, from luxury limos to maxi taxis, private cars to shuttle buses, depending on your journey, along with passenger reviews and star ratings. Soon as I walk in to the hotel I was greeted and Got my room fast. The Windy City, the third-largest in the US, is located alongside majestic Lake Michigan. Which the best airport near BEST WESTERN GRANT PARK HOTEL? 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2020 o hare airport to best western grant park


The annual celebration of the city's favorite type of food is back for 2019! As every year, the spread will feature an array of different seafood from local purveyors – crab, shrimp, fish, crab legs, (and for the non-seafood eaters) even gyros, pizza, and Asian cuisine. After you've grabbed your favorites, browse around the tents of local vendors and artists or settle in to hear some awesome music. There are rides for the whole family and a kids fun zone with special games and exciting presentations. General admission is $5.


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The world is blanking me again – approaches ignored – emails unanswered – proposals rejected – deals stalled – even friends not returning calls; – feel powerless. Sudden plague of bluebottles in my cottage – Ronnie the postman thinks it's a dead rat in the attic – says he can smell it; – feel disgusted. Optician on Monday says my deteriorating eyesight is due to cataracts in both eyes – not bad enough for surgery yet; – feel decrepit. Letter from Urology dept. at St Johns Hospital informing that they have installed new kit – a 'green light laser' – offering to excavate a new channel through my prostate, so I can pee properly; – feel apprehensive.
Wednesday sunlight gets me into active mode – collecting fieldstones from a nearby pile. Crabbit gamekeeper challenges me – altercation – phone assistant factor who comes down. "It's ok with the stones, Laurence, but you've been spotted taking timber from the woods – that's not on." "But it's left to rot!" – feel angry. Dark mood erupts during telephone misunderstanding. Phone back to apologise – colleague says you can't force this Laurence – you need to chill. He's right – too tense – need to step back. My mind drifts to a Spanish beach and suddenly I know that's where I need to be – paddling in the Med – mixed fried fish at Marisa's – sun in my face. Feel delinquent but determined. Next bulletin from Andalucia.
Following the Civil Society Summit on 18th Feb – Stephen Maxwell comments that some responses to the societal crises of our time give cause for optimism. But, he argues, such shifts are disparate and incoherent; can civil society rise to the challenge of combining them into a common platform?
Final call for donations to Senscot's annual appeal. We'll probably end up with around £5,000 and reach 80 full company members. Many thanks to all who took the trouble.
Kenji Miyazama (1896 – 1933) was a Japanese poet, Buddhist and social activist. This is his poem 'Strong in the Rain'.


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Brighton withdraws from SELCRA
Brighton City Council votes to withdraw from SELCRA
Brighton withdraws from SELCRA Brighton City Council votes to withdraw from SELCRA Check out this story on livingstondaily.com: http://ldpaa.com/1QjVVkO
Noe Hernandez, Livingston Daily Published 7:48 a.m. ET Dec. 4, 2015 | Updated 7:03 p.m. ET Dec. 4, 2015
Vote does not surprise SELCRA director
Brighton Mayor Jim Muzzin, center, reviews his notes as Green Oak Township's Rich Everett, far right, discusses SELCRA's future during a November intergovernmental meeting.(Photo: Noe Hernandez)
The Brighton City Council voted Thursday to withdraw from the Southeast Livingston County Recreation Authority, becoming the first municipality to abandon the struggling agency.
The council voted 6-0 to separate from SELCRA. Councilman Jim Bohn suggested the withdrawal and Mayor Pro Tem Shawn Pipoly seconded the motion. Councilwoman Ricci Bandkau did not attend the meeting.
"As the SELCRA liaison for the city, I would have preferred a consensus among the four municipalities that make up SELCRA to dissolve SELCRA and reintegrate under Brighton Area Schools," Councilwoman Susan Gardner said after the council meeting. "It is my hope that reintegration is still open for consideration in early 2016."
SELCRA provides youth and adult sports programs, including baseball, softball, basketball and soccer. It also coordinates community events and other activities, including karate and fencing.
Brighton will remain a part of SELCRA until July 1.
With the vote, Brighton will no longer have a member on the SELCRA Board of Directors and the city will not contribute funds toward the agency's operational costs.
Brighton currently contributes some $55,000 a year to SELCRA. Brighton Township, Genoa Township and Green Oak Township also contribute funds to the agency.
The council's vote did not shock SELCRA Director Derek Smith.
"I'm not surprised," Smith said. "We already knew what their intentions were."
Brighton residents will still have access to SELCRA programs, but will have to pay 50 percent more in participation fees if the three townships decide to remain part of SELCRA. For example, a soccer player now paying $100 to play would pay $150 beginning July 1.
The SELCRA board meets again Wednesday.
Contact Livingston County business reporter Noe Hernandez at (517) 552-2854 or nhernandez@livingstondaily.com. Follow him on Twitter @sayyesnoe.
Read or Share this story: http://ldpaa.com/1QjVVkO

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L'Atlas – Phosphenes
December 15, 2015 · by Jill Thayer, Ph.D.
By Jill Thayer, Ph.D., Contributing Writer, ARTPULSE Magazine, Miami, December, 2015
1 December 2015 – 6 February 2016, Lelia Mordoch Gallery, Miami
"Persistence 3," 2013, Mixed-media on canvas, 47 x 47 inches (120 x 120cm)
I think life is energy and also calligraphy is energy, painting is the translation of pure energy[1]. – L'Atlas
Phosphenes, L'Atlas' solo exhibition at Leila Mordoch Gallery, Miami features 14 canvases and a selection of pieces from his recent show at Galerie Magda Danysz, Paris. The series align with the artist's methodologies in the expanded field of public art.
French artist Jules Dedet Granel, aka. L'Atlas creates vector-based installations that form a semiological language of Op Art and Geometric Abstraction. His paintings evoke the grid patterns of Mondrian, the optical illusions of M.C. Escher and Vasarely, and the perceptual juxtapositions of Bridget Riley. L'Atlas, much like Riley, engages the viewer in the process of observation, but also, in the interaction of it.
The work is produced with flat and fluorescent pigments applied in a mixed-technique using aérosol on laque. Multi-colored stripes of horizontal and vertical lines criss-cross in a dynamic color field of dayglo and metallic paints. The optical illusions are mesmerizing in vibrating hues that channel Albers' language of color.
"Close-Up No. 5," 2013, Mixed-media on canvas, 47 inches diameter (120cm).
Close-Up No. 5, 2013, 120cm diameter offers a visual dialog set on a circular plane. A base pattern of white and grey vertical bars connect here and there with short horizontal paths. Diagonal stripes of warm and cool neutrals add tint and tone to geometric forms like shadows passing over the undulations of a building in late afternoon.
Persistence 3, 2013, 120 x 120cm appears random in its devised motif composed on a background of bright apple green. A layer of silver horizontal lines mediates an aura of intensity, further defined by a black ruled code.
"Persistance 4," 2013, Mixed-media on canvas, 47 x 47 inches (120 x 120cm)
Persistence 4, 2013, 120 x 120cm breaks tradition, as an under painting of orange stripes glows with soft edges that gently rests behind vertical columns of silver. A top layer of bold black lines unifies the piece in a De Stijl-like presence of abstract and constructivist elements. The columns resemble metal framework exposing a sunlit bank of office windows across the way.
The installation features daylight and blacklight views to optimize fluorescent properties of the series. The gallery becomes a sacred space of perceptual phenomena, as a psychedelic era is reincarnated.
L'Atlas deftly traverses the resonance of his projects in an en pointe approach that challenges inquiry. Phosphenes is a statement in perception and visual culture––drawing upon the cuneiforms of history in the earliest systems of writing. L'Atlas' offers a new paradigm that transpires language in the contemporary contexts of a globalized world.
L'Atlas' recent show at Galerie Magda Danysz, Paris.
[1] Hinke, Fernanda. 'Interview with spiritual-mystic Paris graffiti artist L'Atlas," Underground Paris. 2002. Online. Retrieved from http://undergroundparis.org/latlas-interview.
Jill Thayer, Ph.D. is an artist, educator, and curatorial archivist. She is faculty at Santa Monica College in Art History: Global Visual Culture, and Post University, Malcolm Baldrige School of Business MBA program in Integrative Strategies. Her oral history narratives, "In Their Own Words: Oral Histories of CGU Art," featuring Claremont Graduate University professors emeritus and contemporary artists are included in Archives of American Art at The Smithsonian Institution.
This entry was posted in Art History, Artist Profiles, Exhibition Theory, Jill Thayer, Miami Artists, Ph.D., Rosario Bond, Visual Culture and tagged L'Atlas, Paris Artists, Phosphenes. Bookmark the permalink.
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Long Meadow news 09-11-12
By Sylvia Weller |
PUBLISHED: September 13, 2012 at 11:15 a.m. | UPDATED: April 24, 2019 at 1:54 a.m.
Long Meadow Community Club will meet Wednesday, Sept. 19, at the Home Plate Restaurant in Fort Morgan at noon. Joyce Hueske is the hostess.
Mary Gahagen had her grandchildren come for a visit on Saturday. Enjoying the day were Jaycee and Savanah from Hillrose, Amy Smith, Holden and Lily from Denver, and Elizabeth Meyer, J. R. and Peyton of Loveland.
Mary and Rob Pachek attended a wedding at Salida over the weekend.
Charles B. Clark and John A. Kimberly in 1872, built a paper mill. Their first product was newsprint made from rags. Clark, a Civil war veteran, and Kimberly and two other men invested $7,500 each to form Kimberly, Clark and Co. In 1928, after the four founders had died, the company's chemists developed Cellucotton, a revolutionary breakthrough in absorbent material, used extensively during World War I for gas mask filters and surgical dressings. In 1924, they marketed Kleenex facial tissue, the sanitary cold cream remover. Stars such as Helen Hayes endorsed the product. Six years later, surveys showed that the most consumers used Kleenex tissues as disposable handkerchiefs rather than as a cold cream remover. Advertising slogan: "Don't put a cold in your pocket." Sales boomed. Sales today are over $4 billion.
Thirty Days Hath September
Thirty days has September,
April, June and November.
All the rest have thirty-one,
Excepting February alone,
And that has twenty-eight days clear,
And twenty-nine in each leap year.
Pond's Cold Cream
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DEADLINE ALERT: Kessler Topaz Meltzer &…
DEADLINE ALERT: Kessler Topaz Meltzer & Check, LLP Announces Deadline in Athira Pharma, Inc. Securities Fraud Class Action Lawsuit
By Kessler Topaz Meltzer & Check LLP |
August 19, 2021 at 11:14 a.m.
RADNOR, Pa., Aug. 19, 2021 /PRNewswire/ — The law firm of Kessler Topaz Meltzer & Check, LLP reminds investors of Athira Pharma, Inc. (NASDAQ: ATHA) ("Athira") that securities fraud class action lawsuits have been filed on behalf of those who purchased or acquired Athira common stock: a) pursuant and/or traceable to the registration statement and prospectus (collectively, the "Registration Statement") issued in connection with Athira's September 2020 initial public offering ("IPO"); and/or b) between September 18, 2020 and June 17, 2021, inclusive (the "Class Period").
Deadline Reminder: Investors who purchased or acquired Athira common stock pursuant to or traceable to the IPO and/or during the Class Period may, no later than August 24, 2021, seek to be appointed as a lead plaintiff representative of the class. For additional information or to learn how to participate in this litigation please contact Kessler Topaz Meltzer & Check, LLP: James Maro, Esq. (484) 270-1453; toll free at (844) 887-9500; via e-mail at info@ktmc.com; or click https://www.ktmc.com/athira-pharma-class-action-lawsuit?utm_source=PR&utm_medium=link&utm_campaign=athira
Athira is a late-stage clinical biopharmaceutical company that focuses on developing small molecules to restore neuronal health and stop neurodegeneration. On September 18, 2020, Athira filed its prospectus on a Form 424B4, which forms part of the Registration Statement. In the IPO, Athira sold approximately 13,397,712 shares of common stock at a price of $17.00 per share. Athira received proceeds of approximately $208.5 million from the IPO, net of underwriting discounts and commissions.
According to the complaints, on June 17, 2021, after the market closed, Athira announced in a press release that it had placed its president and Chief Executive Officer, Dr. Leen Kawas, on leave pending a review of actions stemming from doctoral research she conducted while at Washington State University ("WSU"). According to Athira's press release, Athira's Board "formed an independent special committee to undertake this review."
The same day, the scientific publication STAT published an article stating that WSU was investigating claims that Dr. Kawas "published several papers containing altered images while she was a graduate student." These papers "are foundational to Athira's efforts to treat Alzheimer's" because they "established that a particular molecule affects the activity of HGF." Though Athira is developing a different molecule than the one Dr. Kawas examined in the papers at issue, her "doctoral work laid the biological groundwork that Athira continues to use in their approach to treating Alzheimer's." Specifically, "[i]mages of Western blots, used to determine the presence of specific proteins in biological samples, look as though they've been altered from their original state." According to experts cited in the article, "If the Western blots are inaccurate, then the whole study must be redone."
Following this news, Athira's share price fell $7.09, or approximately 39%, to close at $11.15 per share on June 18, 2021.
The complaints allege that in the Registration Statement and/or throughout the Class Period the defendants made materially false and misleading statements and omitted to state that: (1) Dr. Kawas had published research papers containing improperly altered images while she was a graduate student; (2) this purported research was foundational to Athira's efforts to develop treatments for Alzheimer's because it laid the biological groundwork that Athira was using in its approach to treating Alzheimer's; (3) as a result, Athira's intellectual property and product development for the treatment of Alzheimer's was based on invalid research; and (4) as a result of the foregoing, the defendants' positive statements about Athira's business, operations, and prospects, were materially misleading and/or lacked a reasonable basis.
Athira investors may, no later than August 24, 2021, seek to be appointed as a lead plaintiff representative of the class through Kessler Topaz Meltzer & Check, LLP or other counsel, or may choose to do nothing and remain an absent class member. A lead plaintiff is a representative party who acts on behalf of all class members in directing the litigation. In order to be appointed as a lead plaintiff, the Court must determine that the class member's claim is typical of the claims of other class members, and that the class member will adequately represent the class. Your ability to share in any recovery is not affected by the decision of whether or not to serve as a lead plaintiff.
Kessler Topaz Meltzer & Check, LLP prosecutes class actions in state and federal courts throughout the country involving securities fraud, breaches of fiduciary duties and other violations of state and federal law. Kessler Topaz Meltzer & Check, LLP is a driving force behind corporate governance reform, and has recovered billions of dollars on behalf of institutional and individual investors from the United States and around the world. The firm represents investors, consumers and whistleblowers (private citizens who report fraudulent practices against the government and share in the recovery of government dollars). The complaint in this action was not filed by Kessler Topaz Meltzer & Check, LLP. For more information about Kessler Topaz Meltzer & Check, LLP please visit www.ktmc.com.
Kessler Topaz Meltzer & Check, LLP
James Maro, Jr., Esq.
280 King of Prussia Road
Radnor, PA 19087
info@ktmc.com
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text:
\section{Introduction}
\label{sec:introduction}

The quantum marginal problem (QMP) is interested in characterizing the space of reduced/marginal states of a multipartite quantum state, and is widely regarded as being an important, albeit challenging problem to solve.

Connections and applications of the QMP to other topics in quantum theory (and beyond) include multipartite entanglement and separability~\cite{coffman2000distributed,walter2013entanglement}, quantum error correction~\cite{ocko2011quantum,huber2017quantum,yu2021complete}, entropic constraints~\cite{majenz2018constraints,christandl2018recoupling,kim2020entropy,osborne2008entropic}, other forms of quantum compatibility problems~\cite{heinosaari2016invitation,haapasalo2021quantum,doherty2008quantum}, the asymptotics of representations of the symmetric groups~\cite{daftuar2005quantum,christandl2006spectra,christandl2018recoupling}, the asymptotic restriction problem for tensors~\cite{christandl2018universal}, random matrix theory~\cite{collins2021projections,christandl2014eigenvalue}, and many-body physics~\cite{coleman2001reduced}.

The quantum marginal problem, and its associated terminology, is derived from an analogous problem in probability theory called the \textit{classical} marginal problem. The classical marginal problem is concerned with characterizing the relationships between the various marginal distributions of a joint, multivariate distribution~\cite{fritz2012entropic,vorob1962consistent,malvestuto1988existence}. As a joint probability distribution and its marginals can always be faithfully represented by the eigenvalues of a joint quantum state and its marginals using a product eigenbasis, the quantum marginal problem subsumes the classical marginal problem, and consequently, any of its applications~\cite{fritz2012entropic,liang2011specker,abramsky2011sheaf,fraser2018causal}.

One of the earliest formulations of the problem dates back to the early 1960s, when, for the purposes of simplifying calculations involving the atomic and molecular structure, quantum chemists became interested in characterizing the possible reduced density matrices of a system of $N$ interacting fermions~\cite{coulson1960present,coleman1963structure}. This version of the problem, referred to as the $N$-representability problem, has a long history~\cite{coleman2000reduced,coleman2001reduced,lude2013functional,borland1972conditions,ruskai2007connecting,klyachko2009pauli} that continues to evolve~\cite{mazziotti2012structure,mazziotti2012significant,klyachko2006quantum}.

Now the QMP comes in a variety of flavours which can be broadly organized by considering any additional assumptions or constraints that are imposed on either i) the properties of the joint state, and/or ii) the properties of the set of candidate density operators~\cite{tyc2015quantum}.

When focusing on the joint state, specializations of the QMP exist where the joint state is assumed to be fermionic~\cite{coleman2000reduced}, bosonic~\cite{wei2010interacting}, Gaussian~\cite{eisert2008gaussian,vlach2015quantum}, separable~\cite{navascues2021entanglement}, or having symmetric eigenvectors~\cite{aloy2020quantum}. Generally speaking, the QMP is difficult in the sense that it is a QMA-complete problem~\cite{liu2006consistency,liu2007quantum,wei2010interacting,bookatz2012qma}. For the purposes of this paper, the only restriction imposed on joint states will be that they live in a finite-dimensional Hilbert space, with a finite and fixed number of subsystems.

Regarding the list of candidate density operators, e.g., $(\rho_{AB}, \rho_{BC}, \rho_{AC})$, the list of subsystems they correspond to, e.g., $(AB, BC, AC)$, is known as the \textit{marginal scenario}, while the individual elements, e.g., $AB$, $BC$, or $AC$, will be referred to as \textit{marginal contexts}. A key consideration for understanding previous work on the QMP is whether the marginal contexts are disjoint. When the marginal contexts are disjoint, a complete solution to the QMP is known~\cite{klyachko2004quantum,klyachko2006general}. For a given specification of Hilbert space dimensions, this solution takes the form of a finite list of linear inequality constraints on the spectra of the candidate density operators. These solutions furthermore recover earlier results pertaining to low dimensional Hilbert-spaces~\cite{higuchi2003one,higuchi2003qutrit,bravyi2003requirements,han2005compatibility}.

In contrast, when the marginal scenario involves overlapping marginal contexts, existing results are comparatively more sporadic and typically weaker, being only applicable to low-dimensional systems, small numbers of parties, or only yielding necessary but insufficient constraints~\cite{chen2014symmetric,carlen2013extension,butterley2006compatibility,hall2007compatibility,chen2016detecting}. One promising approach, developed in~\cite{christandl2018recoupling} for relating marginal spectra to the recoupling theory of the symmetric group, appears limited to situations where the marginal contexts do not overlap too much.

Whenever a candidate set of density operators is explicitly given, one strategy to decide their compatibility is to use convex optimization techniques, e.g., semidefinite programming~\cite{vandenberghe1996semidefinite}. If the joint state is not necessarily pure, compatibility can be decided with a single semidefinite program~\cite{hall2007compatibility}. Additionally, when the joint state is assumed pure, compatibility can still be decided by an infinite hierarchy of semidefinite programs~\cite{yu2021complete}. In either case, analytic inequality constraints that serve as witnesses for incompatibility can be extracted from the outputs of such semidefinite programs~\cite{hall2007compatibility}.

The objective of this paper is to improve our understanding of the QMP, in particular for the case of overlapping marginal contexts, by i) deriving inequality constraints that are necessarily satisfied by all compatible collections of density operators, and ii) proving that if a collection of density operators satisfies these inequalities, then they are compatible. This paper begins by formally defining the QMP and then reformulating it from a different perspective. The primary advantage of this reformulation of the QMP is that it exposes an implicit symmetry of the problem which is helpful in deriving our main result.

\section{A Reformulation of the QMP}
\label{sec:reform_qmp}

This section introduces some notation and terminology that is sufficient to formally define both the QMP and an equivalent reformulation that is better suited for the techniques developed in subsequent sections.

First and foremost, every Hilbert space considered will be complex, finite-dimensional, and labeled by some subscript $X$, e.g., $H_{X}$. For each labeled Hilbert space, $H_{X}$, we will implicitly assume there exists some canonical orthonormal basis such that $H_{X} \cong \mathbb C^{d_X}$ where $d_X = \dim(H_{X})$. The corresponding set of linear operators, density operators (states), and pure states of $H_{X}$ are respectively denoted $\mathcal L(H_{X})$, $\mathcal D(H_{X})$, and $\mathcal P(H_{X})$. The identity operator on $\mathcal H_{X}$ is denoted $I_{X}$.

Given a list of labels, $S = (X_1, \ldots, X_k)$, which we identify with the concatenated string, $S \simeq X_1\cdots X_k$, the composite Hilbert space $H_{X_1} \otimes \cdots \otimes H_{X_k}$, will be labeled by $S$ itself and thus denoted $H_{S}$. For instance, if $S = ABC$, then $H_{ABC} = H_{A} \otimes H_{B} \otimes H_{C}$. Additionally, for any positive integer $n$ and label $X$, the list of labels consisting of $n$ copies of $X$ will be abbreviated by
\begin{equation}
    nX \coloneqq (X, \stackrel{n}{\ldots}, X) \simeq X\stackrel{n}{\cdots}X
\end{equation}
Using this notational convention, the $n$th tensor-power of a Hilbert space $H_{X}$ can be written as $H_{X}^{\otimes n} = H_{nX} = H_{X\stackrel{n}{\cdots}X}$.

Associated to any instance of the QMP, is a \textit{joint} (or \textit{global}) Hilbert space $H_{J}$ where $J = (X_1, \ldots, X_p)$ is a given finite list of labels called the \textit{joint context}. Every non-empty sublist $S$ of $J$ will be called a \textit{marginal context}. For each $S \subseteq J$, the partial trace from $H_{J}$ onto $H_{S}$ will be denoted by
\begin{equation}
    \label{eq:partial_trace}
    \mathrm{Tr}_{J \setminus S} : \mathcal L(H_{J}) \to \mathcal L(H_{S}).
\end{equation}
A finite, non-empty tuple of marginal contexts,
\begin{equation}
    \mathcal M = (S_1, \ldots, S_m),
\end{equation}
is called a \textit{marginal scenario}. The cardinality of a marginal scenario will always be denoted by $m = \abs{\mathcal M}$.
\begin{prob}[QMP]
    \label{prob:qmp}
    Given a marginal scenario, $\mathcal M = (S_1, \ldots, S_m)$, and list of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ where $\rho_{S_i} \in \mathcal D(H_{S_i})$ for each $i \in \{1, \ldots, m\}$, decide if there exists a joint pure\footnotetext{The assumption of purity in the joint state can be made without loss of generality (see \cref{sec:pure_vs_mixed_qmp}).} state $\psi_{J} \in \mathcal P(H_J)$ such that
    \begin{equation}
        \label{eq:qmp}
        \forall S_i \in \mathcal M : \rho_{S_i} = \mathrm{Tr}_{J\setminus S_i}(\psi_{J}).
    \end{equation}
\end{prob}

Whenever such a pure state $\psi_{J}$ exists, the $m$-tuple of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ is said to be \textit{compatible} and otherwise they are \textit{incompatible}.

For the sake of brevity, unless otherwise specified, a joint Hilbert space $H_{J}$ with joint context $J$ will always be implicitly given together with a particular marginal scenario $\mathcal M = (S_1, \ldots, S_m)$ of length $m$.

Our first step is to reinterpret the $m$ linear constraints imposed on the joint state $\psi_{J}$ by \cref{eq:qmp} as a single linear constraint on the $m$th tensor-power state, $\psi_{J}^{\otimes m}$. Specifically, the $m$-tuple of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ satisfies \cref{eq:qmp} if and only if
\begin{equation}
    \label{eq:parallel}
    \rho_{S_1} \otimes \cdots \otimes \rho_{S_m} = \mathrm{Tr}_{J \setminus S_1}(\psi_{J}) \otimes \cdots \otimes \mathrm{Tr}_{J \setminus S_m}(\psi_{J}).
\end{equation}
This observation motivates the following definitions.
\begin{defn}
    The \textit{Hilbert space on $\mathcal M$}, denoted $H_{\mathcal M}$, is
    \begin{equation}
        H_{\mathcal M} \coloneqq H_{S_1} \otimes \cdots \otimes H_{S_m}.
    \end{equation}
    The \textit{partial trace from $mJ$ onto $\mathcal M$}, denoted $\mathrm{Tr}_{mJ\setminus \mathcal M}$, is
    \begin{equation}
        \mathrm{Tr}_{mJ\setminus \mathcal M} \coloneqq \mathrm{Tr}_{J\setminus S_1} \otimes \cdots \otimes \mathrm{Tr}_{J\setminus S_m}.
    \end{equation}
\end{defn}
Note that the partial trace from $mJ$ onto $\mathcal M$, $\mathrm{Tr}_{mJ\setminus \mathcal M}$, is simply the partial trace operation mapping elements of $\mathcal L(\mathcal H_{mJ}) = \mathcal L(H_{J}^{\otimes m})$ to elements of $\mathcal L(H_{\mathcal M}) = \mathcal L(H_{S_1} \otimes \cdots \otimes H_{S_m})$.
\begin{defn}
    An \textit{$\mathcal M$-product state} is any state, $\rho_{\mathcal M} \in \mathcal D(H_{\mathcal M})$, of the form
    \begin{equation}
        \rho_{\mathcal M} = \rho_{S_1} \otimes \cdots \otimes \rho_{S_m}
    \end{equation}
    where each component, $\rho_{S_i}$, is a state in $\mathcal D(H_{S_i})$.
\end{defn}
Since there is a bijection between $m$-tuples of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ and $\mathcal M$-product states $\rho_{S_1} \otimes \cdots \otimes \rho_{S_m}$, the QMP can be equivalently restated entirely in terms of $\rho_{\mathcal M}$.
\begin{prob}
    \label{prob:cqmp}
    Given an $\mathcal M$-product state, $\rho_{\mathcal M}$, determine whether or not there exists a pure state $\psi_{J}$ such that
    \begin{equation}
        \label{eq:cqmp}
        \rho_{\mathcal M} = \mathrm{Tr}_{mJ\setminus \mathcal M}(\psi_{J}^{\otimes m}).
    \end{equation}
\end{prob}
The equivalence between \cref{prob:qmp} and \cref{prob:cqmp} follows directly from \cref{eq:parallel}; moreover, whenever such a pure state $\psi_{J}$ exists in either formulation of the QMP, it satisfies both \cref{eq:qmp} and \cref{eq:cqmp}. Pursuant to this equivalence, an $\mathcal M$-product state, $\rho_{S_1} \otimes \cdots \otimes \rho_{S_m}$, is said to be \textit{compatible} whenever the $m$-tuple of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ is compatible (and \textit{incompatible} otherwise). The set of all compatible $\mathcal M$-product states will be denoted $\mathcal C_{\mathcal M}$.

\section{Necessary and Sufficient Inequality Constraints}
\label{sec:necc_suff}

In this section, we construct inequalities that are necessarily satisfied by \textit{all} compatible $\mathcal M$-product states, $\rho_{\mathcal M}$.
These inequalities, therefore, can be used to answer the QMP in the negative; if an $\mathcal M$-product state violates any of the forthcoming inequalities, then it is incompatible.
In addition, it will be shown that if an $\mathcal M$-product state, $\rho_{\mathcal M}$, satisfies all of the forthcoming inequalities, then it must be compatible.

These inequalities emerge from considering the permutation symmetry of the $k$th tensor power, $\psi_J^{\otimes k}$, of a pure state $\psi_J \in \mathcal P(H_{J})$. For each $k \in \mathbb N$, let $\mathfrak{S}_{k}$ be the symmetric group on $k$ symbols, and let $T_{J} : \mathfrak{S}_{k} \to \mathcal L(H_{J}^{\otimes k})$ be the representation of $\mathfrak{S}_k$ acting on $H_{J}^{\otimes k}$ by permutation of its $k$ factors.
\begin{defn}
    \label{defn:symsub}
    The $k$th symmetric subspace $\vee^{k}H_{J} \subseteq H_{J}^{\otimes k}$ is defined as
    \begin{equation}
        \vee^{k}H_{J} = \{ \ket{\phi} \in H_{J}^{\otimes k} \mid \forall \pi \in \mathfrak{S}_k, T_{J}(\pi) \ket{\phi} = \ket{\phi} \}.
    \end{equation}
    The orthogonal projection operator onto $\vee^{k}H_{J}$ will be denoted by $\Pi_{J}^{(k)}$.
\end{defn}
Given any vector $\ket{\psi_{J}} \in H_{J}$, it is straightforward to verify that $\ket{\psi_{J}}^{\otimes k}$ is an element of the $k$th symmetric subspace $\vee^{k}H_{J} \subseteq H_{J}^{\otimes k}$.
\begin{prop}
    \label{prop:copy_sym}
    Let $\psi_{J} = \ket{\psi_J}\bra{\psi_J}$ be a pure state, let $k \in \mathbb N$. Then\footnotetext{Throughout this paper, an inequality $A \geq B$ between Hermitian operators $A$ and $B$ always indicates that $A - B$ is positive semidefinite, i.e. $A - B \geq 0$. See \cite[Section V]{bhatia1997matrix}.}
    \begin{equation}
        \label{eq:kth_inclusion}
        \psi_{J}^{\otimes k} \leq \Pi_{J}^{(k)},
    \end{equation}
    where $\Pi_{J}^{(k)}$ is defined in \cref{defn:symsub}.
\end{prop}
By comparing \cref{eq:cqmp} with \cref{eq:kth_inclusion}, and recalling that partial traces are positive channels, it becomes possible to \textit{eliminate} $\psi_{J}$ from \cref{eq:cqmp}. For example, when $k=m$, the partial trace $\mathrm{Tr}_{mJ \setminus \mathcal M}$ applied to \cref{eq:kth_inclusion} yields $\mathrm{Tr}_{mJ\setminus\mathcal M}(\psi_{J}^{\otimes m}) \leq \mathrm{Tr}_{mJ\setminus\mathcal M}(\Pi_{J}^{(m)})$ and thus we obtain the following corollary.
\begin{cor}
    If $\rho_{\mathcal M}$ is a compatible $\mathcal M$-product state, then
    \begin{equation}
        \label{eq:n1}
        \rho_{\mathcal M} \leq \mathrm{Tr}_{mJ\setminus\mathcal M}(\Pi_{J}^{(m)}).
    \end{equation}
\end{cor}
In \cref{sec:n1}, it is shown that the utility of this constraint is quite sensitive to the marginal scenario under consideration. For certain marginal scenarios, \cref{eq:n1} happens to be satisfied by \textit{all} $\mathcal M$-product states, and thus is useless for the purposes of the QMP. Nevertheless, for other marginal scenarios, \cref{eq:n1} happens to be violated by some $\mathcal M$-product states, and thus is a \textit{non-trivial} condition for the compatibility of an $\mathcal M$-product state $\rho_{\mathcal M}$.

Analogous reasoning can be used to construct stronger inequality constraints for the QMP. When $k$ is a multiple of $m$, $k=nm$, one can apply the $n$th tensor power of $\mathrm{Tr}_{mJ\setminus\mathcal M}$ to both sides of \cref{eq:kth_inclusion}. While it is clear from the preceding discussion that this will yield inequality constraints necessarily satisfied by all $\mathcal M$-product states, we will additionally show that their satisfaction for all $n \in \mathbb N$ is \textit{sufficient} to conclude that $\rho_{\mathcal M}$ must be an $\mathcal M$-product state.
\begin{thm}
    \label{thm:main}
    An $\mathcal M$-product state, $\rho_{\mathcal M}$, is compatible if and only if for all $n \in \mathbb N$,
    \begin{equation}
        \label{eq:nth_order}
        \rho_{\mathcal M}^{\otimes n} \leq \mathrm{Tr}_{mJ\setminus\mathcal M}^{\otimes n}(\Pi^{(nm)}_{J}).
    \end{equation}
\end{thm}
Note that $\mathrm{Tr}_{mJ\setminus\mathcal M}^{\otimes n}$ is the partial trace operation taking elements of $\mathcal L(H_{J}^{\otimes nm})$ to elements of $\mathcal L(H_{\mathcal M}^{\otimes n})$.

To prove \cref{thm:main} we use the following lemma, proven in \cref{sec:proof_finetti_lemma}, that the upper-bound in \cref{eq:nth_order}, up to normalization, represents the expected value of $\sigma_{\mathcal M}^{\otimes n}$ when $\sigma_{\mathcal M}$ is sampled according to a probability measure, $\nu_{\mathcal M}$, whose support is precisely the set of compatible $\mathcal M$-product states, denoted $\mathcal C_{\mathcal M}$.
\begin{lem}
    \label{lem:de_finetti_compatible}
    There exists a probability measure, $\nu_{\mathcal M}$, over $\mathcal D(H_{\mathcal M})$, with support $\mathcal C_{\mathcal M}$, such that for all $n \in \mathbb N$,
    \begin{equation}
        \mathrm{Tr}_{mJ\setminus\mathcal M}^{\otimes n}(\Pi^{(nm)}_J) = \tbinom{nm+d_{J}-1}{nm} \int_{\mathcal C_{\mathcal M}} \hspace{-1em}\nu_{\mathcal M}(\mathop{}\!\mathrm{d} \sigma_{\mathcal M}) \sigma_{\mathcal M}^{\otimes n},
    \end{equation}
    where $d_J = \dim(H_J)$.
\end{lem}
\begin{proof}
    See \cref{sec:proof_finetti_lemma}.
\end{proof}
Now consider a measurement effect, $E_{n}$, acting on $H_{\mathcal M}^{\otimes n}$, i.e. a Hermitian operator $E_n \in \mathcal L(H_{\mathcal M}^{\otimes n})$ such that $0 \leq E_n \leq I_{\mathcal M}^{\otimes n}$. If \cref{eq:nth_order} is satisfied by some state $\rho_{\mathcal M} \in \mathcal D(H_{\mathcal M})$, and $\mathrm{Tr}(E_n\rho_{\mathcal M}^{\otimes n}) \neq 0$, then \cref{lem:de_finetti_compatible} implies
\begin{equation}
    \label{eq:prob_ratio_lower}
    \sup_{\sigma_{\mathcal M} \in \mathcal C_{\mathcal M}}\frac{\mathrm{Tr}(E_n\sigma_{\mathcal M}^{\otimes n})}{\mathrm{Tr}(E_n\rho_{\mathcal M}^{\otimes n})} \geq \tbinom{nm+d_{J}-1}{nm}^{-1}.
\end{equation}
Therefore, to show that every $\mathcal M$-product state, $\rho_{\mathcal M}$, eventually violates \cref{eq:nth_order}, and thus prove \cref{thm:main}, it suffices to prove the existence of a sequence of measurement effects, $n \mapsto E_n$, such that the above ratio of probabilities, as a function of increasing $n$, approaches zero faster than $\tbinom{nm+d_{J}-1}{nm}^{-1}$, thus violating \cref{eq:prob_ratio_lower}.

The particular problem of finding a sequence of measurements such that $\mathrm{Tr}(E_n\rho_{\mathcal M}^{\otimes n})$ stays reasonably large while simultaneously minimizing $\mathrm{Tr}(E_n\sigma_{\mathcal M}^{\otimes n})$ for all $\sigma_{\mathcal M}$ distinct from $\rho_{\mathcal M}$ is related to the problems of asymmetric quantum state discrimination and quantum hypothesis testing~\cite{hiai1991proper,ogawa2005strong,pereira2022analytical,hayashi2001asymptotics,hayashi2002two}. Broadly speaking, existing results in these fields are sufficiently strong to establish the claimed violation of \cref{eq:prob_ratio_lower} for each incompatible $\rho_{\mathcal M} \not \in \mathcal C_{\mathcal M}$. Our specific approach relies on ideas developed by \citeauthor{keyl2006quantum}~\cite{keyl2006quantum} for the purposes of quantum state estimation\footnote{Our estimation-theoretic approach is largely inspired by applications of the \textit{spectral} estimation technique~\cite{keyl2005estimating} to the \textit{spectral} QMP~\cite{christandl2006spectra,christandl2007nonzero,christandl2018recoupling}.}.

In the interest of being constructive and non-asymptotic, in \cref{sec:keyl_exclusive} (specifically \cref{rem:state_discrim}) we show how to construct, for each $\rho_{\mathcal M}$, an explicit sequence of projection operators, $n \mapsto E_n$, such that for all $n \in \mathbb N$,
\begin{equation}
    \label{eq:incompatibility_measure}
    \sup_{\sigma_{\mathcal M} \in \mathcal C_{\mathcal M}}\frac{\mathrm{Tr}(E_{n} \sigma_{\mathcal M}^{\otimes n})}{\mathrm{Tr}(E_{n} \rho_{\mathcal M}^{\otimes n})} \leq \exp( - (n-d_{\mathcal M}^2) \infkeyl{\rho_{\mathcal M}} + c(\rho_{\mathcal M}))
\end{equation}
where $c(\rho_{\mathcal M}) \geq 0$ and $\infkeyl{\rho_{\mathcal M}} \geq 0$ are quantities independent of $n$ (but dependent on $\rho_{\mathcal M}$) and $d_{\mathcal M} = \dim(H_{\mathcal M})$. Additionally, it is shown that the exponential rate, $\infkeyl{\rho_{\mathcal M}}$, vanishes if and only if $\rho_{\mathcal M}$ is compatible and thus its positivity can serve as a witness of the incompatibility of $\rho_{\mathcal M}$.

\begin{proof}[Proof of \cref{thm:main}]
    The discussion preceding \cref{thm:main} has already established the ``only if'' portion of \cref{thm:main}: applying $\mathrm{Tr}_{mJ\setminus\mathcal M}^{\otimes n}$ to \cref{eq:kth_inclusion} when $k = nm$ yields
    \begin{align}
        \rho_{\mathcal M}^{\otimes n}
        = \mathrm{Tr}_{mJ\setminus\mathcal M}^{\otimes n}(\psi_J^{\otimes nm})
        \leq \mathrm{Tr}_{mJ\setminus\mathcal M}^{\otimes n}(\Pi^{(nm)}_{J}).
        \label{eq:nth_disco}
    \end{align}
    Therefore, all that remains is to prove the ``if'' portion of \cref{thm:main}.
    Suppose $\rho_{\mathcal M} \in \mathcal D(H_{\mathcal M})$ is a state that satisfies \cref{eq:nth_order} for some particular value of $n$. By combining \cref{eq:incompatibility_measure} with \cref{eq:prob_ratio_lower}, we conclude
    \begin{align}
        \label{eq:incompatibility_upper_bound}
        \infkeyl{\rho_{\mathcal M}}
        &\leq \frac{\ln \tbinom{nm + d_{J} - 1}{nm} + c(\rho_{\mathcal M})}{n-d_{\mathcal M}^{2}}.
    \end{align}
    Since $\tbinom{nm + d_{J} - 1}{nm} \in O(n^{d_{J}-1})$ is polynomial of degree $d_{J}-1$ in $n$, the upper bound above approaches zero in the limit as $n \to \infty$. For finite $n$, the inequality in \cref{eq:incompatibility_upper_bound} merely implies that $\infkeyl{\rho_{\mathcal M}}$ must be small. For the purposes of \cref{thm:main}, if a state $\rho_{\mathcal M}$ satisfies \cref{eq:nth_order} for all $n$, \cref{eq:incompatibility_upper_bound} implies that $\infkeyl{\rho_{\mathcal M}} = 0$ and thus $\rho_{\mathcal M} \in \mathcal C_{\mathcal M}$ must be a compatible $\mathcal M$-product state.
\end{proof}

\section{Conclusion}
\label{sec:conclusion}

This paper makes progress toward an analytic solution to the quantum marginal problem (QMP) by constructing an countably-infinite family of necessary operator inequalities whose satisfaction by a given tuple of density operators is sufficient to conclude their compatibility.
The primary advantage of this approach is its generality: for any finite Hilbert space dimension(s) and any number of subsystems with arbitrary overlap, the corresponding family of necessary inequalities is shown to be sufficient.
The results of this paper, therefore, constitute the first quantifier-free solution to the QMP for overlapping marginal contexts.
However, the characterization of compatible density operators produced by this approach is not finite, and thus inherently more challenging to compute.
Evidently, further insights will be required to produce a finite set of necessary and sufficient conditions for the overlapping QMP.

\begin{acknowledgments}
    I would like to thank J. Davis and R. Spekkens for numerous helpful discussions.
    The majority of this research was conducted while T.C.F. was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), grant 411301803.
\end{acknowledgments}

\bibliographystyle{apsrev4-1}

redpajama_set_name: RedPajamaArXiv


text:
Tag: First Amendment
Categories : Land Use NJ Supreme Court ReporterTags : First AmendmentFree speechLeave a comment
NJ Supreme Court Rules Co-Op Board's Leaflet Ban Violated Resident's Free Speech Rights
On December 19, 2014 January 14, 2019 By Donald Scarinci
In Dublirer v. 200 Linwood Avenue Owners, the Supreme Court of New Jersey recently held that a co-op's leaflet ban was unconstitutional. In reaching its decision, the court clarified the standard to evaluate restrictions on free speech in a common-interest community like the building in this case. The Facts of
Categories : Municipal LawTags : BillboardFirst AmendmentNew Jersey Zoning LawLeave a comment
NJ Appeals Court Upholds Municipal Ban on Digital Billboards
On November 14, 2014 November 8, 2014 By Patrick J. McNamara
The Appellate Division of the New Jersey Superior Court recently held that banning digital billboards, while allowing static ones, does not violate the U.S. Constitution. The trial court had previously held that the Franklin Township ordinance violated the First Amendment. The Facts of the Case The dispute in E&J Equities
Categories : Municipal LawTags : City CouncilFirst AmendmentNew BrunswickNew Jersey liquor license transferLeave a comment
New Brunswick to Pay $1.4 Million to Resolve Bar Owner's First Amendment Claims
On August 26, 2014 August 27, 2014 By Kara Kaczynski
The City of New Brunswick impermissibly blocked a bar owner's efforts to open a restaurant and sports bar to be named "Buck Foston's," according to a New Jersey jury. It awarded plaintiff Lawrence D. Blatterfein more than $1.5 million for the violation of his First Amendment rights. The Facts of
Categories : Constitutional Law Municipal LawTags : First AmendmentMunicipal sign ordinanceU.S. Supreme CourtLeave a comment
U.S. Supreme Court to Review Municipal Sign Ordinance Next Term
On August 5, 2014 August 4, 2014 By Patrick J. McNamara
The U.S. Supreme Court recently agreed to consider whether an Arizona town's sign ordinance violates the First Amendment, as applied to a local church. Under long-standing First Amendment precedent, content-based restrictions on speech are closely scrutinized and upheld only if they are "narrowly tailored" to a "compelling government interest." Reed
Categories : Public Employment LawTags : First AmendmentU.S. Supreme CourtLeave a comment
U.S. Supreme Court Addresses First Amendment Rights of Public Employees
On July 18, 2014 August 21, 2014 By sdon
New Jersey municipalities should be aware of an important U.S. Supreme Court decision addressing the expanded free speech rights of public employees. In Lane v. Franks, a unanimous Court held that the First Amendment protected a public employee when he testified truthfully about fraud in the work place pursuant to
Categories : Municipal LawTags : Christianitycouncil meetingsFirst Amendmentnational traditionPrayerreligious equalityLeave a comment
Supreme Court Upholds Prayer Before Council Meeting
On May 13, 2014 May 9, 2014 By Sheri Siegelbaum
On May 5, 2014, the Supreme Court upheld Christian prayers at the beginning of local town meetings in the case Town of Greece v. Galloway. The Court declared the prayer in line with long-standing, national traditions, even though the nation has grown more religiously diverse. The Court was narrowly divided
Categories : Constitutional Law Election LawTags : aggregate limitcampaign contributionscampaign donationscampaign fundingFirst Amendmentpolitical committeesLeave a comment
Aggregate Campaign Contribution Limits Violate First Amendment
On April 18, 2014 April 21, 2014 By Donald Scarinci
The recent opinion of the U.S. Supreme Court in McCutcheon v. Federal Election Committee has struck a blow to the regulation of money in elections. The Court struck down the federal limits on the total amounts individual donors may contribute to candidates and political committees. The Court reasoned that donating


See pricing info, deals and product reviews for Truvia® Nectar Stevia Sweetener 300g at Quill.com. Order online today and get fast, free shipping for your business.
Truvia® Nectar has 50% fewer calories per serving than sugar and adds a touch of sweetness wherever you would use agave, sugar or honey. Introducing NEW Truvia® Nectar blend, the reduced calorie way to love honey. Blended with stevia leaf extract and honey, Truvia® Nectar has 50% fewer calories per serving than sugar and adds a touch of sweetness wherever you would use agave, sugar or honey. Because Truvia® Nectar is twice as sweet as honey, you only need to use half as much. Try in coffee, tea, on oatmeal, yogurt and fruit and in your favorite recipes - all with 50% fewer calories from sugar. Life with less sugar is just as sweet with Truvia® Nectar.


C is also for Chrysocolla; my most recent love. I adore it for its marvellous and intricate patterns in all shades of blue and green. Found close to large copper deposits, it is intermingled with a combination of some or all of these: turquoise, malachite, azurite, jasper and quartz, and it is this combination of minerals that gives each stone its fascinating and unique patterns. It's not expensive either; you can go large on these without breaking the bank. I have just listed, for the first time, 8mm and 10mm cabochons, at $3.50 and $5.50 per stone respectively. I'll stock larger and different shapes too, if there is a demand. If you want to know what it's good for, it's supposed to be a stone of reconciliation and reassurance, and has been known as the 'wise stone', as those who wore it were believed to be good at the kind of win-win negotiation we're all supposed to strive, coming up with clever compromises and solutions. I could do with a bit of that. To shop 8mm chrysocolla cabochons, click here; to shop 10mm, click here.
It's the kind of stone that looks great with silver, and I often see it in quite chunky settings. But below I've found a collection of pieces whose designers have let their imaginations run riot, from the delicate, branch-like construction of Lucie Veilleux's sterling silver bracelet (bottom left) to the fantastical blooming of Amy Buettner's Chrysocolla, sterling silver and 18k gold centerpiece (top left) to the openwork on ZYGO Jewelry's sterling silver ring, that lightens the entire appearance.
It's not the most well-known gemstone, but Chrysocolla is endlessly fascinating, with its patterning and array of different colours. It's the copper in it that is responsible for the vivid blue-green colours and it often combines with other copper-bearing minerals which give it its unique mix of colours and characteristic mottled patterns, such as turquoise, blue azurite and malachite. Just listed I have these gloss finish 7-8mm rondelle beads, at $20 per strand. These are B grade, and I think that the range of colours, textures and patterns on these stones are just amazing. I also have these in a matte finish, same size, which are equally multi-coloured, but with an amazing soft and silky finish (see left). These are also $20 per strand. And finally, some AB grade beads, which are much more uniform in terms of colour leaning firmly towards the blue-green end of the spectrum, again same size. These are also matte finish, and these are $37 per strand. I can see these looking great with some chunky silver accents – let your creativity go wild! Beads are a relatively new venture for me, and all of the ones I have are funky and wonderful – to check out the full range, click here. To shop check out Chrysocolla beads click here.


We continue to be overwhelmed with how God is pulling all of this together, and with everyone who has contributed to our adoption fund. We now only have 148 more pieces of the puzzle to sell.
Anyway, we've also been thinking of having a Yard Sale to raise a little money and also get rid of some things. Then we also have been thinking of doing a breakfast or dinner fundraiser or something. Anyone have experience with this, thoughts or advice or want to help put it together?


GlusterFS allows you to replicate files across a number of servers giving higher performance and additional resiliency. Also, almost instantaneous replication to off-site servers makes it a good option for disaster recovery and business continuity.
In this example, we'll have two storage nodes to replicate the data between, and also one client node to mount the network filesystem from remotely.
Update: As of CentOS 6.6, these RPMs are available in the base repo.
# do this inside a directory to contain all the RPMs, replace with your release and arch.
# all others can be omitted for now but may be useful in the future.
# configured, so it might be good to disable those repos temporarily.
Disable firewall (or open it up to addresses on subnet) so it doesn't block attempts to connect to gluster. We'll just disable in this example.
# this prints out the hostname or ip of the other nodes in the cluster, UUID, and status - connected, etc.
Hard disks are referred to as "bricks" in gluster. We have /dev/sdb on both nodes which we'll use as the bricks for gluster. First we'll partition /dev/sdb with fdisk, then format, mount and add to fstab.
In this example, both gluster01 and gluster02 have 8GB /dev/sdb disks. We want to mirror the data across the disks, so if a node dies all the data will be accessible. It's like a raid1 across two hosts instead of two disks in 1 host.
Mount the volume. From any machine that has glusterfs-fuse installed, you should be able to mount it.
Write to mount point, and verify it on both servers.
# verify it got put on the bricks, you'll see it on both servers.
At this point, you can mount the volume (pointing to any of the peer IP/Hostnames) from a client computer and read/write to it like any network volume. Say you mount glusterfs01:/gv0 from a client and glusterfs01 dies, the client will seemlessly failover to glusterfs02:/gv0 and will continue reading/writing without having to do anything. When you get glusterfs01 fixed up and bring it back online, as soon as glusterd starts, whatever updates occurred while it was offline will be synchronized.
Now if we need to push the contents of this volume to another datacenter, we can use glusterfs-geo-replication which basically rsyncs the volume across the WAN.


Noel is an instructional assistant professor of educational leadership. He completed his doctorate in educational policy and theory at the University of Rochester (N.Y.), where his dissertation focused on how student-teacher relationships are impacted when teachers are required to live in the district where they teach. Noel earned his undergraduate degree in psychology and Africana studies and a master of arts in liberal studies focusing on education and English literature from Rutgers University. He authored a book chapter on advanced placement and college readiness in "The Plight of Invisibility: A Community-Based Approach to Understanding the Educational Experiences of Urban Latina/os," edited by D.M. Harris and J.M. Kiyama (Peter Lang).
Parries is an assistant professor of counseling. She comes to the COE from the University of North Alabama, where she taught in the clinical mental health program of the department of counselor education. She also worked as a counselor with The Balanced Life LLC and as a volunteer counselor with several nonprofits, all in Florence, Ala. Parries received her undergraduate degree in English from Cleveland State University. She earned master's degrees in English and education and her doctoral degree in counselor education and supervision from Old Dominion University.
Sacra is an instructional assistant professor of counseling. She recently completed her doctorate in counselor education and supervision at the University of Montana. Her dissertation focused on clients' counseling experiences in Bhutan, where she worked as a college counselor. Sacra earned her undergraduate degree in brain and cognitive sciences with an emphasis on neurobiology from the University of Rochester and a master of arts in counselor education from the University of Montana. She has worked as a counselor at an elementary and middle school and a job corps in Montana.
Scott is an assistant professor of leadership, language and curriculum. Previously he worked as a term faculty member in the COE and DePaul's School of New Learning. Scott earned his undergraduate degree in economics and philosophy and a master's degree in measurement, evaluation, and assessment from the University of Illinois at Chicago. He completed his doctorate in adult learning and leadership at Columbia University Teacher's College. Scott edited "Learning Cities for Adult Learners," the spring 2015 issue of the New Directions for Adult & Continuing Education journal.


Saturday (6 April 2019) was the start of the new 2019/20 tax year, bringing into effect changes to income tax bands and allowances.
The personal allowance is now £12,500, up £650 from £11,850 in 2018/19. However, there remains a reduction in the personal allowance for those with 'adjusted net income' over £100,000. The reduction is £1 for every £2 of income above £100,000. For 2019/20 this means there is no personal allowance available where an individual's adjusted net income exceeds £125,000.
The tax on income is different for Scottish resident taxpayers compared to taxpayers resident elsewhere in the UK (except for savings and dividend income which are taxed at normal UK bands and rates). The Scottish income tax rates and bands apply to income such as employment income, self-employed trade profits and property income.
For taxpayers elsewhere in the UK the basic rate of tax is 20% (7.5% for dividends). For 2019/20 the basic rate band increases to £37,500 (up from £34,500) so that the threshold at which the 40% higher rate band (32.5% for dividends) applies is £50,000 for those who are entitled to the full personal allowance (up from £46,350). Individuals pay tax at 45% (38.1% for dividends) on their income over £150,000.
From April 2019, the Welsh Government has the right to vary the rates of income tax payable by Welsh taxpayers. The UK government has reduced the rates of income tax paid by Welsh taxpayers by 10 pence. However, the Welsh Government has set the Welsh rate of income tax at 10 pence which is added to the reduced rates, meaning the tax payable by Welsh taxpayers continues to be the same as that payable by English and Northern Irish taxpayers.
We have put together a tax card with all the tax rates for 2019/2020. You can download this as a PDF or you can request for an EQ Tax Card here. These are a handy size to keep in your wallet or purse for use whenever you need it.
If you would like to discuss your own circumstances, require support or further information on income tax bands and allowances, please speak to one of our EQ Taxperts or call your local office contact.


text:
Q: How to insert variable in `validationSchema` Formik and Yup? I try to validate <FieldArray /> in Formik with Yup. I generate a bunch of user names after click on "add user". Every Field looks like:
<Field
   type="text"
   name={`users.${index}.name`}
/>
<ErrorMessage
   name={`users.${index}.name`}
   component="div"
   className="invalid-feedback"
/>

How should my validationSchema look like? In my opinion I should add index to shape method instead of  simple name. But how can I do this?
validationSchema={yup.object().shape({
   // ?????
   name: yup.string().required('First Name is required'),
})}

A: Edit:
What you want to do is to update the Formik component when your initial values changes. set enableReinitialize to true

You can do 2 things:

*

*Append new key/val pair to object like this

var eatables = {vegetable: 'Carrot'}
var fruit_var = 'fruit'
eatables[fruit_var] = 'Apple'
console.log(eatables) // {fruit: 'Apple', vegetable: 'Carrot'}

*In ES6, you can use shorthand computed property names.

var fruit_var = 'fruit'
var eatables = {[fruit_var]: 'Apple', vegetable: 'Carrot'}
console.log(eatables) // {fruit: 'Apple', vegetable: 'Carrot'}

Source - Javascript: Computed Property Names (ES6)

redpajama_set_name: RedPajamaStackExchange


New Mountain Design consistently exceeds my client's expectations. Jason's Design and Visualization Skills are superb. His products are among the most unique and highest quality available. He consistently delivers on time and budget, and adds tremendous value to every one of our projects.
Jason understands our clients and delivers creative and exceptional results, from Design through Execution, every time.
Uncompromising, enthusiastic, and talented are a few adjectives that initially come to mind to describe New Mountain Design. Jason's ideas, abilities, and products are tremendous.
New Mountain Design has worked with us on our home being built in Pole Creek. We live in Des Moines, IA - which means we are building long distance. In addition to making everything run seamlessly (despite the logistics), Jason has solidly demonstrated a keen interest to detail, uniqueness, quality, and a strong desire to please us, the customer. He is always open to exploring new ideas, down to the minute details. I honestly have never run across a Designer with such an eye for Design - He offered us countless, innovative, yet functional suggestions.
Thank you so much for helping us create our dream kitchen. It has exceeded our expectations. This was clearly the most trouble free renovation we've ever experienced (out of many...) Time for a toast!
Just wanted to give you guys a HUGE thank you for the amazing work and the finished product at the almost complete Portal Road house. Everything is INCREDIBLE and even though it is currently very dirty, it looks so fine! We can't wait to move in and sure appreciate the cabinetry, and more importantly your creativity.
Once again I can't thank you enough for being so flexible and easy to work with. I have enjoyed working with NMD immensely!


text:
//
//  AMClient+MenuSection.m
//  AirMenuKit
//
//  Created by Robert Lis on 12/02/2014.
//  Copyright (c) 2014 air-menu. All rights reserved.
//

#import "AMClient+MenuSection.h"
#import "AMObjectBuilder.h"

@implementation AMClient (MenuSection)

-(NSURLSessionDataTask *)findMenuSectionWithIdentifier:(NSString *)identifier
                                            completion:(MenuSectionCompletion)completion
{
    NSAssert(identifier, @"identifier cannot be nil");
    NSString *urlString = [@"menu_sections/" stringByAppendingString:identifier];
    return [self GET:urlString
          parameters:nil
             success:^(NSURLSessionDataTask *task, id responseObject) {
                 AMMenuSection *section = [[AMObjectBuilder sharedInstance] objectFromJSON:responseObject];
                 if(completion) completion(section, nil);
             }
             failure:^(NSURLSessionDataTask *task, NSError *error) {
                 if(completion) completion(nil, error);
             }];
}

-(NSURLSessionDataTask *)updateMenuSection:(AMMenuSection *)section
                               withNewName:(NSString *)name
                            newDescription:(NSString *)description
                    newStaffKindIdentifier:(NSString *)identifier
                                completion:(MenuSectionCompletion)completion
{
    NSAssert(identifier, @"identifier cannot be nil");
    NSMutableDictionary *params = [NSMutableDictionary dictionary];
    if(name) [params setObject:name forKey:@"name"];
    if(description) [params setObject:description forKey:@"description"];
    if(identifier) [params setObject:identifier forKey:@"staff_kind_id"];

    NSString *urlString = [@"menu_sections/" stringByAppendingString:section.identifier.description];
    return [self PUT:urlString
          parameters:params
             success:^(NSURLSessionDataTask *task, id responseObject) {
                 AMMenuSection *section = [[AMObjectBuilder sharedInstance] objectFromJSON:responseObject];
                 if(completion) completion(section, nil);
             }
             failure:^(NSURLSessionDataTask *task, NSError *error) {
                 if(completion) completion(nil, error);
             }];
}

-(NSURLSessionDataTask *)deleteMenuSection:(AMMenuSection *)section
                                completion:(MenuSectionCompletion)completion
{
    NSAssert(section.identifier, @"identifier cannot be nil");
    NSString *urlString = [@"menu_sections/" stringByAppendingString:section.identifier.description];
    return [self DELETE:urlString
             parameters:nil
                success:^(NSURLSessionDataTask *task, id responseObject) {
                    AMMenuSection *section = [[AMObjectBuilder sharedInstance] objectFromJSON:responseObject];
                    if(completion) completion(section, nil);
                }
                failure:^(NSURLSessionDataTask *task, NSError *error) {
                    if(completion) completion(nil, error);
                }];
}

/*
 Menu Section > Menu Item
 */

-(NSURLSessionDataTask *)createItemOfSection:(AMMenuSection *)section
                                    withName:(NSString *)name
                                 description:(NSString *)description
                                       price:(NSNumber *)price
                                    currency:(NSString *)currency
                                 staffKindId:(NSString *)staffKindIdentifier
                                      avatar:(UIImage *)avatar
                                  completion:(MenuSectionItemCompletion)completion
{
    NSAssert(section.identifier, @"sections identifier cannot be nil");
    NSAssert(name, @"name identifier cannot be nil");
    NSAssert(price, @"price identifier cannot be nil");
    NSAssert(description, @"description identifier cannot be nil");
    NSAssert(currency, @"currency identifier cannot be nil");
    NSMutableDictionary *params = [@{@"name" : name, @"description" : description, @"price" : price, @"currency" : currency} mutableCopy];
    if(staffKindIdentifier) [params setObject:staffKindIdentifier forKey:@"staff_kind_id"];
    NSString *urlString = [@"menu_sections/" stringByAppendingFormat:@"%@/%@", section.identifier, @"menu_items"];
    if(avatar)
    {
        return [self POST:urlString
               parameters:params
constructingBodyWithBlock:^(id<AFMultipartFormData> formData) {
                      [formData appendPartWithFormData:UIImagePNGRepresentation(avatar) name:@"avatar"];
                  }
                  success:^(NSURLSessionDataTask *task, id responseObject) {
                      AMMenuItem *menuItem = [[AMObjectBuilder sharedInstance] objectFromJSON:responseObject];
                      if(completion) completion(menuItem, nil);
                  }
                  failure:^(NSURLSessionDataTask *task, NSError *error) {
                      if(completion) completion(nil, error);
                  }];
    }
    else
    {
        return [self POST:urlString
               parameters:params
                  success:^(NSURLSessionDataTask *task, id responseObject) {
                      AMMenuItem *menuItem = [[AMObjectBuilder sharedInstance] objectFromJSON:responseObject];
                      if(completion) completion(menuItem, nil);
                  }
                  failure:^(NSURLSessionDataTask *task, NSError *error) {
                      if(completion) completion(nil, error);
                  }];
    }
}

-(NSURLSessionDataTask *)findItemsOfSection:(AMMenuSection *)section
                                 completion:(MenuSectionItemsCompletion)completion
{
    NSAssert(section.identifier, @"sections identifier should not be nil");
    NSString *urlString = [@"menu_sections/" stringByAppendingFormat:@"%@/%@", section.identifier, @"menu_items"];
    return [self GET:urlString
          parameters:nil
             success:^(NSURLSessionDataTask *task, id responseObject) {
                 NSArray *menuItems = [[AMObjectBuilder sharedInstance] objectFromJSON:responseObject];
                 if(completion) completion(menuItems, nil);
             }
             failure:^(NSURLSessionDataTask *task, NSError *error) {
                 if(completion) completion(nil, error);
             }];
}

@end

redpajama_set_name: RedPajamaGithub


text:
Q: How can I improve the results of HoughLinesP() in my OpenCV Python script I am trying to get all the lines in this image:

This is the code that I'm using:
threshold = 30
minLineLength =10
maxLineGap = 10
lines = cv2.HoughLinesP(img,1,np.pi/360, threshold, minLineLength, maxLineGap)

The problem is that I'm getting too many lines (~300):

But if I increase the threshold value it starts to miss some lines:

Is there any way of reducing the number of lines while keeping line-detection accurate?
Thanks in advance!

A: It works (mostly) fine for me in Python/OpenCV. Adjust your HoughP line arguments as appropriate.
I think you need to threshold your image first. And perhaps thin the white lines.
Input:

import cv2
import numpy as np

# read image as color not grayscale
img = cv2.imread("lines.png", cv2.IMREAD_COLOR)

# convert img to grayscale
gray = cv2.cvtColor(img, cv2.COLOR_BGR2GRAY)

# do threshold
thresh = cv2.threshold(gray, 30, 255, cv2.THRESH_BINARY)[1]

# get hough line segments
threshold = 30
minLineLength =10
maxLineGap = 10
lines = cv2.HoughLinesP(thresh, 1, np.pi/360, threshold, minLineLength, maxLineGap)

# draw lines
results = img.copy()
for [line] in lines:
    print(line)
    x1 = line[0]
    y1 = line[1]
    x2 = line[2]
    y2 = line[3]
    cv2.line(results, (x1,y1), (x2,y2), (0,0,255), 1)

# show lines
cv2.imshow("lines", results)
cv2.waitKey(0)

# write results
cv2.imwrite("lines_hough.png",results)

Resulting Hough lines in red:

You get a lot of parallel very close lines that you may want to merge somehow or thin out the list.