text:
DNA Exonerations in Georgia Result in Disparate Compensation Awards
Loaded on July 15, 2010 by David Reutter published in Prison Legal News July, 2010, page 24
Filed under: Searches, DNA Testing/Samples, Wrongful Conviction, Wrongful Imprisonment. Location: Georgia.
by David M. Reutter
The disparity in compensation awards for prisoners exonerated by DNA evidence in Georgia demonstrates the need for evenhanded compensation laws.
Five wrongly convicted prisoners, Clarence Harrison, Robert Clark, Douglas Echols, Samuel Scott and Willie Otis "Pete" Williams, spent a combined 82 years in Georgia prisons for crimes they didn't commit. All were proven innocent by DNA evidence.
Three of those five received compensation from the state – Harrison, who served 17 years for the abduction, robbery and rape of a woman waiting for a bus; Clark, who served almost 24 years for rape, kidnapping and robbery; and Williams, who served 21 years for kidnapping, rape and aggravated sodomy before being exonerated in 2007.
Harrison, who was 44 when released from prison in 2004, was awarded $1 million by state lawmakers. Clark re-ceived $1.2 million after the Georgia legislature passed a resolution approving the payment in March 2007. He had contracted hepatitis C while in prison. Williams was awarded $1.2 million in compensation.
"Can you even fathom in your wildest imagination what it must be like for this man to lose his entire adult life, until now, incarcerated in prison, literally excommunicated from free society, beat down emotionally and probably physically, too?" asked state Rep. Larry O'Neal, who sponsored the resolution for Clark's compensation payment.
Echols and Scott, however, had a prosecutor who still insists they are not "factually innocent" despite the DNA evi-dence. They were cleared in 2002 of raping a woman in Scott's home.
The victim claimed Scott held her down while Echols raped her. A third assailant was never identified. Scott served 5 years before being paroled, while Echols was imprisoned for 15 years.
Their prosecutor, Spencer Lawton, was made famous by John Brendt's book, Midnight in the Garden of Good and Evil. Lawton wrote a letter to state legislators asserting that Echols and Scott were never exonerated and are not factu-ally innocent despite DNA evidence to the contrary.
The lawmaker who sponsored the legislation to compensate Scott and Echols said he felt "blindsided" by Lawton's let-ter. "He wasn't man enough to admit he may have convicted the wrong guys," observed former state Rep. Tom Bordeaux. "It was wrong in the most fundamental way. It was indecent."
Echols has sued Lawton for violating his constitutional rights by lobbying against Bordeaux's compensation legisla-tion. See: Echols v. Lawton, U.S.D.C. (S.D. Ga.), Case No. 4:08-cv-00023-WTM-GRS. Meanwhile, Echols and Scott have said the fact that their conviction records were not expunged has made it hard for them to find work, as they are still technically convicted felons.
Georgia is one of 23 states that do not have laws governing compensation for the wrongly convicted. Instead, exonerees have to find a lawmaker willing to introduce legislation requesting compensation on their behalf – a process that has had disparate results.
"Rather than the emotional aspect on a case-by-case basis, I think we ought to have some guidelines to help us in the future," said Georgia Governor Sonny Perdue.
Beyond fair compensation laws, one state lawmaker, Rep. Stephanie Stuckey Benfield, repeatedly introduced meas-ures to reform Georgia's eyewitness identification procedures, since faulty eyewitness testimony has been implicated in a high percentage of wrongful convictions. Although her legislation failed to pass, on December 10, 2008 the Georgia Peace Officers Standards and Training Council voted to require training for law enforcement officers on new eyewitness identification rules.
"We needed to step up to the plate," said Ken Vance, the Council's director. "This is good for law enforcement in Georgia." It's also good for defendants who risk being wrongly convicted based on faulty eyewitness identification.
Other recent wrongful convictions in Georgia include those of John Jerome White and Michael Marshall. White served a total of 22 years for the rape, robbery and assault of a 74-year-old woman before being exonerated by DNA evidence on December 10, 2007. In April 2009 state lawmakers approved $500,000 in compensation for White; how-ever, it included provisions requiring him to undergo drug testing, remain employed and stay out of trouble. Also, despite clear evidence that he had been wrongly convicted, his compensation award just barely passed the state senate on a vote of 29 to 21 (a minimum of 29 votes were required).
The most recent Georgia DNA exoneree, Michael Marshall, was proven innocent in December 2009 after serving 1½ years of a 4-year sentence for vehicle theft. He had been homeless at the time of the crime, and was arrested because he looked like a sketch of a suspect who stole a truck at gunpoint. His compensation request is pending.
There have been eight DNA exonerations in Georgia since 1999, most involving the efforts of the Georgia Innocence Project.
Sources: The News Tribune, www.jacksonville.com, Atlanta Journal-Constitution, www.ga-innocenceproject.org
Celebrity Justice: Prison Lifestyles of the Rich and Famous, by Matthew Clarke
California Uses $1.08 Billion in Federal Stimulus Funds to Pay Prison Guard Salaries, by Michael Brodheim
New Medical Director at Texas Jail Previously Sanctioned
U.S. Supreme Court to Review California Prison Population Reduction Orders, by John Dannenberg
Florida Prison Psychiatrist Resigns; License Revoked Over Sex with Patient, by David Reutter
Dallas County Jail Settles Three Medical-Related Suits for $795,000
Florida's Civil Rights Restoration Process Insufficiently Funded, by David Reutter
Convictions Upheld in Appeal of Lynne Stewart, Attorney to Blind Sheikh, but Case Remanded for Resentencing, by Justin Miller
Texas Youth Commission Ombudsman Resigns Following Smuggling Indictment, by Matthew Clarke
Kentucky Lethal Injection Protocol Adopted in Violation of APA, by Brandon Sample
Pennsylvania County Jail Settles Medical Indifference Suit for $55,000
Eighth Circuit Upholds $2,501 Retaliation Judgment Against Arkansas Prison Guard
Indiana Sex Offender Registration Law Can Not Be Retroactively Applied
Valley Fever Cases at California Prison Increase in 2009
Release Conditions Requiring Defendant to Tell Probation Officer about Romantic Relationships Vacated
Only Three States in Compliance with Unfunded Federal Sex Offender Mandates, by Matthew Clarke
DNA Exonerations in Georgia Result in Disparate Compensation Awards, by David Reutter
Guilty Pleas in Angola Horse Selling Scheme, by David Reutter
San Francisco Settles Wrongful Incarceration Cases for $7.5 Million, by Michael Brodheim
Canyon County Jail in Idaho Settles Conditions Suit With Consent Decree and $190,000 in Attorney's Fees, by Brandon Sample
U.S. Senator's Girlfriend, In-Law Get Department of Justice Jobs, by Brandon Sample
U.S. Department of Justice Releases Report on HIV in Prisons
South Carolina Settles Prisoner Stabbing Death Lawsuit for $47,500
California County Jail Settles Wrongful Death Suit for $600,000
Virginia Sheriff's Office, PHS Settle Wrongful Death Suit for $1.6 Million
CCA Pays $70,000 in Damages, Attorney Fees to Settle PLN Censorship Suit, by Alex Friedmann
Prison-based Call Centers Open in Austria, India, by Matthew Clarke
New York Prisoner Beaten, Guards Convicted, GEO Settles Suit for $80,000, by Matthew Clarke
California Counties Vie to House ICE Prisoners, by Michael Brodheim
New Jersey DOC Agrees to Let Prisoner Preach
Erroneously Released Texas Prisoner Entitled to Credit on Sentence, by Matthew Clarke
Media Agencies Intervene to Unseal Records in Prisoner's Wrongful Death Suit, by Brandon Sample
Sacramento County Partially Settles Taxpayer Suit Alleging Il-legal Conditions of Confinement in Juvenile Facilities, by Michael Brodheim
Oregon Offers Early Release to Illegal Immigrants Who Consent to Deportation, by Michael Brodheim
The Politics of Death: Throwing Mumia Abu-Jamal Under the Bus, by Dave Lindorff
CMS Nurse Denied Summary Judgment for Failure to Treat Prisoner for Heat Illness;$400,000 Settlement Following Sixth Circuit Ruling, by David Reutter
Wisconsin County Pays $750,000 to Settle Jail Sex Abuse Suit
Marsy's Law Enjoined in California, by Michael Brodheim
Arkansas Federal Jury Awards $261,000 to Male Prisoner Raped by Male Guard
One in Six HIV-Infected Americans Spent Time in Prison or Jail in 2006, by Michael Brodheim
Houston Police Department Conducted Blood Draw Training on Prisoners, by Greg Dober
Second Circuit Ruling in Post-9/11 Immigration Detention Case, by Matthew Clarke
Massachusetts Supreme Judicial Court: Sheriff May Not Charge Jail Fees, by Matthew Clarke
Former New York Corrections Commissioner Receives Four-Year Prison Sentence, by Matthew Clarke
Sixth Circuit: No Eleventh Amendment Immunity When ADA Claim Includes Fourteenth Amendment Violations, by Matthew Clarke
Georgia Officials Receive Prison Sentences in Charge-Fixing Scheme
News in Brief:
More from David Reutter:
Ninth Circuit: Federal Sentencing Court Must Hear Defendant Before Determining If Acceptance of Responsibility Reduction Applies, Nov. 18, 2019
Massachusetts Supreme Court Suppresses Evidence Obtained After Miranda Warnings Translated into Spanish Deemed Incapable of Conveying Meaningful Advice, Nov. 18, 2019
Tennessee Supreme Court Abandons Doctrine of Abatement Ab Initio, Oct. 16, 2019
Michigan Supreme Court: Reaching Out Door of Home to Retrieve ID Inadequate to Surrender Fourth Amendment Rights, Sept. 17, 2019
New York Court of Appeals: Jury Trial Right Attaches to Deportable Crimes Punishable by Less Than Six Months in Jail, July 17, 2019
Ohio Supreme Court: Plea Defendant Must Be Informed of Maximum Penalty for Postrelease-Control Violation Prior to Pleading Guilty to a New Felony, June 17, 2019
Indiana Supreme Court: IAC Where Lawyer Marks 'Not Applicable' to Immigration Consequences Warning on Court's Advisement Form Without Knowing Client's Immigration Status, June 17, 2019
Arkansas Supreme Court Rules Justification Defense Available When Charged With Manslaughter, June 17, 2019
New York Court of Appeals: Jury Trial Right Attaches to Deportable Crimes Punishable by Less Than Six Months in Jail, May 15, 2019
Fourth Circuit: 9-Year Increase in Guidelines Range Due to Misclassification as Career Offender Warrants § 2241 Petition to Be Heard on Merits When § 2255 Relief Unavailable, May 15, 2019
$1.5 million for Michigan man wrongfully convicted, then exonerated, Jan. 18, 2020. Wrongful Conviction, Money/Property.
Rikers Prisoner Smuggles DNA Evidence of Rape Out of Jail; $500,000 Settlement, Jan. 9, 2020. Sexual Assault, DNA Testing/Samples, Settlements.
National Fingerprint Database Frees Man After 36 Years, Dec. 18, 2019. Databases, Wrongful Conviction, Wrongful Imprisonment.
Georgia Supreme Court: Warrantless Search of Vehicle's Airbag Control Module is Unconstitutional, Dec. 18, 2019. Misconduct/Corruption, Police Misconduct, Searches, Police Searches.
Police Use of Rapid DNA Machines Unregulated, Dec. 17, 2019. DNA Testing/Samples, Police.
Man Held in Jail Almost Three Months for Possession of ... Honey, Dec. 10, 2019. Misconduct/Corruption, Crime Labs, Wrongful Imprisonment.
Texas Counties Waste Millions by Jailing Defendants Charged with Citation-Eligible Misdemeanors, Dec. 10, 2019. Jail Specific, Wrongful Imprisonment.
The Cost of Wrongful Convictions, Dec. 9, 2019. Wrongful Conviction.
Suits Demand Cops be Held Accountable for Thousands of Rape Test Kits Sitting Untested, Dec. 8, 2019. DNA Testing/Samples, Prison Rape Elimination Act.
The Two-Edged Sword of DNA Exonerates Another Prisoner, Nov. 19, 2019. DNA Testing/Samples, Release and Reentry.


text:
The Sailing frigate classification system used during the 19th Century was a classification scheme used in several western countries.  It was officially adopted by the United States Navy.  The system classified sailing frigates according to their gun rating.

Ratings

Use
The United States Navy used this classification system officially, beginning at least by 1825. The Royal Institution of Naval Architects, an international organization of naval architects, also adopted the system, beginning in 1860.

The United States adopted a new rating system during the American Civil War, based on the thrown weight of broadsides.  By 1875, this system was replaced by a system of steam warship classification based on tonnage.

References

United States Navy

redpajama_set_name: RedPajamaWikipedia


text:
Business rates replaced by VAT hike – proposed pie in the sky A Tory MP is calling for business rates to...
Stuart Hicks
Dunlop Heywood
Rating + Commercial Property
Investment / Capital Markets Surveyor The Company Our client has been providing expert advice...
TBC, Manchester / North West
Liverpool 2015 office review paints mixed picture
Liverpool's 2015 Commercial Office Market Review revealed office take-up across the city region totalled 576,847 sq ft last year, a 2.4% increase on the previous year. But city centre take-up dipped more than 20%.
The report, which is jointly produced by member agents of Professional Liverpool's Property Group and supported by Liverpool BID Company, highlighted that smaller lettings dominated the market in the Commercial District with a record number of deals; 156 in 2015 compared to 121 in 2014.
Take-up in the commercial district was 301,956 sq ft, down 21% on 2014.
Out-of-town take-up rose by 73% on 2014, reaching 193,824 sq ft with significant deals including 22,900 sq ft to Deyes High School at Vortex House and 22,140 sq ft at Liverpool Innovation Park to AIMES Grid Services.
Demand for office space in the city fringe also improved, with a 20% rise in take-up. This was boosted by the letting of 33,433 sq ft to the NHS Liverpool CCG at The Department, the largest letting in the commercial district and city fringe combined.
This latest report also revealed take-up in Grade B and Grade B* categories was dominant across the city region, accounting for 90 of all office take-up in 2015.
Tony Reed, chairman of Professional Liverpool's Property Group, said: "The statistics are extremely encouraging with a 2.4% rise in overall take-up across the region, reflecting that the market has continued to benefit from the economic upturn. While the growth of the creative/IT/media sectors helps to cement Liverpool's reputation as a truly entrepreneurial city.
"But there are still challenges as we see a continued lack of Grade A office stock across the Liverpool City Region. With no new Grade A space in planning, with the exception of Peel Land & Property's Liverpool Waters scheme, this is set to continue."
Bill Addy, chief executive of Liverpool BID Company, which is currently overseeing the renewal ballot of the Commercial District BID, said: 'The market in Liverpool city centre is vibrant and shows a healthy demand for space. The number of start-up businesses is an encouraging underlying trend with the rise of the Digital and Creative sector in the Commercial District a highly positive sign of the confidence in the city. The BID Company is focused on nurturing and developing these trends over the next five years, but this report is clear; Grade A supply needs to improve and as a city we must strive to stimulate the market – especially by investing in our transport and digital infrastructure.''
The review was launched at a breakfast event held at the recently launched 25,000 sq ft Tempest building by Capital & Centric on Tithebarn Street.
DOWNLOAD REPORT: Commercial Office Market Review 2015
It isn't correct to say that city centre take up reduced by 20%, the report states:
"Total office take-up in the Commercial District in 2015 was 301,956 sq ft which is down by 21% from 2014 but similar to the 316,964 sq ft of 2013."
The Commercial District isn't the city centre, but only a pocket of it.
It also goes on to say:
"As with 2014, the decreasing supply can be attributed to an increasing number of building uses being converted from offices to hotels, serviced apartments and residential, including Martins bank building, 60 Old Hall Street, Orleans House and Mersey House."
Given some of these are very considerable office blocks, the loss of these and yet the city still posting a rise in overall take up I feel is not a "mixed picture" at all but instead one of a city doing well against all the odds.
March 02, 2016 at 12:39 pm By Mike
We need grade A office space, the big question is when if ever Peel, will get off their backsides and start Liverpool Waters, they have been very quiet since the announcements this time last year about the Priness Dock, very worrying that no news has been released since. They do not inspire confidence at all in their scheme.
March 02, 2016 at 1:06 pm By Man on bicycle
We need to push ahead with our plans for HS2/3 to future proof a level playing field.
March 03, 2016 at 4:22 pm By Gwydion

redpajama_set_name: RedPajamaCommonCrawl


text:
AnnouncementsApp Manager, camera, demo, e-learning, Maya, mobile applications, physics, programming, Puzzles, releases, texture compression, Verge3D, WebGL, Woocommerce, Wordpress
We are excited to present a new release of Verge3D for Maya and Verge3D Ultimate. Version 4.2 brings a new E-learning demo, the possibility to load 3D view on demand in WooCommerce, new connector and map range puzzles, draggable 3D interfaces, and restructured settings panels in Maya. E-learning Demo We've…
Announcements3ds Max, App Manager, camera, demo, e-learning, mobile applications, physics, programming, Puzzles, releases, texture compression, Verge3D, WebGL, Woocommerce, Wordpress
We are excited to present a new release of Verge3D for 3ds Max! Version 4.2 brings a new E-learning demo, the possibility to load 3D view on demand in WooCommerce, new connector and map range puzzles, draggable 3D interfaces, and restructured settings panels in 3ds Max. E-learning Demo We've created…
AnnouncementsApp Manager, Blender, camera, demo, e-learning, mobile applications, nodes, physics, programming, Puzzles, releases, texture compression, Verge3D, WebGL, Woocommerce, Wordpress
We are excited to present a new release of Verge3D for Blender! Version 4.2 brings two highly interactive demos, the mobile controls for the first-person mode, the possibility to load 3D view on demand in WooCommerce, support for Alpha Mode settings in Image Texture node, new connector and map range…
AnnouncementsAndroid, App Manager, batching, clipping planes, desktop applications, global illumination, iOS, light probes, lighting, Maya, mobile applications, physics, Puzzles, releases, Ultimate, Verge3D, wireframe
We are excited to announce the 3.7 release of Verge3D for Maya and Verge3D Ultimate. We had our reasons for the prolonged release cycle, and here they are. Verge3D 3.7 for Maya introduces support for many more shader nodes, the global illumination technique to drastically improve the realism of your…
Announcements3ds Max, Android, App Manager, batching, clipping planes, desktop applications, global illumination, iOS, light probes, lighting, mobile applications, physics, Puzzles, releases, Verge3D, wireframe
The long-awaited update has finally arrived! We had our reasons for the prolonged release cycle, and here they are. Verge3D 3.7 for 3ds Max introduces support for Arnold renderer, the global illumination technique to drastically improve the realism of your scenes, brings the clipping planes to show off the internals…
AnnouncementsAndroid, App Manager, batching, Blender, clipping planes, desktop applications, global illumination, iOS, light probes, lighting, mobile applications, physics, Puzzles, releases, Verge3D, wireframe
The long-awaited update has finally arrived! We had our reasons for the prolonged release cycle, and here they are. Verge3D 3.7 for Blender introduces the global illumination technique to drastically improve the realism of your scenes, brings the clipping planes to show off the internals of complex objects, integrates tools…
Announcementsgame, JavaScript, Maya, physics, plugins, Puzzles, releases, shaders, soft-body, Verge3D
We are excited to announce the 3.5 release of Verge3D for Maya and Verge3D Ultimate. This version includes a plugin system for Verge3D, soft-body simulation, physical constraints, advanced mouse controls, dozen Maya shaders, Coat settings support, shadow catchers, exporting custom properties, lots of new puzzles, and more. There is also…
Announcements3ds Max, constraints, game, JavaScript, physics, plugins, Puzzles, releases, soft-body, Verge3D
Here is the 3ds Max version of Verge3D 3.5. This huge release brings many new features: Verge3D plugins, soft-body simulation, physical constraints, advanced mouse controls, various 3ds Max integrations, new puzzles for code-less workflow, and more. There is also a new game example included in this version. Arcade Racing We…
AnnouncementsBlender, constraints, game, JavaScript, physics, plugins, Puzzles, releases, soft-body, Verge3D
The long-awaited version 3.5 is here! This is a huge release with a myriad of new features, most important of which are Verge3D plugins, soft-body simulation, physical constraints, specular workflow support, meta balls, advanced mouse controls, several new puzzles the users asked for, and many many more. We have also…
Verge3D 3.1 for Maya Released
AnnouncementsMaya, physics, Puzzles, release, releases, Verge3D, VR
Here is a new release of Verge3D for Maya, an artist-friendly toolkit for creating 3D web interactives. Version 3.1 brings new demos and Maya integrations, means for operating VR controllers, many new features in the physics engine, logic blocks for Vector operations, faster image-based lighting modes, React.js and Vue.js templates,…


text:
Ctenophthalmus stirps är en loppart som beskrevs av Beaucournu et Orhan 1983. Ctenophthalmus stirps ingår i släktet Ctenophthalmus och familjen mullvadsloppor. Inga underarter finns listade i Catalogue of Life.

Källor

Mullvadsloppor
stirps


At Kage Design Group we believe that good design can be the most effective way to enrich people's lives. Our concepts evolve from the unique conditions that each project presents. As a full service licensed Architecture / Interior Design firm, we place our focus on creation at the intersection of art and function. It is our mission to deliver an innovative product that embraces efficient, practical, creative solutions with a focus on genuinely understanding each client's needs. Our architects and interior designers work closely together to combine their talents, expertise, skills and resources in an effort to provide our clientele with a well rounded, quality product.
Dyehouse Comeriato Architect is a service-oriented architectural firm located in Naples, Florida. Our staff have been serving Southwest Florida since 1978 with award winning architecture and client satisfaction. Our architecture services are used for both residential and commercial clients. We have a proven track record and encourage you to browse our galleries on our website, and contact us to set up a consultation.
Founded in 2001 by Tyler Petersen, Petersen Design Studio merged a wealth of experience and skill into a company that would earn a remarkable reputation as a leading design firm. For over ten years, Petersen Design Studio has been designing unique and innovative structures, each project is handled with unique perspective and integrity. In 2012 George and Tyler revamped Petersen Design Studio into PDS Architecture Inc., and pushed forward with their commitment to operating on a foundation of unwavering ethics and quality results. PDS Architecture will provide you with the best design possible. We approach every project with a focus on aesthetics, cost, and functionality. Tyler Petersen has worked on many award-winning projects throughout his career. He began as an Associate at Sheeley Architects in 2000 and was quickly responsible for the design and construction oversight of several 20+ story high-rise condominiums in Southwest Florida. That same year Tyler started Petersen Design Studio with an emphasis on providing one-on-one design services for high-end residential clients. Since then his clientele has evolved to include commercial and institutional projects. His portfolio ranges from small medical offices and restaurant design to high-rise condominiums and heavy industrial facilities. His goal of building relationships with clients continues to this day, as evident by the numerous repeat clients.
We offer 3D Drafting & Rendering done in Chief Architect; Whether you're remodeling or building new, 3-D renderings can help you save money and get exactly what you want on your next home project. We are a custom home design firm who prides itself in getting the details right. We have a list of satisfied customers and our strive to retain satisfied clients is our number one goal.
Albert Dambrose is the owner and Florida licensed architect (AR93689) of Studio A.D. The firm specializes in modern residential design throughout Florida. Studio A.D. provides services from consulting, schematic design, design development, construction documents and construction administration of any project.
Award winning Design Firm with no boundries for designing. Working with some of the most successful and respected builders, the firm's projects include high-density multi-family, high-density single family, luxury single family, custom homes, as well as active adult communities and signature recreational clubhouse buildings. Joe McHarris is personally involved in every project and leads his highly motivated and professional team with a retail approach and sensitivity toward today's marketplace. McHarris Planning & Design's creative and functional solutions have resulted in widespread industry recognition, countless awards and have built a reputation on their consistent delivery of communities that appeal to builders and homebuyers alike.
Thinking about building a new home? Southwest Florida offers the ultimate warm-weather lifestyle with year-round mild temperatures, an endless variety of outdoor activities, and convenient shopping and dining. HomeQwest Development, your liaison to everything involving your new home. Building your new home with us has many advantages in price, "customability" options, our experience, and quality. You can choose one of our designs, or bring us a design of your own choice, or we will help you design the home you really want. We also include a four-year home warranty policy to help protect your new home after the first year. You won't find a better option in our area. Why deal with template home designs, minimal upgrade options, and higher prices.
Established over 4 decades ago, Timothy A. Krebs has designed well over 1000 structures, ranging from single family homes to 100,000 sq. ft. office buildings. His expertise in design and construction methods has created projects that span from the North East United States to the Southern Parts of Florida.


See everything Port Macquarie has to offer! Fly to the Hastings River mouth and track south along the beaches, then cut inland to touch down at Bago and enjoy the maze, a chocolate tasting, a wine tasting and a cheese and meat platter with wine.
Fly back to the airport via the Bago Bluff and Wauchope.


text:
Q: Microsoft Sql server 2016 installation error Hello I have tried installing SQL Server 2016 and I have had errors that I could not fix.

Object reference is not set to an instance of an object

this is the error that I receive and I think the problem comes from databases engine services because it is the component that keeps failing during the installation. If someone has encountered a similar problem and knows how to solve it, I would be grateful if he shared it.
     Overall summary:
      Final result:                  Failed: see details below
      Exit code (Decimal):           -2147467261
      Start time:                    2017-05-26 14:59:12
      End time:                      2017-05-26 15:17:17
      Requested action:              Install

    Setup completed with required actions for features.
    Troubleshooting information for those features:
      Next step for RS:              Use the following information to resolve the error, uninstall this feature, and then run the setup process again.
      Next step for SQLEngine:       Use the following information to resolve the error, uninstall this feature, and then run the setup process again.
Product features discovered:
  Product              Instance             Instance ID                    Feature                                  Language             Edition              Version         Clustered  Configured

Package properties:
  Description:                   Microsoft SQL Server 2016
  ProductName:                   SQL Server 2016
  Type:                          RTM
  Version:                       13
  Installation location:         D:\x64\setup\
  Installation edition:          Developer

  Slipstream:                    True
  SP Level                       1

Product Update Status:
  Success: KB 3182545

Product Updates Selected for Installation:
  Title:                         Microsoft SQL Server 2016  with SP1
  Knowledge Based Article:       KB 3182545
  Version:                       13.1.4001.0
  Architecture:                  x64
  Language:                      1033

  Update Source:                 Slipstream

User Input Settings:
  ACTION:                        Install
  ADDCURRENTUSERASSQLADMIN:      false
  AGTSVCACCOUNT:                 NT Service\SQLSERVERAGENT
  AGTSVCPASSWORD:                *****
  AGTSVCSTARTUPTYPE:             Manual
  ASBACKUPDIR:                   Backup
  ASCOLLATION:                   Latin1_General_CI_AS
  ASCONFIGDIR:                   Config
  ASDATADIR:                     Data
  ASLOGDIR:                      Log
  ASPROVIDERMSOLAP:              1
  ASSERVERMODE:                  MULTIDIMENSIONAL
  ASSVCACCOUNT:                  <empty>
  ASSVCPASSWORD:                 <empty>
  ASSVCSTARTUPTYPE:              Automatic
  ASSYSADMINACCOUNTS:            <empty>
  ASTELSVCACCT:                  <empty>
  ASTELSVCPASSWORD:              <empty>
  ASTELSVCSTARTUPTYPE:           0
  ASTEMPDIR:                     Temp
  BROWSERSVCSTARTUPTYPE:         Disabled
  CLTCTLRNAME:                   <empty>
  CLTRESULTDIR:                  <empty>
  CLTSTARTUPTYPE:                0
  CLTSVCACCOUNT:                 <empty>
  CLTSVCPASSWORD:                <empty>
  CLTWORKINGDIR:                 <empty>
  COMMFABRICENCRYPTION:          0
  COMMFABRICNETWORKLEVEL:        0
  COMMFABRICPORT:                0
  CONFIGURATIONFILE:             C:\Program Files\Microsoft SQL Server\130\Setup Bootstrap\Log\20170526_145857\ConfigurationFile.ini
  CTLRSTARTUPTYPE:               0
  CTLRSVCACCOUNT:                <empty>
  CTLRSVCPASSWORD:               <empty>
  CTLRUSERS:                     <empty>
  ENABLERANU:                    false
  ENU:                           true
  EXTSVCACCOUNT:                 <empty>
  EXTSVCPASSWORD:                <empty>
  FEATURES:                      SQLENGINE, RS, CONN, IS
  FILESTREAMLEVEL:               0
  FILESTREAMSHARENAME:           <empty>
  FTSVCACCOUNT:                  <empty>
  FTSVCPASSWORD:                 <empty>
  HELP:                          false
  IACCEPTROPENLICENSETERMS:      false
  IACCEPTSQLSERVERLICENSETERMS:  true
  INDICATEPROGRESS:              false
  INSTALLSHAREDDIR:              C:\Program Files\Microsoft SQL Server\
  INSTALLSHAREDWOWDIR:           C:\Program Files (x86)\Microsoft SQL Server\
  INSTALLSQLDATADIR:             <empty>
  INSTANCEDIR:                   C:\Program Files\Microsoft SQL Server\
  INSTANCEID:                    MSSQLSERVER
  INSTANCENAME:                  MSSQLSERVER
  ISSVCACCOUNT:                  NT Service\MsDtsServer130
  ISSVCPASSWORD:                 <empty>
  ISSVCSTARTUPTYPE:              Automatic
  ISTELSVCACCT:                  NT Service\SSISTELEMETRY130
  ISTELSVCPASSWORD:              <empty>
  ISTELSVCSTARTUPTYPE:           Automatic
  MATRIXCMBRICKCOMMPORT:         0
  MATRIXCMSERVERNAME:            <empty>
  MATRIXNAME:                    <empty>
  MRCACHEDIRECTORY:
  NPENABLED:                     0
  PBDMSSVCACCOUNT:               <empty>
  PBDMSSVCPASSWORD:              <empty>
  PBDMSSVCSTARTUPTYPE:           0
  PBENGSVCACCOUNT:               <empty>
  PBENGSVCPASSWORD:              <empty>
  PBENGSVCSTARTUPTYPE:           0
  PBPORTRANGE:                   <empty>
  PBSCALEOUT:                    false
  PID:                           *****
  QUIET:                         false
  QUIETSIMPLE:                   false
  ROLE:
  RSINSTALLMODE:                 DefaultNativeMode
  RSSHPINSTALLMODE:              DefaultSharePointMode
  RSSVCACCOUNT:                  NT Service\ReportServer
  RSSVCPASSWORD:                 <empty>
  RSSVCSTARTUPTYPE:              Automatic
  SAPWD:                         <empty>
  SECURITYMODE:                  <empty>
  SQLBACKUPDIR:                  <empty>
  SQLCOLLATION:                  SQL_Latin1_General_CP1_CI_AS
  SQLSVCACCOUNT:                 NT AUTHORITY\SYSTEM
  SQLSVCINSTANTFILEINIT:         false
  SQLSVCPASSWORD:                *****
  SQLSVCSTARTUPTYPE:             Automatic
  SQLSYSADMINACCOUNTS:           Moni-Laptop\SimeonGerginov
  SQLTELSVCACCT:                 NT Service\SQLTELEMETRY
  SQLTELSVCPASSWORD:             <empty>
  SQLTELSVCSTARTUPTYPE:          Automatic
  SQLTEMPDBDIR:                  <empty>
  SQLTEMPDBFILECOUNT:            8
  SQLTEMPDBFILEGROWTH:           64
  SQLTEMPDBFILESIZE:             8
  SQLTEMPDBLOGDIR:               <empty>
  SQLTEMPDBLOGFILEGROWTH:        64
  SQLTEMPDBLOGFILESIZE:          8
  SQLUSERDBDIR:                  <empty>
  SQLUSERDBLOGDIR:               <empty>
  SUPPRESSPRIVACYSTATEMENTNOTICE: false
  TCPENABLED:                    0
  UIMODE:                        Normal
  UpdateEnabled:                 true
  UpdateSource:                  Slipstream
  USEMICROSOFTUPDATE:            false
  X86:                           false

  Configuration file:            C:\Program Files\Microsoft SQL Server\130\Setup Bootstrap\Log\20170526_145857\ConfigurationFile.ini

Detailed results:
  Feature:                       Client Tools Connectivity
  Status:                        Passed

  Feature:                       Reporting Services - Native
  Status:                        Failed: see logs for details
  Reason for failure:            An error occurred for a dependency of the feature causing the setup process for the feature to fail.
  Next Step:                     Use the following information to resolve the error, uninstall this feature, and then run the setup process again.
  Component name:                SQL Server Database Engine Services Instance Features
  Component error code:          -2147467261
  Error description:             Object reference not set to an instance of an object.
  Error help link:               http://go.microsoft.com/fwlink?LinkId=20476&ProdName=Microsoft+SQL+Server&EvtSrc=setup.rll&EvtID=50000&ProdVer=13.0.4001.0&EvtType=0x9C8C9C81%400x44A889F9&EvtType=0x9C8C9C81%400x44A889F9

  Feature:                       Database Engine Services
  Status:                        Failed: see logs for details
  Reason for failure:            An error occurred during the setup process of the feature.
  Next Step:                     Use the following information to resolve the error, uninstall this feature, and then run the setup process again.
  Component name:                SQL Server Database Engine Services Instance Features
  Component error code:          -2147467261
  Error description:             Object reference not set to an instance of an object.
  Error help link:               http://go.microsoft.com/fwlink?LinkId=20476&ProdName=Microsoft+SQL+Server&EvtSrc=setup.rll&EvtID=50000&ProdVer=13.0.4001.0&EvtType=0x9C8C9C81%400x44A889F9&EvtType=0x9C8C9C81%400x44A889F9

  Feature:                       Integration Services
  Status:                        Passed

  Feature:                       SQL Browser
  Status:                        Passed

  Feature:                       SQL Writer
  Status:                        Passed

  Feature:                       Setup Support Files
  Status:                        Passed

Rules with failures:

Global rules:

Scenario specific rules:

Rules report file:               C:\Program Files\Microsoft SQL Server\130\Setup Bootstrap\Log\20170526_145857\SystemConfigurationCheck_Report.htm

redpajama_set_name: RedPajamaStackExchange


Boleh's six pack on show!
Hidden away below the floor of Boleh's cabin but revealed in this photograph is the auxiliary power pack which will drive Boleh's 16.5 tonnes at 6 knots. These twin electric motors, manufactured by Lynch Motors in Devon, weigh only 15 kilos each yet deliver 13 kilowatts (17.4 horsepower) of power to each of Boleh's 2 propellers.
Robin Kilroy, Boleh's original designer, produced an innovative auxiliary power system for Boleh in 1950. This relied on a single 8.5 horsepower Stuart Turner petrol engine working through a unique 'drive' to a propeller lowered in to the water when required.
Boleh's new 35 horsepower diesel electric system rivals this in design innovation and is modeled on that fitted in the Royal Barge 'Gloriana' for which our Naval Architect, Graham Westbrook was also responsible.


This paper discusses some practical applications of chapter five of the Nan Jing as found in Tu Ju Wang Shuhe Mai Jue Ba Shi Yi Nan Jing Bien Zhen (Pictorial Notes on the Pulse Songs of Wang Shuhe).1 The five-depth system is also discussed in other important texts of Chinese traditional medicine, including the Mai Jing (Pulse Classic)2 and Li Zhi Shen's Pulse Diagnosis.3 While each text discusses the method of pressure to assess the five depths, none of these sources discuss the practical application of the method.
In chapter five, beans are used to signify the depth of pressure. "With a very light hand press superficially and then press harder; 1st depth is 3 beans pressure to the lungs and skin, 2nd depth is 6 beans to the heart and the vessels, 3rd depth is nine beans to the spleen and flesh, 4th depth is 12 beans to the liver and sinews, 5th depth is 15 beans to touch the kidney and bones."4 Please note: depth is based on light pressure at the skin and deep pressure close to the bone, rather than using the vessel as the starting and ending point. For instance, the fingers are not merely pressing at six beans of pressure to examine the heart; one is pressing with an intention to examine the vessels.
Application The ability of an organ to control the tissues is another indirect suggestion that occurs in chapter five of the Nan Jing. For instance, the spleen's capacity to control the flesh is evaluated at the middle depth. To examine the biceps, explore the middle depth in the right distal position. This provides information about the muscles and flesh along the trajectory of the lung vessel. As an alternative, use the middle depth (earth and spleen sector) in the proximal position to examine the musculature of the lower back.
Needle depth: The standard needle depth can be divided into five regions, and the region corresponding to the tissue layer is needled. The pulse can be palpated while the needle depth is adjusted to refine the depth of insertion. The distance within the point range can be divided into five sectors. The area closest to the bony landmark is the Water area - look for the most tender spot or palpate for the spot that corrects the pulse depth.
I would like to thank Christine Chang, a licensed acupuncturist and pharmacist at Emperor's College of Traditional Oriental Medicine, for her assistance in translating chapter five of the Tu Ju Wang Shuhe Mai Jue Ba Shi Yi Nan Jing Bien Zhen.


text:
OPEC would miss 'friend Trump', wary of strains under Biden, sources say
By Alex Lawler, Rania El Gamal
(This November 7 story has been refiled to add credit in signoff)
FILE PHOTO: OPEC logo is pictured ahead of an informal meeting between members of the Organization of the Petroleum Exporting Countries (OPEC) in Algiers, Algeria September 28, 2016. REUTERS/Ramzi Boudina/File Photo
LONDON/DUBAI (Reuters) - Key members of OPEC are wary that strains in the OPEC+ alliance could reemerge with Joe Biden as U.S. President, sources close to the organisation said, and would miss President Donald Trump who went from criticising the group to helping bring about a record oil output cut.
Biden could modify U.S. diplomatic relations with three members of OPEC - de facto leader Saudi Arabia, and sanctioned countries Iran and Venezuela, as well as with key non-OPEC producer Russia. Russia is the leader of oil producers allied with OPEC, a group known as OPEC+.
Strict enforcement of U.S. sanctions on Iran and Venezuela has kept millions of barrels of oil per day off the market, and if Biden should relax measures on either in years to come an increase in production could make it harder for OPEC to balance supply with demand.
Biden has said he would prefer multilateral diplomacy to the unilateral sanctions Trump has imposed, although that may not mean a relaxation in sanctions any time soon. In his campaign, Biden said he'd return to Iran's 2015 nuclear deal if Tehran resumes compliance with the pact.
Trump quit the pact in 2018, reimposing sanctions that cut Iran's oil exports. Some in OPEC fear that a return of Iranian volumes will add to oversupply without cutbacks elsewhere and worry about Moscow's continued participation in OPEC+.
"Iran sanctions can be re-evaluated and then Iran will be back to the market, so again there would be oversupply and the current cut deal will be at risk," an OPEC source said before the election result was known.
"There is the risk of Russia leaving the OPEC+ deal too which means a collapse of the agreement, as it was Trump who brought Moscow on board," the source said.
Biden has named Russia as Washington's most serious global threat. During his campaign, he also pledged to reassess ties with Saudi Arabia.
Trump in April was involved in talks that led to a deal in which the Organization of the Petroleum Exporting Countries and Saudi Arabia worked with allied producers led by Russia to agree a record oil supply cut as the coronavirus outbreak hammered demand.
Trump stepped in to put political pressure on Saudi Arabia and Russia to end a dispute that had sparked a price war and resulted in both countries planning to raise output just as the pandemic led to restrictions on travel - and consequently in demand for fuel.
The result was an unprecedented global deal to cut oil supply by about 20 million barrels per day, or around 20 percent. OPEC+ alone agreed to cut 9.7 million bpd.
For Trump, the motivation was to raise global oil prices and prevent bankruptcies and hundreds of thousands of job losses in the U.S. energy industry as the election loomed.
Trump has been a proponent of the oil and gas industry, rolling back environmental regulations and rejecting mainstream science on how emissions are causing global warming.
Earlier in his presidency, he had criticised OPEC for seeking higher prices and urged members to pump more. Anti-OPEC U.S. legislation known as NOPEC - first introduced years ago - didn't become law despite having gained some momentum earlier in his presidency.
"Trump is now our friend - after the historic U-turn," said a senior OPEC source from a U.S. ally OPEC member, declining to be named. "From NOPEC to Art of the Deal," he added, in a reference to the April OPEC+ pact and a 1987 book by Trump.
SIGNIFICANT IMPLICATIONS
Trump developed a close relationship with top OPEC producer Saudi Arabia's de facto ruler Mohammed bin Salman, or "MbS", who relies on the United States for weapons and protection against regional rivals such as Iran.
The OPEC+ alliance has been propping up oil prices since 2017 and any developments that threaten the future of the alliance could weaken the market, with significant implications for OPEC and other producers, governments and traders.
Trump engaged more actively with OPEC than his predecessors, often taking to Twitter to comment on production decisions and oil price moves. Biden is seen as more likely to keep the cartel at arm's length.
"My view is that Biden would rely more on professional advice from his advisers and would not micromanage as Trump does today," said Chakib Khelil, Algeria's oil minister for a decade and a former OPEC president.
"Biden would not have the cosy relationship with Putin that Trump appears to have," Khelil added.
Still, despite Biden's campaign comments on the U.S.-Saudi relationship, a radical reset is viewed as unlikely. Gulf regional sources and diplomats have told Reuters a Biden win would not upend decades-long alliances.
And a source familiar with Iranian oil thinking welcomed Biden's win but said he doubted sanctions would be lifted swiftly. This would give OPEC+ members long enough to adjust their deal to make room for more Iranian oil.
"Even if Iran's sanctions are lifted, it will take two to four months for Iran's oil exports to return to pre-sanctions levels due to technical issues," he said. "Therefore, OPEC+ has enough time to decide on a new production ceiling."
Additional reporting by Rania El Gamal; Editing by David Holmes


The Place Where Kids Want to Take Their Parents!
We had a great 2018 Season.
We are now closed and reopening on Friday, May 17/2019.
Have a great winter and we are looking forward to having you camp with us in 2019.
May 1/2019 via email or by telephone.
Contact us for more information or to reserve your stay.


St. Patrick and the symbols associated with him can be seen in some of the objects from our Decorative Arts collection, Collins Barracks, Dublin.
Triptych, depicts scenes of St Patrick's life in Ireland, enamels and brass inlaid with silver and painted with enamels, London, c. 1903, by Alexander Fisher.
Clay pipe from Frongoch Internment Camp, North Wales. Decorated with harps and shamrock, painted green in places. Dated to from mid-1916 up to the release at Christmas 1916, but possibly extending to 1917.
Evening dress of olive green shantung silk and antique Irish crochet with matching silk coat, designed by Neillí Mulcahy for her aunt Phyllis, Mrs. Sean T. O'Kelly. Phyllis O'Kelly wore the dress to a reception in The White House on the evening of St. Patrick's Day 1959, during the first ever Irish Presidential visit to the U.S.A.. The designer supplied Beverly Bags of Bray, Co. Wicklow, with remnants of the silk and crochet and the company made a bag to match.


Dr. Yuhsun Edward Shih earned his Education PhD degree from Capella University with the specialization in Instructional Design for Online Learning. His dissertation researched the effectiveness of mobile learning. Dr. Shih was the instructional Designer at California State University, San Bernardino and the CSUSB's Blackboard system administrator, and have been teaching online courses at several universities since 2007.


Hello I am , June 11th, 2018, This is my post about Car Trunk Storage Ideas Wonderful 10 Genius Ways To Organize Everything In Your Spaces Decorating 0, on this page you can see so beautiful design about Car Trunk Storage Ideas Wonderful 10 Genius Ways To Organize Everything In Your Spaces Decorating 0, the first picture in this post is the best picture we use about Car Trunk Storage Ideas Wonderful 10 Genius Ways To Organize Everything In Your Spaces Decorating 0, detail on photo's above, you can see in the second paragraph.
The first picture is a picture Car Trunk Storage Ideas Wonderful 10 Genius Ways To Organize Everything In Your Spaces Decorating 0, the image has been in named with : Car Trunk Storage Ideas Wonderful 10 Genius Ways To Organize Everything In Your Spaces Decorating 0, This image was posted on category : Uncategorized, have resolution : 580x800 pixel. This posting entitled Car Trunk Storage Ideas Wonderful 10 Genius Ways To Organize Everything In Your Spaces Decorating 0, and many people looking for about Car Trunk Storage Ideas Wonderful 10 Genius Ways To Organize Everything In Your Spaces Decorating 0 from search engine.


text:
Police say no charges for Florida father who asked clerk if he could pawn his infant child
The man told police this was part of a prank video for social media
By ABC7 Staff | May 7, 2019 at 11:04 PM EDT - Updated May 8 at 5:38 PM
SARASOTA, Fla. (WWSB) - Police in Sarasota, Florida say a 43-year-old father will not face charges after asking a clerk from a local jewelry store if he could pawn his infant child.
Sarasota Police say the father walked into the A&F Jewelry and Loan store on North Washington Boulevard around 5:15 p.m. on Tuesday, May 7 with his infant child in a carrier and asked the clerk if he could pawn a baby. The clerk told police he didn't respond and eventually the man left, but while leaving he reportedly said, "I just need a babysitter for like two weeks."
Police say no charges for father who asked clerk if he could pawn his infant child
Based on statements the man made, the clerk was concerned for the safety of the child and contacted Sarasota Police. Police then released video of the man Tuesday night to ask for the public's help identifying him. At the time, police did not know the relationship between the man and the infant.
Within two hours of posting the video to social media, police say the man saw himself on social media and went to police to explain what happened. He told officers that he had involved the clerk in a prank and that he had intended to post video of that prank on social media.
Detectives say they determined the child was safe but they contacted the Florida Department of Children and Family Services as a routine part of their investigation.
At this time, the man is not facing any charges.
Copyright 2019 WWSB. All rights reserved.
Richmond police search for missing woman who may need medication
The Richmond Police Department is searching for a missing woman who may be in need of medication.


text:
\section{Introduction}
Quantum entanglement is a non-local quantum correlation providing
a lot of useful applications in the field of quantum
communications and computations such as quantum teleportation and
quantum key distribution~\cite{tele, qkd1, qkd2}. This important
role of quantum entanglement has stimulated intensive study in
both way of its quantification and qualification.

One of the essential differences of quantum correlations (especially,
quantum entanglement) from other classical ones is that it cannot be
freely shared among the parties in multipartite quantum systems. In
particular, a pair of components that are maximally entangled cannot
share entanglement~\cite{CKW,ov} nor classical correlations~\cite{KW}
with any part of the rest of the system, hence the term {\em Monogamy
  of Entanglement}~(MoE)~\cite{T04}. Monogamy of entanglement was
shown to have a complete mathematical characterization for multi-qubit
systems~\cite{ov} using a certain entanglement measures, the
concurrence~\cite{ww}.

Whereas MoE shows the restricted sharability of multi-party quantum
entanglement, the distribution of entanglement, or {\em Entanglement
  of Assistance}~(EoA)~\cite{d, cohen} in multipartite quantum systems
was shown to have a dually monogamous (or {\em Polygamous})
property. Using {\em Concurrence of Assistance}~(CoA)~\cite{lve} as
the measure of distributed entanglement, it was also shown that
whereas monogamy of entanglement inequalities provide an upper bound
for bipartite sharability of entanglement in a multipartite system,
the same quantity provides a lower bound for distribution of bipartite
entanglement in a multipartite system~\cite{gbs}. In this paper, by
introducing the concept of {\em One-way Unlocalizable
  Entanglement}~(UE), we provide a polygamy inequality of entanglement
in tripartite quantum systems of arbitrary dimension using entropic
entanglement measure. Based on the functional relation between
concurrence and entropic measure in two-qubit systems, we provide a
polygamy inequality in multi-qubit systems. We also provide several
trade offs between UE and other correlations such as EoA, and
localizable entanglement.

The paper is organized as follows. In Sec.~\ref{sec: qf}, we provide
the definition of UE, and its basic properties. In Sec.~\ref{sec:
  3dual}, we provide a polygamy inequality of distributed entanglement
in tripartite quantum systems in terms of entropy and EoA. In
Sec.~\ref{sec: ndual}, we generalize the polygamy inequality of
entanglement into multi-qubit systems, and provide a more tight
polygamy inequality for three-qubit systems. In Sec.~\ref{sec: other},
we provide several trade offs between UE and other correlations, and
we summarize our results, in Sec.~\ref{sec: Conclusion}.

\section{One-Way Unlocalizable Entanglement}
\label{sec: qf}
\subsection{Definition}
\label{subsec: def}
For any bipartite quantum state $\rho_{AB}$,
its one-way distillable common randomness~\cite{DV}
is defined as
\begin{equation}
  C_D^\leftarrow(\rho_{AB})=\lim_{n\rightarrow \infty}
                              \frac{1}{n}I^\leftarrow(\rho_{AB}^{\otimes n}),
\end{equation}
where, the function $I^\leftarrow(\rho_{AB})
$~\cite{Henderson-Vedral01} is
\begin{equation}\begin{split}
  I^\leftarrow(\rho_{AB}) &=  \max_{\{M_x\}} \left[S(\rho_A)-\sum_x p_x S(\rho^x_A)\right],

\end{split}
\label{HVI}
\end{equation}
and where the maximum is taken over all the measurements $\{M_x\}$
applied on system $B$.  Here, $S(\rho_A)$ is the von Neumann entropy
of $\rho_A\equiv \mbox{$\mathrm{tr}$}_B(\rho_{AB})$, $p_x\equiv \mbox{$\mathrm{tr}$}[(I_A\otimes
M_x)\rho_{AB}]$ is the probability of the outcome $x$, and
$\rho^x_A\equiv \mbox{$\mathrm{tr}$}_B[(I_A\otimes {M_x})\rho_{AB}]/p_x$ is the state of
system $A$ when the outcome was $x$.

For a tripartite pure state $\ket{\psi}_{ABC}$ with
$\rho_{A}=\mbox{$\mathrm{tr}$}_{BC}\ket{\psi}_{ABC}\bra{\psi}$, $\rho_{AB}
=\mbox{$\mathrm{tr}$}_{C}\ket{\psi}_{ABC}\bra{\psi}$, and $\rho_{AC}
=\mbox{$\mathrm{tr}$}_{B}\ket{\psi}_{ABC}\bra{\psi}$, it was shown that~\cite{KW}
\begin{align}
    S(\rho_A)&=I^\leftarrow(\rho_{AB})+E_f(\rho_{AC})

    \label{KWmain1}
\end{align}
Here, $E_f(\rho_{AC})$ is the {\em Entanglement of
Formation}~(EoF) of $\rho_{AC}$ defined as~\cite{bdsw}
\begin{equation}
E_f(\rho_{AC})=\min \sum_{i}p_i S(\rho^{i}_{A}),
\label{eof}
\end{equation}
where the minimization is taken over all pure state decomposition of
$\rho_{AC}$ such that,
\begin{equation}
\rho_{AC}=\sum_{i} p_i |\phi^i\rangle_{AC}\langle\phi^i|,
\label{decomp}
\end{equation}
with $\mbox{$\mathrm{tr}$}_{C}|\phi^i\rangle_{AC}\langle\phi^i|=\rho^{i}_{A}$.

As a dual quantity to EoF, EoA is defined by the maximum average
entanglement of $\rho_{AC}$,
\begin{equation}
E_a(\rho_{AC})=\max \sum_{i}p_i S(\rho^{i}_{A}),
\label{EoA}
\end{equation}
over all possible pure state decompositions of $\rho_{AC}$.

\begin{definition} The \emph{one-way unlocalizable entanglement}~(UE)
  of a bipartite state $\rho_{AB}$ is defined as follows:
\begin{equation}
  E_u^{\leftarrow}(\rho_{AB}) := S(\rho_A)-E_a(\rho_{AC}),
  \label{puremain1}
\end{equation}
where $\rho_{AC}$ denotes the reduced state of a purification
$\ket{\psi}_{ABC}$ of $\rho_{AB}$.
\end{definition}

The one-way unlocalizable entanglement can be equivalently
characterized as follows:

\begin{Lem}
  For any given bipartite state $\rho_{AB}$, its one-way unlocalizable
  entanglement is given by
\begin{equation}
  E_u^{\leftarrow}(\rho_{AB})=\min_{\{M_x\}}\left[S(\rho_A)-\sum_xp_xS(\rho_A^x)\right],
\label{fragility}
\end{equation}
where the minimum is taken over all possible rank-1 measurements
$\{M_x\}$ applied on subsystem $B$.
\label{Lem: puremain1}
\end{Lem}
\begin{proof}
Eq.~(\ref{fragility}) can be rewritten as
\begin{equation}
E_u^{\leftarrow}(\rho_{AB}) =  S(\rho_A)- \max_{\{M_x\}}\sum_x p_x S(\rho^x_A),
\label{unlocal2}
\end{equation}
where the maximum is taken over all possible rank-1 measurements
$\{M_x\}$ applied on system $B$.

Since $\ket{\psi}_{ABC}$ is a pure state, all possible pure state
decompositions of $\rho_{AC}$ can be realized by rank-1 measurements
of subsystem $B$, and conversely, any rank-1 measurement can be
induced from a pure state decomposition of $\rho_{AC}$. Thus, the
second term on the right hand side of Eq.~(\ref{unlocal2}) is the
maximum average entanglement over all possible pure state
decomposition of $\rho_{AC}$, which is the definition of
$E_a(\rho_{AC})$, and this completes the proof.
\end{proof}

By definition, the UE of $\rho_{AB}$ is the difference between
$S(\rho_A)$ and $E_a(\rho_{AC})$. Here, $S(\rho_A)$ quantifies the
entanglement of the pure state $\ket{\psi}_{A(BC)}$ with respect to
the bipartite cut $A$--$BC$, whereas $E_a(\rho_{AC})$ measures the
maximum average entanglement that can be localized on the subsystem
$AC$ with the assistance of $B$. The terminology used is then
clear. Figure~\ref{relation} graphically illustrates this separation.

\begin{figure}

 \includegraphics[width=6.5cm]{figure1.eps}\\
  \caption{The entanglement of $\ket{\psi}_{A(BC)}$ ((a) in the figure)
   consists of two different parts: One is the localizable entanglement onto $AC$
   represented by $E_a(\rho_{AC})$ ((b) in the figure), and the other part
   is the unlocalizable entanglement represented by $E_u^{\leftarrow}(\rho_{AB})$ ((c) in the figure). }\label{relation}
\end{figure}

\subsection{Properties}
\label{subsec: prop}

\subsubsection{Subadditivity}
\label{subsub: subadditivity}
\begin{Lem}
For all bipartite states $\rho_{AB}$ and $\sigma_{A'B'}$,
\begin{equation}
E_u^{\leftarrow}(\rho_{AB}\otimes
\sigma_{A'B'})\leq E_u^{\leftarrow}(\rho_{AB})+E_u^{\leftarrow}(\sigma_{A'B'}),
\label{subadd}
\end{equation}
where
\begin{equation}
\begin{split}
E_u^{\leftarrow}(\rho_{AB}\otimes \sigma_{A'B'})= \min_{\{L_z\}}
 \left[S(\rho_A \otimes \sigma_{A'})-\sum_z r_z S(\tau^z_{AA'})\right],\\
\end{split}
\end{equation}
with $r_z=\mbox{$\mathrm{tr}$}[(I_{AA'}\otimes L_{z})\rho_{AB}\otimes
\sigma_{A'B'}]$, $\tau^z_{AA'}=\mbox{$\mathrm{tr}$}_{BB'}[(I_{AA'}\otimes
L_{z})\rho_{AB}\otimes \sigma_{A'B'}]/r_{z}$, and the minimum is
taken over all possible rank-1 measurements $\{L_z\}$ applied on
subsystem $BB'$. \label{lem: subadd}
\end{Lem}

\begin{proof}
Let $\{M_x \}$ and $\{N_y \}$ be the optimal rank-1 measurements
on subsystems $B$ and $B'$ for $E_u^{\leftarrow}(\rho_{AB})$ and
$E_u^{\leftarrow}(\sigma_{A'B'})$ respectively, then, we have

\begin{align}
E_u^{\leftarrow}&(\rho_{AB})+E_u^{\leftarrow}(\sigma_{A'B'})\nonumber\\
&=S(\rho_A)+S(\sigma_{A'})-\sum_{x}p_x S(\rho^{x}_{A})-\sum_{y}q_y S(\sigma^{y}_{A'})\nonumber\\
&=S(\rho_{A}\otimes \sigma_{A'})-\sum_{xy}p_x q_y S(\rho^x_{A}\otimes \sigma^y_{A'})\nonumber\\
&\geq E_u^{\leftarrow}(\rho_{AB}\otimes \sigma_{A'B'}),
\end{align}
where $p_x\rho^x_A=\mbox{$\mathrm{tr}$}_{B}[(I_A\otimes M_{x})\rho_{AB}]$,
$q_y\sigma^y_{A'}=\mbox{$\mathrm{tr}$}_{B'}[(I_{A'}\otimes N_{y})\sigma_{A'B'}]$,
and the second equality is due to the additivity of von Neumann
entropy and the definition of
$E_u^{\leftarrow}(\rho_{AB}\otimes \sigma_{A'B'})$.
\end{proof}

By Lemma~\ref{lem: subadd}, we can assure the existence of the
{\em regularized} UE
\begin{equation}
E_{u,\infty}^\leftarrow(\rho_{AB}):=\lim_{n\rightarrow \infty}\frac{E_u^{\leftarrow}(\rho_{AB}^{\otimes n})}{n},
\label{regular}
\end{equation}
which satisfies
\begin{equation}
E_{u,\infty}^\leftarrow(\rho_{AB})\leq E_u^{\leftarrow}(\rho_{AB}).
\label{regular2}
\end{equation}

\subsubsection{Simple Lower Bound}
\label{subsub: bounds}
\begin{Lem}
For any bipartite state $\rho_{AB}$,
\begin{equation}
E_u^{\leftarrow}(\rho_{AB})\geq \max\{I_{c}^{\leftarrow}(\rho_{AB}),~ 0 \},
\label{lower}
\end{equation}
where $I_{c}^{\leftarrow}(\rho_{AB}):=S(\rho_A)-S(\rho_{AB})$ is
the coherent information of $\rho_{AB}$. \label{lem: lower}
\end{Lem}
\begin{proof}
Let $\ket{\psi}_{ABC}$ be a purification of $\rho_{AB}$, then due
to the monotonicity of entanglement, we have
\begin{equation}
E_a(\rho_{AC})\leq \min\{S(\rho_A),~S(\rho_C)\},
\label{EoAupper}
\end{equation}
where $\rho_{AC}=\mbox{$\mathrm{tr}$}_{B}\ket{\psi}_{ABC}\bra{\psi}$.

Thus, together with Lemma~\ref{Lem: puremain1}, we have
\begin{align}
E_u^{\leftarrow}(\rho_{AB})&=S(\rho_A)-E_a(\rho_{AC})\nonumber\\
&\geq \max\{S(\rho_A)-S(\rho_C),~0\}\nonumber\\
&=\max\{I_{c}^{\leftarrow}(\rho_{AB}),~ 0 \},
\end{align}
where the last equality is due to the purity of $\ket{\psi}_{ABC}$, that is,
$S(\rho_C)=S(\rho_{AB})$.
\end{proof}

Since $\ket{\psi}_{ABC}^{\otimes n}$ is a purification of both
$\rho_{AB}^{\otimes n}$ and $\rho_{BC}^{\otimes n}$, we have
\begin{equation}
  E_u^{\leftarrow}\left(\rho_{AB}^{\otimes n}\right)+E_a\left(\rho_{AC}^{\otimes n}\right)=nS(\rho_A).
\end{equation}
By taking the limit $n\to\infty$, and due to the
relation~\cite{Smolin-Ver-Win}
\begin{equation}
\lim_{n\to\infty}\frac{E_a\left(\rho_{AC}^{\otimes n}\right)}{n}=\min\{S(\rho_A),S(\rho_C)\},
\end{equation}
we have that
\begin{equation}
{E_{u,\infty}^{\leftarrow}}(\rho_{AB})=\max\{I_c^{\leftarrow}(\rho_{AB}),0\}.
\label{asymp}
\end{equation}

Eq.~(\ref{asymp}) implies that, in the asymptotic limit of many
copies, separable states do not exhibit quantumness in their
correlations, or their correlations are completely erasable. This
is a strong evidence that the distinction between separable and
entangled states is {\em operational} only in asymptotic sense,
since separable states can exhibit non-zero UE in finite
case.

\section{Polygamy of entanglement in tripartite quantum systems}
\label{sec: 3dual}
For any bipartite pure state $\ket \phi_{AB}$, its concurrence, $\mathcal{C}(\ket
\phi_{AB})$ is defined as~\cite{ww}
\begin{equation}
\mathcal{C}(\ket \phi_{AB})=\sqrt{2(1-\mbox{$\mathrm{tr}$}\rho^2_A)},
\label{pure state concurrence}
\end{equation}
where $\rho_A=\mbox{$\mathrm{tr}$}_B(\ket \phi_{AB}\bra \phi)$.
For any mixed state $\rho_{AB}$, its concurrence is defined via {\em convex-roof extension}, that is,
\begin{equation}
\mathcal{C}(\rho_{AB})=\min \sum_k p_k \mathcal{C}({\ket {\phi_k}}_{AB}),
\label{mixed state concurrence}
\end{equation}
where the minimum is taken over all possible pure state
decompositions, $\rho_{AB}=\sum_kp_k{\ket {\phi_k}}_{AB}\bra
\phi_k$.

As a dual value to concurrence, CoA~\cite{lve} of $\rho_{AB}$ is defined as
\begin{equation}
\mathcal{C}^a(\rho_{AB})=\max \sum_k p_k \mathcal{C}({\ket {\phi_k}}_{AB}),
\label{CoA}
\end{equation}
where the maximum is taken over all possible pure state decompositions of
$\rho_{AB}$.

By using concurrence and CoA as the quantification of bipartite entanglement, it was
shown that there exists a polygamy relation of
entanglement in multi-qubit systems~\cite{gbs}. More precisely,
for any pure state $\ket{\psi}_{A_1 \cdots A_n}$ in an $n$-qubit
system $\mathcal H_{A_1} \otimes \cdots \otimes \mathcal H_{A_n}$
where $\mathcal H_{A_i} \cong
\mathbb{C}^2$ for $i=1,\ldots,n$,
\begin{equation} \mathcal{C}_{A_1 (A_2 \cdots
A_n)}^2  \leq  (\mathcal{C}^a_{A_1 A_2})^2
+\cdots+(\mathcal{C}^a_{A_1 A_n})^2, \label{ndual}
\end{equation}
where $\mathcal{C}_{A_1 (A_2 \cdots A_n)}$ is the concurrence of
$\ket{\psi}_{A_1 \cdots A_n}$ with respect to the bipartite cut $A_1$
and $A_{2}\cdots A_{n}$, and $\mathcal{C}^a_{A_1 A_i}$ is the CoA of
$\rho_{A_1A_i}=\mbox{$\mathrm{tr}$}_{A_2\cdots A_{i-1}A_{i+1}\cdots A_n}\left(\ket{\psi}_{A_1 \cdots A_n}\bra{\psi}\right)$ for $i=2,
\ldots ,n$.

In this section, we provide an analytic upper bound of UE
in Eq.~(\ref{fragility}), and derive a polygamy inequality of
entanglement in terms of von-Neumann entropy and EoA
for tripartite quantum systems of arbitrary dimension.

First, for an upper bound of UE, we have the following
theorem.

\begin{Thm}
For any bipartite state $\rho_{AB}$ in a bipartite quantum system
$\mathcal{H}_A \otimes \mathcal{H}_B$,
\begin{equation}
E_u^{\leftarrow}(\rho_{AB})\leq\frac{I(\rho_{AB})}{2}, \label{upper}
\end{equation}
where $I(\rho_{AB})=S(\rho_A)+S(\rho_B)-S(\rho_{AB})$ is the
mutual information of $\rho_{AB}$. \label{Thm: upper}
\end{Thm}

\begin{proof}
  Let $\rho_B=\sum_{i=0}^{d_B-1}\lambda_{i}\ket{e_i}_B\bra{e_i}$ be a
  spectral decomposition of $\rho_B=\mbox{$\mathrm{tr}$}_{A}(\rho_{AB})$ where $d_B$ is
  the dimension of the subsystem $\mathcal{H}_B$. The proof method
  follows the construction used in~\cite{christandl}.

For any state $\sigma \in \mathcal{H}_B$, define the channels
\begin{align}
M_{0}(\sigma):&=\sum_{i=0}^{d_B-1}\ket{e_i}\bra{e_i}\sigma\ket{e_i}\bra{e_i}\nonumber\\
M_{1}(\sigma):&=\sum_{i=0}^{d_B-1} |\tilde e_j\rangle\langle\tilde
e_j|\sigma|\tilde e_j\rangle\langle\tilde e_j|, \label{channels1}
\end{align}
where $\{ |\tilde e_j \rangle \}_{j} $ is the {\em Fourier basis}
such that,
\begin{equation}
|\tilde e_j \rangle = \frac{1}{\sqrt{d}}\sum_{k=0}^{d_B-1}
\omega_d^{jk}\ket{e_k},~j=0,\ldots ,d_B-1, \label{fourier}
\end{equation}
and $\omega_d = e^{\frac{2\pi i}{d}}$ is the $d$-th root of unity.

Notice that $M_0(\rho_B)=\rho_B$, and
$M_1(\rho_B)=\frac{1}{d_B}I_B$, so that
$M_1(M_0(\rho_B))=M_0(M_1(\rho_B))=\frac{1}{d_B}I_B$. We can also
write
\begin{equation}
M_0(\sigma)=\frac{1}{d_B}\sum_{b=0}^{d_B-1}Z^b\sigma Z^{-b},~
M_{1}(\sigma)=\frac{1}{d_B}\sum_{a=0}^{d_B-1}X^a\sigma X^{-a},
\label{channels2}
\end{equation}
where $Z$ and $X$ are generalized $d_B$-dimensional Pauli operators,
\begin{align}
Z=&\sum_{j=0}^{d_B-1}\omega_d^j\ket{e_j}\bra{e_j},\nonumber\\
X=&\sum_{j=0}^{d_B-1}\ket{e_{j+1}}\bra{e_j}=\sum_{j=0}^{d_B-1} \omega_d^{-j}|\tilde
e_j \rangle \langle \tilde e_j |.\label{paulis}
\end{align}

In the following, we will write
\begin{align}
(I_A\otimes M_0)(\rho_{AB})&=\sum_{i=0}^{d_B-1} \sigma_A^i \otimes \lambda_i\ket{e_i}_B\bra{e_i},\nonumber\\
(I_A\otimes M_1)(\rho_{AB})&=\sum_{j=0}^{d_B-1} \tau_A^j \otimes
\frac{1}{d_B}|\tilde e_j \rangle_B \langle \tilde e_j|,
\end{align}
where $\lambda_i\sigma_A^i=\mbox{$\mathrm{tr}$}_B [(I_A \otimes
\ket{e_i}_B\bra{e_i})\rho_{AB}]$, and $\tau_A^j/d_B=\mbox{$\mathrm{tr}$}_B[(I_A
\otimes |\tilde e_j\rangle_B \langle\tilde e_j|)\rho_{AB}]$ for $i,~j \in \{0,\cdots, d_B-1\}$.

The induced ensembles on $A$ by the channels $M_0$ and $M_1$ will be denoted by $\mathcal
E_0:=\{\lambda_i,\sigma_A^i\}_i$ and $\mathcal E_1:=\{\frac
{1}{d_B},\tau_A^j\}_j$, and the entropy defects of the induced ensembles
on $A$ will be denoted as
\begin{align}
\chi(\mathcal E_0)=&S(\rho_A)-\sum_{i=0}^{d_B-1}\lambda_i S(\sigma_A^i),\nonumber\\
\chi(\mathcal E_1)=&S(\rho_A)-\frac{1}{d_B}\sum_{i=0}^{d_B-1}S(\tau_A^j).
\label{chi}
\end{align}

By defining a four-partite quantum state $\Omega_{XYAB}$ in
$\mathcal B \left( \mbox{$\mathbb C$}^{d_B} \otimes \mbox{$\mathbb C$}^{d_B} \otimes \mbox{$\mathbb C$}^{d_A} \otimes \mbox{$\mathbb C$}^{d_B}\right)$
such that
\begin{widetext}
\begin{equation}
  \Omega_{XYAB}:=\frac 1{d^2_B}\sum_{x,y=0}^{d_B-1}\ket{x}_X
  \bra{x}\otimes\ket{y}_Y\bra{y}\otimes(I_A\otimes X^x_BZ^y_B)\rho_{AB}(I_A\otimes
  Z^{-y}_BX^{-x}_B),
\label{XYAB}
\end{equation}
we have
\begin{align}
\Omega_{XAB}=&\frac
1{d_B}\sum_{x=0}^{d_B-1}\ket{x}_X\bra{x}\otimes X^x_B
\left(\sum_{i=0}^{d_B-1} \sigma_A^i \otimes \lambda_i\ket{e_i}_B\bra{e_i}\right)X_B^{-x},\nonumber\\
\Omega_{YAB}=&\frac
1{d_B}\sum_{y=0}^{d_B-1}\ket{y}_Y\bra{y}\otimes
Z_B^y\left(\sum_{j=0}^{d_B-1} \tau_A^j \otimes
\frac{1}{d_B}|\tilde e_j \rangle_B \langle \tilde
e_j|\right)Z_B^{-y}, \label{XAB, YAB}
\end{align}
\end{widetext}
and
\begin{equation}
  \Omega_{AB}=\rho_A\otimes\frac{I_B}{d_B}.
\end{equation}
By straightforward calculation, we can obtain
\begin{align}
I(\Omega_{X(AB)})=&S(\Omega_X)+S(\Omega_{AB})-S(\Omega_{XAB})\nonumber\\
=&\log d_B+\log d_B+S(\rho_A)-\log d_B\nonumber\\
    &~~~~~~~~~~~~- S\left(\sum_i \sigma_A^i \otimes \lambda_i\ket{e_i}_B\bra{e_i}\right)\nonumber\\
=&\log d_B+S(\rho_A)-H(\vec\lambda)-\sum_i\lambda_iS(\sigma_A^i)\nonumber\\
=&\log d_B-S(\rho_B)+\chi(\mathcal E_0), \label{Ixab}
\end{align}
where $I(\Omega_{X(AB)})$ is the mutual information of $\Omega_{X(AB)}$ with respect to the bipartite cut
$X-AB$, and the second, third equalities are due to the {\em joint
entropy theorem}~\cite{nc}. Analogously, we have
\begin{align}
&I(\Omega_{Y(AB)})=\chi(\mathcal E_1),\nonumber\\
&I(\Omega_{(XY)(AB)})=\log d_B+S(\rho_A)-S(\rho_{AB})\nonumber\\
&~~~~~~~~~~~~~~~~~~=\log d_B+I_c^{\leftarrow}(\rho_{AB}). \label{Iyab Ixyab}
\end{align}

Due to the independence of subsystems $X$ and $Y$, we have
$I(\Omega_{(XY)(AB)})\geq I(\Omega_{X(AB)})+I(\Omega_{Y(AB)})$,
which implies
\begin{equation}
\chi(\mathcal E_0)+\chi(\mathcal E_1)\leq I(\rho_{AB}).
\label{ensembleineq}
\end{equation}

Since $\chi(\mathcal E_0)$ and $\chi(\mathcal E_1)$ of
Eq.~(\ref{ensembleineq}) can be obtained, respectively, from
$\rho_{AB}$ by rank-1 measurements $\{\ket{e_i}_B\bra{e_i} \}_i$
and $\{ |\tilde e_j \rangle_B \langle \tilde e_j| \}_j$ of
subsystem $B$, by defining a rank-1 measurement
\begin{equation}
  \left\{ \frac{\ket{e_i}_B\bra{e_i}}{2}, \frac{|\tilde e_j
    \rangle_B \langle \tilde e_j|}{2}\right\}_{i,j}, \label{povm}
\end{equation}
we have
\begin{equation}
E_u^{\leftarrow}(\rho_{AB})\leq  \frac{\chi(\mathcal
E_0)}{2}+\frac{\chi(\mathcal E_1)}{2}\leq \frac{I(\rho_{AB})}{2},
\label{uppermain}
\end{equation}
which completes the proof.
\end{proof}

\begin{Cor}
For any tripartite pure state $\ket{\psi}_{ABC}$, we have
\begin{equation}
S(\rho_A)\leq E_a(\rho_{AB})+E_a(\rho_{AC}). \label{poly}
\end{equation}
\label{Cor: dual}
\end{Cor}
\begin{proof}
By Lemma~\ref{puremain1}, we have
\begin{align}
E_a(\rho_{AC})&=S(\rho_A)- E_u^{\leftarrow}(\rho_{AB}) ,\nonumber\\
E_a(\rho_{AB})&=S(\rho_A)- E_u^{\leftarrow}(\rho_{AC}),
\end{align}
and thus,
\begin{equation}
E_a(\rho_{AC})+E_a(\rho_{AB})=2S(\rho_A)-
E_u^{\leftarrow}(\rho_{AB}) - E_u^{\leftarrow}(\rho_{AC}).
\end{equation}
Now, by Theorem~\ref{Thm: upper}, we have
\begin{align}
E_a(\rho_{AC})+& E_a(\rho_{AB})\nonumber\\
 \geq & 2S(\rho_A)-\frac{I(\rho_{AB})}{2}-\frac{I(\rho_{AC})}{2}\nonumber\\
=&2S(\rho_A)-S(\rho_A)/2-S(\rho_B)/2+S(\rho_{AB})/2 \nonumber\\
 & -S(\rho_A)/2-S(\rho_C)/2+S(\rho_{AC})/2\nonumber\\
=&S(\rho_A).
\end{align}
\end{proof}

Corollary~\ref{Cor: dual} tells us that for a tripartite pure
state of arbitrary dimension, there exists a polygamy relation of
entanglement in terms of entropy of entanglement and EoA.
Furthermore, this is, we believe, the first result of the
polygamous (or dually monogamous) property of distribution of entanglement in
multipartite higher-dimensional quantum systems rather than qubits.

\section{Polygamy relation of entanglement in multi-qubit quantum systems}
\label{sec: ndual}

In this section, we show that the polygamy inequality of
entanglement in Corollary~\ref{Cor: dual} can be generalized into
multipartite quantum systems for the case when each subsystem is a
two-level quantum system. By investigating the functional relation
between concurrence and EoF in two-qubit systems~\cite{ww}, we
show that there exists a polygamy inequality of entanglement in
terms of entropy and EoA in $n$-qubit systems. We also show that,
in three-qubit systems, we have a more tight polygamy inequality
than Eq.~(\ref{poly}) in Corollary~\ref{Cor: dual}.

First, let us consider the functional relation of concurrence with
EoF in two-qubit systems.  For a 2-qubit mixed state $\rho_{AB}$
(or a pure state $\ket{\psi}_{AB} \in \mathbb{C}^2 \otimes
\mathbb{C}^{d}$), the relation between its concurrence,
$\mathcal{C}_{AB}$ and $E_{f}(\rho_{AB})$ can be given as a
monotone increasing, convex function $\mathcal{E}$~\cite{ww}, such
that
\begin{equation}
 E_f (\rho_{AB}) = {\mathcal E}(\mathcal{C}_{AB}),
\end{equation}
where
\begin{equation} {\mathcal E}(x) = H\Bigl({1\over 2} + {1\over
2}\sqrt{1-x^2}\Bigr), \hspace{0.5cm}\mbox{for } 0 \le x \le 1,
\label{eps}
\end{equation}
and $H(\cdot)$ is the binary entropy function $H(x) = -[x\log_2 x +
(1-x)\log_2 (1-x)]$. The same function $\mathcal{E}(x)$ relates also
the EoA of a bipartite state $\rho_{AB}$ with its CoA
via the equation
\begin{equation}
  E_a(\rho_{AB})\ge\mathcal{E}(\mathcal{C}_{AB}^a),
\label{eoacoa}
\end{equation}
which is due to the convexity of ${\mathcal E}$
and the definition of EoA.
The following lemma shows an important property of the function
$\mathcal{E}(x)$.
\begin{Lem}
\begin{equation}
{\mathcal E}(\sqrt{x^2 + y^2}) \leq {\mathcal E}(x) + {\mathcal
E}(y), \label{Epro}
\end{equation}
for $0 \leq x,~y\leq 1$ such that $0\leq x^2+y^2\leq 1$.
\label{Lem: E}
\end{Lem}
\begin{proof}
By considering \begin{equation} f(x,y)= {\mathcal E}(x) +
{\mathcal E}(y)-{\mathcal E}(\sqrt{x^2 + y^2}), \label{f}
\end{equation} as a two-vairable real-valued function on the
domain $D=\{(x,y)| 0 \leq x,~y \leq 1, 0 \leq \ x^2 +  y^2 \leq 1
\}$, it is enough to show that $f(x,y)\geq 0$ in $D$.

Since $D$ is a compact subset in $\mbox{$\mathbb R$}^2$, whereas $f$ is analytic
on the interior of $D$, and continuous on $D$, the minimum value
of $f$ arises only on the critical points or on the boundary of
$D$. It can be directly checked that $f$ does not have any
vanishing gradient on the interior of $D$, and has non-negative
function values on the boundary of $D$. Thus, $f$ is non-negative
on the domain $D$.
\end{proof}

\subsection{Three-qubit systems}
\label{subsec: 3-qubit}

A direct observation from~\cite{CKW} shows that, for a 3-qubit
pure state $\ket{\psi}_{ABC}$,
\begin{equation}
 \mathcal{C}_{A(BC)}^2 = \mathcal{C}_{AB}^2 + ({\mathcal{C}^{a}_{AC}})^2,

\label{3tangle}
\end{equation}
where $\mathcal{C}_{AB}$ and $\mathcal{C}^{a}_{AC}$ are the
concurrence and concurrence of assistance of $\rho_{AB}$ and
$\rho_{AC}$ respectively. (Later, Eq.~(\ref{3tangle}) was formally
shown in~\cite{ys}.) From Eq.~(\ref{3tangle}) together with
Lemma~\ref{Lem: E}, we have the following theorem.
\begin{Thm}
For a three-qubit pure state $\ket{\psi}_{ABC}$,
\begin{equation}
S(\rho_{A}) \leq  E_f(\rho_{AB}) + E_a(\rho_{AC}).
\label{3qubitineq}
\end{equation}
\label{thm2}
\end{Thm}

\begin{proof}
Since $\ket{\psi}_{ABC}$ is a bipartite pure state in
$\mathbb{C}^2 \otimes \mathbb{C}^{4}$ with respect to the
bipartite cut $A$ and $BC$, we have,
\begin{equation}
S(\rho_{A})= E_f(\psi_{A(BC)})={\mathcal E}(\mathcal{C}_{A(BC)}).
\end{equation}
Thus,
\begin{eqnarray}
S(\rho_{A})
            &=& {\mathcal E}(\mathcal{C}_{A(BC)})\nonumber\\
            &=& {\mathcal E}( \sqrt{\mathcal{C}^{2}_{AB}+ {\mathcal{C}^{a}_{AC}}^2}~)\nonumber\\
            &\leq & {\mathcal E}(\mathcal{C}_{AB}) +  {\mathcal E}(\mathcal{C}^{a}_{AC})\nonumber\\
            &\leq & E_f(\rho_{AB}) + E_a(\rho_{AC}),
\end{eqnarray}
where the first inequality is by Lemma~\ref{Lem: E}, and the
second inequality is by Eq.~(\ref{eoacoa}).
\end{proof}
Thus, the polygamy relation of distributed entanglement in tripartite quantum
systems obtained in Corollary~\ref{Cor: dual} can have a more
tight form in three-qubit systems. Furthermore, the result of
Theorem~\ref{thm2} together with Eqs.~(\ref{KWmain1}) and
(\ref{puremain1}) give us the following corollary.
\begin{Cor}
For any two-qubit mixed state $\rho_{AB}$ with rank less than or equal to two,
\begin{equation}
 I^\leftarrow(\rho_{AB})\leq E_a(\rho_{AB}),
\label{ubound1}
\end{equation}
\begin{equation}
 E_u^{\leftarrow}(\rho_{AB})\leq E_f(\rho_{AB}).
\label{ubound2}
\end{equation}
\label{Cor: 1}
\end{Cor}

\begin{Rmk}\label{Rmk: 1}
Eq.~(\ref{ubound2}) of Corollary~\ref{Cor: 1} implies that any
two-qubit separable state $\rho_{AB}$ of rank less than or equal
to two has zero UE, $E_u^{\leftarrow}(\rho_{AB})=0$. However,
this is not generally true for two-qubit separable states of rank
larger than two. Here, we provide an example of two-qubit
rank-three separable state with non-zero UE.

{\it Example:} Let us consider the following state in $\mbox{$\mathbb C$}^2\otimes
\mbox{$\mathbb C$}^2\otimes \mbox{$\mathbb C$}^3$ quantum system~\cite{CCJKKL},
\begin{eqnarray}
\ket{\Psi}_{ABC}= \frac{1}{\sqrt{2}}\ket{x}_{AC}\ket{0}_B
+\frac{1}{\sqrt{2}}\ket{y}_{AC}\ket{1}_B, \label{eq:ex_Psi}
\end{eqnarray}
where $\ket{x}$ and $\ket{y}$ are two orthogonal states in the
$\mbox{$\mathbb C$}^2\otimes \mbox{$\mathbb C$}^3$ such that
\begin{align}
\ket{x}=&(\ket{02}+\sqrt{2}\ket{10})/{\sqrt{3}},\nonumber\\
\ket{y}=&(\ket{12}+\sqrt{2}\ket{01})/{\sqrt{3}}.
\end{align}

First, since $\rho_{A}=(\ket{0}_{A}\bra{0}+\ket{1}_{A}\bra{1})/2$,
it is clear that $\mathcal{C}_{A(BC)}=\sqrt{2(1-\mbox{$\mathrm{tr}$}\rho^2_A)}=1$,
therefore we have $S(\rho_A)=1$.

Since $\rho_{AC}=(\ket{x}_{AC}\bra{x}+\ket{y}_{AC}\bra{y})/2$,
Hughston-Jozsa-Wootters (HJW) theorem~\cite{HJW} says that for any
decompositions of
$\rho_{AC}=\sum_{i}p_{i}\ket{\phi_{i}}_{AC}\bra{\phi_{i}}$, there
exists an unitary operator $(u_{ij})$ such that
$\sqrt{p_{i}}\ket{\phi_{i}}_{AC}=(u_{i1}\ket{x}_{AC}+u_{i2}\ket{y}_{AC})/\sqrt{2}$
with $2p_{i}=|u_{i1}|^{2}+|u_{i2}|^{2}$. Thus,
\begin{align}
\mbox{$\mathrm{tr}$}_{C}(\ket{\phi_{i}}_{AC}\bra{\phi_{i}})=&\frac{1}{6p_i}
\begin{pmatrix}
  |u_{i1}|^{2}+2|u_{i2}|^{2} & u_{i1}u_{i2}^{*} \\
  u_{i2}u_{i1}^{*} & |u_{i2}|^{2}+2|u_{i1}|^{2}
\end{pmatrix}
\nonumber\\
=&\frac{1}{3}I_{A}+\frac{1}{3}\ket{\psi_{i}}_{A}\bra{\psi_{i}},
\label{eq:decomp_matrix}
\end{align}
with
$\ket{\psi_{i}}=(u_{i2}^*\ket{0}+u_{i1}^*\ket{1})/{\sqrt{2p_{i}}}$,
and we obtain that
$\mathcal{C}(\ket{\phi_{i}}_{AC})=\frac{2\sqrt{2}}{3}$ for any
pure state $\ket{\phi_{i}}_{AC}$ in any pure state decomposition
of $\rho_{AC}$.

Since $\ket{\phi_{i}}_{AC}$ is a $2\otimes 3$ pure state, we have
\begin{eqnarray}
 E_f(\ket{\phi_{i}}_{AC})&=&{\mathcal E}(\mathcal{C}(\ket{\phi_{i}}_{AC}))\nonumber\\
                         &=& H\Bigl( \frac{2}{3} \Bigr)\nonumber\\
                         &=& \log_2 3 -\frac{2}{3},
\end{eqnarray}
and thus $E_f(\rho_{AC})=E_a(\rho_{AC})=\log_2 3 -\frac{2}{3}$.

Now, we have
$E_u^{\leftarrow}(\rho_{AB})=S(\rho_A)-E_a(\rho_{AC})=\frac{5}{3}-\log_2
3 > 0$, whereas, it can be easily seen that $\rho_{AB}$ has a {\em
Positive Partial Transposition}~(PPT) which is equivalent to
separability for two-qubit states~\cite{horo1}. Thus, $\rho_{AB}$
is a two-qubit, rank-three separable state with non-zero
UE.
\end{Rmk}

\subsection{$n$-qubit systems}
\label{subsec: n-qubit} The polygamy inequality of
entanglement in $n$-qubit systems in Eq.~(\ref{ndual}) gives us an
inequality
\begin{equation}
\mathcal{C}_{A_1 (A_2 \cdots A_n)}  \leq \sqrt{
(\mathcal{C}^a_{A_1 A_2})^2 +\cdots+(\mathcal{C}^a_{A_1 A_n})^2 }.
\label{ndual2}
\end{equation}
Thus, together with Lemma~\ref{Lem: E}, we have the following
theorem.
\begin{Thm}
For any $n$-qubit pure state $\ket{\psi}_{A_1 \cdots A_n}$,
\begin{equation}
S(\rho_{A_1})  \leq  E_a(\rho_{A_1 A_2}) +\cdots+E_a(\rho_{A_1
A_n}). \label{nEdua}
\end{equation}
\label{thm: dual}
\end{Thm}

\begin{proof}
First, let us assume that $ (\mathcal{C}^a_{A_1 A_2})^2
+\cdots+(\mathcal{C}^a_{A_1 A_n})^2 \leq 1$, then we have
\begin{align}
S(\rho_{A_1})=&{\mathcal E}(\mathcal{C}_{A_1 (A_2 \cdots A_n)})\nonumber\\
\leq &{\mathcal E}\left(\sqrt{
(\mathcal{C}^a_{A_1 A_2})^2 +\cdots+(\mathcal{C}^a_{A_1 A_n})^2 }\right)\nonumber\\
\leq &{\mathcal E}\left( \mathcal{C}^a_{A_1 A_2}\right) +
{\mathcal E}\left(\sqrt{(\mathcal{C}^a_{A_1 A_3})^2
+\cdots+(\mathcal{C}^a_{A_1 A_n})^2}\right)\nonumber\\
\leq &{\mathcal E}\left( \mathcal{C}^a_{A_1 A_2}\right)+ {\mathcal
E}\left(\mathcal{C}^a_{A_1 A_3}\right) +\cdots + {\mathcal
E}\left(
\mathcal{C}^a_{A_1 A_n}\right)\nonumber\\
\leq& E_a\left(\rho_{A_1 A_2}\right)
+\cdots+E_a\left(\rho_{A_1 A_n}\right),
\end{align}
where the first inequality is due to the monotonicity of the function
$\mathcal E$, the second and third inequalities are obtained by
iterating Lemma~\ref{Lem: E}, and the last inequality is by Eq.~(\ref{eoacoa}).

Now, assume that $ (\mathcal{C}^a_{A_1 A_2})^2
+\cdots+(\mathcal{C}^a_{A_1 A_n})^2 > 1$. Since $S(\rho_{A_1})\leq
1$ for any $n$-qubit pure state $\ket{\psi}_{A_1 \cdots A_n}$, it
is enough to show that $E_a(\rho_{A_1 A_2}) +\cdots+E_a(\rho_{A_1
A_n}) \geq 1$.

Let us first note that there exist $k \in \{2,\ldots ,n-1 \}$ that
satisfies
\begin{align}
&(\mathcal{C}^a_{A_1 A_2})^2 +\cdots+(\mathcal{C}^a_{A_1 A_k})^2
\leq 1,\nonumber\\
&(\mathcal{C}^a_{A_1 A_2})^2 +\cdots+(\mathcal{C}^a_{A_1
A_{k+1}})^2 >1,
\end{align}
and let
\begin{equation}
T=(\mathcal{C}^a_{A_1 A_2})^2 +\cdots+(\mathcal{C}^a_{A_1
A_{k+1}})^2-1.
\end{equation}

Since, $(\mathcal{C}^a_{A_1 A_{k+1}})^2-T=1-(\mathcal{C}^a_{A_1
A_2})^2 -\cdots-(\mathcal{C}^a_{A_1 A_{k}})^2 $, we have
\begin{equation}
0\leq (\mathcal{C}^a_{A_1 A_{k+1}})^2-T \leq 1,
\end{equation}
and
\begin{align}
1 =& \mathcal E \left( \sqrt{(\mathcal{C}^a_{A_1 A_2})^2
+\cdots+(\mathcal{C}^a_{A_1
A_{k+1}})^2-T} \right)\nonumber\\
\leq& \mathcal E \left( \sqrt{(\mathcal{C}^a_{A_1 A_2})^2
+\cdots+(\mathcal{C}^a_{A_1 A_k})^2} \right)\nonumber\\
&~~~~~~~~~~~~~~~~+\mathcal E \left( \sqrt{\left(\mathcal{C}^a_{A_1 A_{k+1}}\right)^2-T} \right)\nonumber\\
\leq& \mathcal E \left( \mathcal{C}^a_{A_1 A_2}\right)+\cdots
+\mathcal E \left( \mathcal{C}^a_{A_1 A_k}\right)+
\mathcal E \left( \mathcal{C}^a_{A_1 A_{k+1}}\right)\nonumber\\
\leq& E_a(\rho_{A_1 A_2})+\cdots + E_a(\rho_{A_1 A_n}),
\end{align}
where the first and second inequalities are by Lemma~\ref{Lem: E} and
by monotonicity of $\mathcal E$, and the last inequality is by Eq.~(\ref{eoacoa}).
\end{proof}

\section{Unlocalizable Entanglement versus Other Measures of
  Correlation}
\label{sec: other} In this section, we provide some properties of
UE concerned with several other correlation
measures. By investigating the relation between UE
and EoF in $2 \otimes 2 \otimes d$ quantum system, we show that
any two-qubit state with zero UE is a separable state. We
also provide a quantitative relation among entropy, localizable
entanglement, and UE for tripartite mixed states.

\subsection{$2\otimes2\otimes d$ pure state}
\label{subsec: 22d}

Let $\ket{\psi}_{ABC}$ be a tripartite pure state in
$\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^d$.

\begin{Thm}
  For any 2-qubit state $\rho_{AB}$,
\begin{equation}
  E_u^{\leftarrow}(\rho_{AB})=0\Longrightarrow  \rho_{AB}\textrm{ separable},
\end{equation}
or, equivalently, if $S(\rho_A)= E_a(\rho_{AC})$, then
$E_f(\rho_{AB})=0$.
\label{thm: 22d}
\end{Thm}

\begin{proof}
Suppose $S(\rho_A)= E_a(\rho_{AC})$, and let
$\rho_{AC}=\sum_{i}p_i\ket{\phi^i}_{AC}\bra{\phi^i}$ be an optimal
decomposition such that,
\begin{eqnarray}
E_a(\rho_{AC})&=&\sum_{i}p_i E(\ket{\phi^i}_{AC})\nonumber\\
              &=&\sum_{i}p_i S(\rho^{i}_A),
\end{eqnarray}
where $\rho^{i}_{A}=\mbox{$\mathrm{tr}$}_{C}(\ket{\phi^i}_{AC}\bra{\phi^i})$ and $\rho_A = \sum_{i}p_i \rho^i_A$.

The concavity of von Neumann entropy says, $S(\rho_{A}) \geq
\sum_{i}p_i S(\rho^{i}_A)$ and the equality holds if and only if
$\rho^i_A$ are  identical for all $i$.  So, by the assumption,
$\rho^i_A$ are  identical for all $i$.

Since $\ket{\phi^i}_{AC}$ is a pure state in $2\otimes d$ system
and its concurrence is
$\mathcal{C}(\ket{\phi^i}_{AC})=2\sqrt{det\rho^i_A}$, we also have
that $\mathcal{C}(\ket{\phi^i}_{AC})$ are identical for all $i$,
say $\mathcal{C}(\ket{\phi^i}_{AC})=\mathcal{C}^*_{AC}$.

Now, we have
\begin{equation}
S(\rho^{i}_A)=E_f(\ket{\phi^i}_{AC})=\mathcal E(\mathcal{C}(\ket{\phi^i}_{AC}))=\mathcal E(\mathcal{C}^*_{AC}),
\end{equation}
for all $i$, and
\begin{eqnarray}
\mathcal E(\mathcal{C}_{A(BC)})&=& E_f(\ket{\psi_{A(BC)}})\nonumber\\
                              &=& S(\rho_A)\nonumber\\
                              &=& \sum_{i}p_i S(\rho^{i}_A)\nonumber\\
                              &=& \sum_{i}p_i \mathcal E(\mathcal{C}(\ket{\phi^i}_{AC}))\nonumber\\
                              &=& \mathcal E(\mathcal{C}^*_{AC}),
\end{eqnarray}
where $\mathcal{C}_{A(BC)}$ is the concurrence of
$\ket{\psi}_{ABC}$ between subsystems $A$ and $BC$, and $\mathcal
E(\cdot)$ is the function in Eq.~(\ref{eps}).

Since $\mathcal E(\cdot)$ is strictly monotone increasing, (the
first derivative $\frac{d}{d\mathcal C}\mathcal E(\mathcal C)$ is
0 at $\mathcal C =0$ and positive elsewhere), we have
\begin{equation}
\mathcal{C}_{A(BC)} = \mathcal{C}^*_{AC},
\end{equation}
therefore
\begin{equation}
\mathcal{C}_{A(BC)} \geq \mathcal C^a_{AC} \geq
\sum_{i}p_i\mathcal C(\ket{\phi^i}_{AC}) =
\mathcal{C}^*_{AC}=\mathcal{C}_{A(BC)},
\end{equation}
and thus,
\begin{equation}
\mathcal{C}_{A(BC)} = \mathcal C^a_{AC}.
\end{equation}

Now, by the Theorem~3 in~\cite{CCJKKL}, we have $\mathcal
C(\rho_{AB})=0$ where $\rho_{AB}$ is a 2-qubit state, which
implies $E_f(\rho_{AB})=0.$
\end{proof}

Any two-qubit state with zero UE is separable by
Theorem~\ref{thm: 22d}, and any two-qubit separable state with
rank less than or equal to two has zero UE by
Corollary~\ref{Cor: 1}. However, the converse of Theorem~\ref{thm:
22d} is not generally true, since Remark~\ref{Rmk: 1} provides us a
two-qubit separable state with non-zero UE.

\subsection{Tripartite Mixed State}
\label{sec: tri mixed}
Since it is known that the EoA is not a bipartite measure nor an {\em
  entanglement monotone}~\cite{gs}, it is not clear yet if there is
any quantitative relation between $E_a(\rho_{AB})$ and
$E_a(\rho_{A(BD)})$ for a tripartite mixed state $\rho_{ABD}$. In
fact, this is equivalent to the quantitative relation between
$E_u^{\leftarrow}(\rho_{AC})$ and
$E_u^{\leftarrow}(\rho_{A(CD)})$. This is because, if we consider a
purification $\ket{\psi}_{ABCD}$ of $\rho_{ABD}$, then any direction
of a quantitative relation between $E_a(\rho_{AB})$ and
$E_a(\rho_{A(BD)})$, say $E_a(\rho_{AB}) \leq E_a(\rho_{A(BD)})$,
would give us
\begin{align}
S(\rho_A)=&E_u^{\leftarrow}(\rho_{A(CD))})+E_a(\rho_{AB})\nonumber\\
=&E_u^{\leftarrow}(\rho_{AC)})+E_a(\rho_{A(BD)})\nonumber\\
\geq& E_u^{\leftarrow}(\rho_{AC)})+E_a(\rho_{AB}),
\end{align}
which implies $E_u^{\leftarrow}(\rho_{AC})\leq E_u^{\leftarrow}(\rho_{A(CD)})$.

In this section, we pay our attentions only to local rank-1
measurements of each subsystems, and we derive a quantitative
relation between localizable entanglement, and UE for
tripartite mixed states.

For $\rho_{ACD}=\mbox{$\mathrm{tr}$}_B \ket{\psi}_{ABCD}\bra{\psi}$, let us define
\begin{equation}
\begin{split}
\widetilde{E}_u^{\leftarrow}(\rho_{A(CD)}) &=
\min_{\{M_x\otimes N_y\}} \left[S(\rho_A)-\sum_{x, y} p_{xy} S(\rho^{xy}_A)\right],\\
\end{split}
\label{unlocal3}
\end{equation}
where $p_{xy}\equiv \mbox{$\mathrm{tr}$}[(I_A\otimes M_x\otimes N_y )\rho_{ACD}]$
is the probability of the outcome $x$ and $y$ on subsystems $C$ and $D$ respectively,
and $\rho^{xy}_A\equiv \mbox{$\mathrm{tr}$}_{CD}[(I_A\otimes M_x\otimes N_y)\rho_{ACD}]/p_{xy}$ is the state
of system $A$ when the outcome were $x$ and $y$.
The minimum in Eq.~(\ref{unlocal3}) is taken over all possible rank-1 measurements
$\{M_x\}$ and $\{N_y\}$ on subsystems $C$ and $D$ respectively.
By definition, we have
\begin{equation}
\widetilde{E}_u^{\leftarrow}(\rho_{A(CD)}) \geq E_u^{\leftarrow}(\rho_{A(CD)}).
\end{equation}
Furthermore, we have the following lemma.
\begin{Lem}
For any tripartite state $\rho_{ACD}$,
\begin{equation}
\widetilde{E}_u^{\leftarrow}(\rho_{A(CD)}) \geq E_u^{\leftarrow}(\rho_{AC}).
\end{equation}
\label{lem1}
\end{Lem}
\begin{proof}
For $\rho_{ACD}$, let $\{M_x \}$ and $\{N_y\}$ are the optimal
rank-1 measurements of $C$ and $D$ respectively, such that
\begin{equation}
\widetilde{E}_u^{\leftarrow}(\rho_{A(CD)})=S(\rho_A)-\sum_{x, y} p_{xy} S(\rho^{xy}_A).
\end{equation}
Due to the concavity of von Neumann entropy, we have
\begin{align}
\widetilde{E}_u^{\leftarrow}(\rho_{A(CD)})&=S(\rho_A)-\sum_{x,y} p_{xy} S(\rho^{xy}_A)\nonumber\\
&= S(\rho_A)-\sum_{x} p_{x}\left[\sum_{y}\frac{p_{xy}}{p_{x}} S(\rho^{xy}_A)\right]\nonumber\\
&\geq S(\rho_A)-\sum_{x} p_{x} S\left(\sum_{y}\frac{p_{xy}}{p_{x}} \rho^{xy}_A\right)\nonumber\\
&=S(\rho_A)-\sum_{x} p_{x} S(\rho^{x}_A)\nonumber\\
&\geq E_u^{\leftarrow}(\rho_{AC}),
\end{align}
where
\begin{align}
p_{x}&=\mbox{$\mathrm{tr}$}_{AC}\left[(I_{A}\otimes M_{x})\rho_{AC} \right]\nonumber\\
&=\mbox{$\mathrm{tr}$}_{ACD}\left[(I_{A}\otimes M_{x}\otimes I_{D})\rho_{ACD} \right]\nonumber\\
&=\sum_{y} \mbox{$\mathrm{tr}$}_{ACD}\left[(I_{A}\otimes M_{x}\otimes N_{y})\rho_{ACD} \right]\nonumber\\
&=\sum_{y}p_{xy},
\end{align}

\begin{align}
\rho^x_A&=\frac{1}{p_x}\mbox{$\mathrm{tr}$}_{C}\left[(I_{A}\otimes M_{x})\rho_{AC} \right]\nonumber\\
&=\frac{1}{p_x}\mbox{$\mathrm{tr}$}_{CD}\left[(I_{A}\otimes M_{x}\otimes I_{D})\rho_{ACD} \right]\nonumber\\
&=\sum_{y}\frac{1}{p_x} \mbox{$\mathrm{tr}$}_{CD}\left[(I_{A}\otimes M_{x}\otimes N_{y})\rho_{ACD} \right]\nonumber\\
&=\sum_{y}\frac{p_{xy}}{p_x}\rho^{xy}_A,
\end{align}
and the second inequality is due to the definition of $E_u^{\leftarrow}(\rho_{AC})$.
\end{proof}

Now, we are ready to have the following theorem.

\begin{Thm}
For any tripartite mixed state $\rho_{ABC}$ with a purification $\ket{\psi}_{ABCD}$,
\begin{align}
S(\rho_A)&\geq \widetilde{E}_a(\rho_{AB})+E_u^{\leftarrow}(\rho_{AC}),
\end{align}
where $\widetilde{E}_a(\rho_{AB})$ is the
localizable entanglement~{\em \cite{PVMC}} of $\rho_{AB}$, defined by
\begin{equation}
\widetilde{E}_a(\rho_{AB})=\max_{\{M_x\otimes N_y\}} \sum_{x, y} p_{xy} S(\rho^{xy}_A)
\end{equation}
over all possible rank-1 measurements
$\{M_x\}$ and $\{N_y\}$ on subsystems $C$ and $D$ respectively.
\label{Thm: 3mix}
\end{Thm}
\begin{proof}

Eq.~(\ref{unlocal3}) can be rewritten as
\begin{align}
\widetilde{E}_u^{\leftarrow}(\rho_{A(CD)})&=S(\rho_A)-\max_{\{M_x\otimes N_{y} \}}
\sum_{x,y} p_{xy} S(\rho^{xy}_A)\nonumber\\
&=S(\rho_A)-\widetilde{E}_a(\rho_{AB}),
\label{unlocal4}
\end{align}
and Lemma~\ref{lem1} completes the proof.
\end{proof}

Theorem~\ref{Thm: 3mix} can be considered as an alternative of
Lemma~\ref{Lem: puremain1} for mixed states case. Furthermore,
Theorem~\ref{Thm: 3mix} together with Lemma~\ref{Lem: puremain1}
give us the following simple corollary.

\begin{Cor}
For any tripartite mixed state $\rho_{ABC}$ with a purification $\ket{\psi}_{ABCD}$,
\begin{equation}
E_a(\rho_{A(BC)})\geq \widetilde{E}_a(\rho_{AB}).
\end{equation}
\label{cor2}
\end{Cor}
\begin{proof}
By Theorem~\ref{Thm: 3mix}, we have
\begin{equation}
S(\rho_A) \geq \widetilde{E}_a(\rho_{AB})+E_u^{\leftarrow}(\rho_{AD})
\end{equation}
for any pure state $\ket{\psi}_{ABCD}$, whereas
\begin{equation}
S(\rho_A)=E_a(\rho_{A(BC)})+E_u^{\leftarrow}(\rho_{AD}),
\end{equation}
for the tripartite partition $A-BC-D$.
\end{proof}

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

In this paper, we have proposed the concept of UE, and shown that the
polygamous nature of distributed quantum entanglement in multipartite systems is
strongly due to this unlocalizable character. As the mathematical
interpretation for this polygamous nature of quantum entanglement, we
have established polygamy inequalities of entanglement in tripartite
quantum systems with arbitrary dimension, and multi-qubit systems.  We
have also provided several trade offs between UE and other
correlations such as EoA, and localizable entanglement.

This is the first result where polygamous property of quantum
entanglement in multipartite higher-dimensional quantum systems is
provided.  Furthermore, the proposed inequalities are in terms of the
entropic entanglement measures such as entropy of entanglement for
pure states and EoA.  In other words, the proposed polygamy inequalities
of distributed entanglement have been shown in terms of the actual quantification
of entanglement with operational meanings, rather than using other
entanglement measures such as concurrence.

\section*{Acknowledgments}
JSK would like to thank Soojoon Lee for useful discussion, and acknowledges the support
from iCORE, MITACS (QIP project) and US Army.
GG acknowledges financial support from NSERC and MITACS-QIP.

redpajama_set_name: RedPajamaArXiv


A new episode, entitled What They Died For, airs tonight at 9/8c on ABC. While Locke devises a new strategy, Jack's group searches for Desmond. It was written by Edward Kitsis/Adam Horowitz/Elizabeth Sarnoff and directed by Paul Edwards. Although I liked last week's crazy Allison Janney origin stuff, I'm glad everybody's back in this episode. Maybe we will learn which candidate is getting the job and why Kate's name got crossed off. I love the use of The Doors in the promo. The 2.5 hour series finale airs this Sunday!
this is the best season ever of LOST.


text:
STEPHEN WHYNO
AP Hockey Writer
WASHINGTON -- Coach John Tortorella paced up and down the bench, trying to keep the Blue Jackets' spirits high as they faced their first two-goal deficit since November.
The Jackets just didn't have it Thursday night. Columbus lost a 5-0 clunker to the Washington Capitals, ending its winning streak at 16 games, one shy of the record set by the 1992-93 Pittsburgh Penguins.
The Blue Jackets' fear-inducing, league-leading power play was blanked over five chances, and Columbus lost for the first time since Nov. 26. Goalie Sergei Bobrovsky, rock solid throughout the streak, was chased and replaced by Curtis McElhinney after allowing goals to Daniel Winnik, John Carlson, Nate Schmidt, Andre Burakovsky and Justin Williams on 23 shots.
The Capitals extended their winning streak to four and moved within five points of the Blue Jackets for the top spot in the ultra-competitive Metropolitan Division. They were eager to curb Columbus' run, with Burakovsky saying earlier Thursday, "It's going to be a really fun moment for us to end it." A fan at the game held a sign that read, "Hey Blue Jackets, streaking is not allowed in here."
Tortorella conceded that luck was needed during the streak, and his team didn't get the bounces Thursday. Winnik's goal came off a flurry in front, Carlson's went in off his left skate and Schmidt's on his own rebound off Bobrovsky's chest.
Washington's good fortune and a 29-save performance from reigning Vezina Trophy winner Braden Holtby combined for a blowout reminiscent of when the 1979-80 Philadelphia Flyers ended their 35-game unbeaten streak with a 7-1 loss at the Minnesota North Stars.
Tortorella had said he wanted his club to tie the record, but added, "It's not the end of the world if we don't. It's been a really good process for us to handle a little bit of scrutiny that we're not used to at certain times. This is all good."
Tortorella said he'd measure his club more by how it handled the lull when the streak ended. As captain Nick Foligno pointed out before the game, this was only contest No. 37 for Columbus, and the Blue Jackets have a long way to go to show they're more than just a team with a memorable midseason run.
"I'm proud of the group that we've hung together and stayed true to what our mantra is, so to speak, and we've got to continue to do that for a lot more games than just 37," Foligno said.
NOTES: Bobrovsky, who had a 1.64 goals-against average and .941 save percentage during the streak, was pulled for just the second time this season. ... Schmidt's goal was his first in 67 games dating to Jan. 7, 2016. ... Williams has 10 goals and eight assists in his past 23 games after starting the season with one and one in his first 15. ... McElhinney stopped all four shots he faced after replacing Bobrovsky in the third period.
Blue Jackets: Try to start a new streak Saturday at home against the New York Rangers, who are only three points back of them.
Capitals: Visit the Ottawa Senators on Saturday looking to extend their less-talked-about winning streak to five.
Follow AP Hockey Writer on Twitter at http://www.twitter.com/SWhyno.


We have an incredible Pandora's box of breathtaking furnishings and décor accessories, including extravagant table settings, candelabras, ambient light fittings, ottomans, wall fixtures, loungers, figurines, dividers and much more! Our creative direction stems from being inspired by our clients, their ideas and preferences, brainstorming concepts, and ultimately the detailed execution to create each unforgettable event.
Alongside the practical coordination and detailed project management of the event, we also manage all appropriate suppliers.
The coordination of caterers, beverage suppliers, entertainment, engineers, transport services, emergency services, staff, and any other external supplier.
The conception and design of stationary and decoration.


text:
Heritage Festival In Ogden Celebrating Golden Spike Anniversary
OGDEN, Utah – This year, the annual Heritage Festival will be a special anniversary edition, celebrating the completion of the Transcontinental Railroad 150 years ago.
The festival began Thursday with the meeting of two historic steam engines — The Living Legend 844 and The Big Boy 4014 — at Ogden Union Station, 2501 Wall Ave.
Celebrations continue Friday starting at 10 a.m. at the Union Station with food, vendors, performances, and exhibits.
The festival will continue through Saturday.
For information, go to visitogden.com.
Carter Williams
1 of the 4 spikes driven at Promontory Summit fetches $2.2M at auction
The Arizona spike, which was ceremoniously driven in Utah to mark the transcontinental railroad completion in 1869, sold for a little more than $2 million during an auction that closed Thursday.
John Hollenhorst, KSL TV
Colorado Town Disputes Utah's Transcontinental Claim To Fame
When Utah celebrated the 150th anniversary of the driving of the golden spike, the cheering section was pretty much statewide. But in a tiny Colorado town, some people think the whole thing in Utah is a historical fraud.
Mike Anderson, KSL TV
Spike 150 Brings Boost To Weber, Box Elder Economy
A large influx of visitors into Northern Utah is expected to translate into millions of dollars being spent, especially in Box Elder, and Weber Counties.
Keith McCord, KSL TV
Iconic 'Champagne Photo,' Photographers Remembered On Golden Spike Anniversary
As the transcontinental railroad was nearing completion, fortunately the Union Pacific and Central Pacific Railroads made sure the event was documented. They hired 3 photographers to preserve that great day in 1869.
Keri Wilcox, KSL TV
Spike 150 Celebrations Honor Native American Culture and Legacy
As Utah and the nation celebrate the 150th anniversary of the completion of the Transcontinental Railroad, Native Americans hope we also remember and honor an often forgotten legacy of progress.
BRADY McCOMBS Associated Press
Cannons, Bells, Music Mark 150th Anniversary Of Railroad
Visitors traveled from as far as China and dressed in old-fashioned dresses, top hats and bonnets for the celebration featuring music, a flyover, the firing of cannons and ringing of bells at the Golden Spike National Historic Park.