Document Zbl 1338.81104

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1338.81104 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">×</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools & Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/1338.81104?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open — the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span> <a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:1338.81104" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>  </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>  </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>  </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>  </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>  </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>  </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>  </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>  </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>  </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>  </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>  </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>  </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>  </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>  </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>  </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>  </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>  </span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab   </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>  </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki & ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/pan.jun" title="Author Profile">Pan, Jun</a>; <a href="/authors/zhou.lan" title="Author Profile">Zhou, Lan</a>; <a href="/authors/gu.shi-pu" title="Author Profile">Gu, Shi-Pu</a>; <a href="/authors/wang.xingfu" title="Author Profile">Wang, Xing-Fu</a>; <a href="/authors/sheng.yu-bo" title="Author Profile">Sheng, Yu-Bo</a>; <a href="/authors/wang.qin.2" title="Author Profile">Wang, Qin</a></div> <h2 class="title"> <strong>Efficient entanglement concentration for concatenated Greenberger-Horne-Zeilinger state with the cross-Kerr nonlinearity.</strong> <i>(English)</i> <a class="label nowrap" href="/1338.81104">Zbl 1338.81104</a> </h2> <div class="source"> <a href="/serials/3019" title="Journal Profile">Quantum Inf. Process.</a> <a href="/?q=in%3A355322" title="Articles in this Issue">15, No. 4, 1669-1687 (2016)</a>. </div> <div class="abstract">Summary: Concatenated Greenberger-Horne-Zeilinger (C-GHZ) state, which encodes physical qubits in a logic qubit, has great application in the future quantum communication. We present an efficient entanglement concentration protocol (ECP) for recovering less-entangled C-GHZ state into the maximally entangled C-GHZ state with the help of cross-Kerr nonlinearities and photon detectors. With the help of the cross-Kerr nonlinearity, the obtained maximally entangled C-GHZ state can be remained for other applications. Moreover, the ECP can be used repeatedly, which can increase the success probability largely. Based on the advantages above, our ECP may be useful in the future long-distance quantum communication.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6591510">Cited in <strong>12</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P45" title="MSC2020">81P45</a> </td> <td class="space"> Quantum information, communication, networks (quantum-theoretic aspects) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81P40" title="MSC2020">81P40</a> </td> <td class="space"> Quantum coherence, entanglement, quantum correlations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A78A60" title="MSC2020">78A60</a> </td> <td class="space"> Lasers, masers, optical bistability, nonlinear optics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aquantum+communication">quantum communication</a>; <a href="/?q=ut%3Aconcatenated+Greenberger-Horne-Zeilinger+state">concatenated Greenberger-Horne-Zeilinger state</a>; <a href="/?q=ut%3Aentanglement+concentration">entanglement concentration</a>; <a href="/?q=ut%3Across-Kerr+nonlinearity">cross-Kerr nonlinearity</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1338.81104" data-ciurl="/ci/06591510" data-biburl="/bibtex/06591510.bib" data-amsurl="/amsrefs/06591510.bib" data-xmlurl="/xml/06591510.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06591510.pdf" title="Zbl 1338.81104 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s11128-016-1246-7" aria-label="DOI for “Efficient entanglement concentration for concatenated Greenberger-Horne-Zeilinger state with the cross-Kerr nonlinearity”" title="10.1007/s11128-016-1246-7">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ekert, A.K.: Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67, 661 (1991) · <a href="/0990.94509" class="nowrap">Zbl 0990.94509</a> · <a href="https://doi.org/10.1103/PhysRevLett.67.661" class="nowrap">doi:10.1103/PhysRevLett.67.661</a></td> </tr><tr> <td>[2]</td> <td class="space">Bennett, C.H., Brassard, G., Crepeau, C., Jozsa, R., Peres, A., Wootters, W.K.: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Phys. Rev. Lett. 70, 1895 (1993) · <a href="/1051.81505" class="nowrap">Zbl 1051.81505</a> · <a href="https://doi.org/10.1103/PhysRevLett.70.1895" class="nowrap">doi:10.1103/PhysRevLett.70.1895</a></td> </tr><tr> <td>[3]</td> <td class="space">Long, G.L., Liu, X.S.: Theoretically efficient high-capacity quantum-key-distribution scheme. Phys. Rev. A 65, 032302 (2002) · <a href="https://doi.org/10.1103/PhysRevA.65.032302" class="nowrap">doi:10.1103/PhysRevA.65.032302</a></td> </tr><tr> <td>[4]</td> <td class="space">Deng, F.G., Long, G.L., Liu, X.S.: Two-step quantum direct communication protocol using the Einstein-Podolsky-Rosen pair block. Phys. Rev. A 68, 042317 (2003) · <a href="https://doi.org/10.1103/PhysRevA.68.042317" class="nowrap">doi:10.1103/PhysRevA.68.042317</a></td> </tr><tr> <td>[5]</td> <td class="space">Xu, J.S., Li, C.F.: Quantum integrated circuit: classical characterization. Sci. Bull. 60, 141-141 (2015) · <a href="https://doi.org/10.1007/s11434-014-0703-x" class="nowrap">doi:10.1007/s11434-014-0703-x</a></td> </tr><tr> <td>[6]</td> <td class="space">Zhang, C., Li, C.F., Guo, G.C.: Experimental demonstration of photonic quantum ratchet. Sci. Bull. 60, 249-255 (2015) · <a href="https://doi.org/10.1007/s11434-014-0710-y" class="nowrap">doi:10.1007/s11434-014-0710-y</a></td> </tr><tr> <td>[7]</td> <td class="space">Zheng, C., Long, G.F.: Quantum secure direct dialogue using Einstein-Podolsky-Rosen pairs. Sci. Chin. Phys. Mech. Astro. 57, 1238-1243 (2014) · <a href="https://doi.org/10.1007/s11433-014-5461-x" class="nowrap">doi:10.1007/s11433-014-5461-x</a></td> </tr><tr> <td>[8]</td> <td class="space">Su, X.L., Jia, X.J., Xie, C.D., Peng, K.C.: Preparation of multipartite entangled states used for quantum information networks. Sci. Chin. Phys. Mech. Astro. 57, 1210-1217 (2014) · <a href="https://doi.org/10.1007/s11433-013-5358-0" class="nowrap">doi:10.1007/s11433-013-5358-0</a></td> </tr><tr> <td>[9]</td> <td class="space">Zou, X.F., Qiu, D.W.: Three-step semiquantum secure direct communication protocol. Sci. Chin. Phys. Mech. Astro. 57, 1696-1702 (2014) · <a href="https://doi.org/10.1007/s11433-014-5542-x" class="nowrap">doi:10.1007/s11433-014-5542-x</a></td> </tr><tr> <td>[10]</td> <td class="space">Cao, D.Y., Liu, B.H., Wang, Z., Huang, Y.F., Li, C.F., Guo, G.C.: Multiuser-to-multiuser entanglement distribution based on 1550 nm polarization-entangled photons. Sci. Bull. 60, 1128-1132 (2015) · <a href="https://doi.org/10.1007/s11434-015-0801-4" class="nowrap">doi:10.1007/s11434-015-0801-4</a></td> </tr><tr> <td>[11]</td> <td class="space">Duan, L.M., Lukin, M.D., Cirac, J.I., Zoller, P.: Long-distance quantum communication with atomic ensembles and linear optics. Nature (London) 414, 413-418 (2001) · <a href="https://doi.org/10.1038/35106500" class="nowrap">doi:10.1038/35106500</a></td> </tr><tr> <td>[12]</td> <td class="space">Bennett, C.H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J.A., Wootters, W.K.: Purification of noisy entanglement and faithful teleportation via noisy channels. Phys. Rev. Lett. 76, 722 (1996) · <a href="https://doi.org/10.1103/PhysRevLett.76.722" class="nowrap">doi:10.1103/PhysRevLett.76.722</a></td> </tr><tr> <td>[13]</td> <td class="space">Pan, J.W., Simon, C., Brukner, C., Zeilinger, A.: Entanglement purification for quantum communication. Nature (London) 410, 1067-1070 (2001) · <a href="https://doi.org/10.1038/35074041" class="nowrap">doi:10.1038/35074041</a></td> </tr><tr> <td>[14]</td> <td class="space">Simon, C., Pan, J.W.: Polarization entanglement purification using spatial entanglement. Phys. Rev. Lett. 89, 257901 (2002) · <a href="https://doi.org/10.1103/PhysRevLett.89.257901" class="nowrap">doi:10.1103/PhysRevLett.89.257901</a></td> </tr><tr> <td>[15]</td> <td class="space">Sheng, Y.B., Zhou, L., Long, G.L.: Hybrid entanglement purification for quantum repeaters. Phys. Rev. A 88, 022302 (2013) · <a href="https://doi.org/10.1103/PhysRevA.88.022302" class="nowrap">doi:10.1103/PhysRevA.88.022302</a></td> </tr><tr> <td>[16]</td> <td class="space">Cao, C., Wang, C., He, L.Y., Zhang, R.: Atomic entanglement purification and concentration using coherent state input-output process in low-Q cavity QED regime. Opt. Express 21, 4093-4105 (2013) · <a href="https://doi.org/10.1364/OE.21.004093" class="nowrap">doi:10.1364/OE.21.004093</a></td> </tr><tr> <td>[17]</td> <td class="space">Sheng, Y.B., Zhou, L.: Deterministic polarization entanglement purification using time-bin entanglement. Laser Phys. Lett. 11, 085203 (2014) · <a href="https://doi.org/10.1088/1612-2011/11/8/085203" class="nowrap">doi:10.1088/1612-2011/11/8/085203</a></td> </tr><tr> <td>[18]</td> <td class="space">Ren, B.C., Du, F.F., Deng, F.G.: Two-step hyperentanglement purification with the quantum-state-joining method. Phys. Rev. A 90, 052309 (2014) · <a href="https://doi.org/10.1103/PhysRevA.90.052309" class="nowrap">doi:10.1103/PhysRevA.90.052309</a></td> </tr><tr> <td>[19]</td> <td class="space">Sheng, Y.B., Zhou, L.: Deterministic entanglement distillation for secure double-server blind quantum computation. Sci. Rep. 5, 7815 (2015) · <a href="https://doi.org/10.1038/srep07815" class="nowrap">doi:10.1038/srep07815</a></td> </tr><tr> <td>[20]</td> <td class="space">Bennett, C.H., Bernstein, H.J., Popescu, S., Schumacher, B.: Concentrating partial entanglement by local operations. Phys. Rev. A 53, 2046 (1996) · <a href="https://doi.org/10.1103/PhysRevA.53.2046" class="nowrap">doi:10.1103/PhysRevA.53.2046</a></td> </tr><tr> <td>[21]</td> <td class="space">Bose, S., Vedral, V., Knight, P.L.: Purification via entanglement swapping and conserved entanglement. Phys. Rev. A 60, 194 (1999) · <a href="https://doi.org/10.1103/PhysRevA.60.194" class="nowrap">doi:10.1103/PhysRevA.60.194</a></td> </tr><tr> <td>[22]</td> <td class="space">Yamamoto, T., Koashi, M., Imoto, N.: Concentration and purification scheme for two partially entangled photon pairs. Phys. Rev. A 64, 012304 (2001) · <a href="https://doi.org/10.1103/PhysRevA.64.012304" class="nowrap">doi:10.1103/PhysRevA.64.012304</a></td> </tr><tr> <td>[23]</td> <td class="space">Zhao, Z., Pan, J.W., Zhan, M.S.: Practical scheme for entanglement concentration. Phys. Rev. A 64, 014301 (2001) · <a href="https://doi.org/10.1103/PhysRevA.64.014301" class="nowrap">doi:10.1103/PhysRevA.64.014301</a></td> </tr><tr> <td>[24]</td> <td class="space">Sheng, Y.B., Deng, F.G., Zhou, H.Y.: Nonlocal entanglement concentration scheme for partially entangled multipartite systems with nonlinear optics. Phys. Rev. A 77, 062325 (2008) · <a href="https://doi.org/10.1103/PhysRevA.77.062325" class="nowrap">doi:10.1103/PhysRevA.77.062325</a></td> </tr><tr> <td>[25]</td> <td class="space">Sheng, Y.B., Zhou, L., Zhao, S.M., Zheng, B.Y.: Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs. Phys. Rev. A 85, 012307 (2012) · <a href="https://doi.org/10.1103/PhysRevA.85.012307" class="nowrap">doi:10.1103/PhysRevA.85.012307</a></td> </tr><tr> <td>[26]</td> <td class="space">Wang, H.F., Sun, L.L., Zhang, S., Yeon, K.H.: Scheme for entanglement concentration of unknown partially entangled three-atom W states in cavity QED. Quantum Inf. Process. 11, 431-441 (2012) · <a href="/1239.81029" class="nowrap">Zbl 1239.81029</a> · <a href="https://doi.org/10.1007/s11128-011-0255-9" class="nowrap">doi:10.1007/s11128-011-0255-9</a></td> </tr><tr> <td>[27]</td> <td class="space">Wang, H.F., Zhang, S., Yeon, K.H.: Linear-optics-based entanglement concentration of unknown partially entangled three-photon W states. J. Opt. Soc. Am. B. 27, 2159-2164 (2010) · <a href="https://doi.org/10.1364/JOSAB.27.002159" class="nowrap">doi:10.1364/JOSAB.27.002159</a></td> </tr><tr> <td>[28]</td> <td class="space">Wang, H.F., Zhang, S., Yeon, K.H.: Linear optical scheme for entanglement concentration of two partially entangled three-photon W states. Eur. Phys. J. D. 56, 271-275 (2010) · <a href="https://doi.org/10.1140/epjd/e2009-00277-3" class="nowrap">doi:10.1140/epjd/e2009-00277-3</a></td> </tr><tr> <td>[29]</td> <td class="space">Liang, B.B., Hu, S., Cui, W.X., An, C.S., Xing, Y., Hu, J.S., Sun, G.Q., Jiang, X.X., Wang, H.F.: Scheme for realizing the entanglement concentration of unknown partially entangled three-photon W states assisted by quantum dot-microcavity coupled system. Laser Phys. Lett. 11, 115202 (2014) · <a href="https://doi.org/10.1088/1612-2011/11/11/115202" class="nowrap">doi:10.1088/1612-2011/11/11/115202</a></td> </tr><tr> <td>[30]</td> <td class="space">Ren, B.C., Du, F.F., Deng, F.G.: Hyperentanglement concentration for two-photon four-qubit systems with linear optics. Phys. Rev. A 88, 012302 (2013) · <a href="https://doi.org/10.1103/PhysRevA.88.012302" class="nowrap">doi:10.1103/PhysRevA.88.012302</a></td> </tr><tr> <td>[31]</td> <td class="space">Sheng, Y.B., Zhou, L.: Quantum entanglement concentration based on nonlinear optics for quantum communications. Entropy 15, 1776-1820 (2013) · <a href="/1297.81039" class="nowrap">Zbl 1297.81039</a> · <a href="https://doi.org/10.3390/e15051776" class="nowrap">doi:10.3390/e15051776</a></td> </tr><tr> <td>[32]</td> <td class="space">Ren, B.C., Long, G.L.: General hyperentanglement concentration for photon systems assisted by quantum-dot spins inside optical microcavities. Opt. Express 22, 6547-6561 (2014) · <a href="https://doi.org/10.1364/OE.22.006547" class="nowrap">doi:10.1364/OE.22.006547</a></td> </tr><tr> <td>[33]</td> <td class="space">Zhou, L.: Consequent entanglement concentration of a less-entangled electronic cluster state with controlled-not gates. Chin. Phys. B 23, 050308 (2014) · <a href="https://doi.org/10.1088/1674-1056/23/5/050308" class="nowrap">doi:10.1088/1674-1056/23/5/050308</a></td> </tr><tr> <td>[34]</td> <td class="space">Li, X.H., Ghose, S.: Hyperentanglement concentration for time-bin and polarization hyperentangled photons. Phys. Rev. A 91, 062302 (2015) · <a href="https://doi.org/10.1103/PhysRevA.91.062302" class="nowrap">doi:10.1103/PhysRevA.91.062302</a></td> </tr><tr> <td>[35]</td> <td class="space">Shukla, C., Banerjee, A., Pathak, A.: Protocols and quantum circuits for implementing entanglement concentration in cat state, GHZ-like state and nine families of 4-qubit entangled states. Quantum Inf. Process. 14, 2077-2099 (2015) · <a href="/1317.81037" class="nowrap">Zbl 1317.81037</a> · <a href="https://doi.org/10.1007/s11128-015-0948-6" class="nowrap">doi:10.1007/s11128-015-0948-6</a></td> </tr><tr> <td>[36]</td> <td class="space">Wang, G.Y., Li, T., Deng, F.G.: High-efficiency atomic entanglement concentration for quantum communication network assisted by cavity QED. Quantum Inf. Process. 14, 1305-1320 (2015) · <a href="/1328.81051" class="nowrap">Zbl 1328.81051</a> · <a href="https://doi.org/10.1007/s11128-015-0938-8" class="nowrap">doi:10.1007/s11128-015-0938-8</a></td> </tr><tr> <td>[37]</td> <td class="space">Choudhury, B.S., Dhara, A.: An entanglement concentration protocol for cluster states. Quantum Inf. Process. 12, 2577-2585 (2013) · <a href="/1270.81025" class="nowrap">Zbl 1270.81025</a> · <a href="https://doi.org/10.1007/s11128-013-0549-1" class="nowrap">doi:10.1007/s11128-013-0549-1</a></td> </tr><tr> <td>[38]</td> <td class="space">Du, F.F., Deng, F.G.: Heralded entanglement concentration for photon systems with linear-optical elements. Sci. Chin. Phys. Mech. Astro. 58, 1-8 (2015)</td> </tr><tr> <td>[39]</td> <td class="space">Zhou, L.: Efficient entanglement concentration for electron-spin W state with the charge detection. Quantum Inf. Process. 12, 2087-2101 (2013) · <a href="/1267.81085" class="nowrap">Zbl 1267.81085</a> · <a href="https://doi.org/10.1007/s11128-012-0511-7" class="nowrap">doi:10.1007/s11128-012-0511-7</a></td> </tr><tr> <td>[40]</td> <td class="space">Zhou, L., Sheng, Y.B., Cheng, W.W., Gong, L.Y., Zhao, S.M.: Efficient entanglement concentration for arbitrary less-entangled NOON states. Quantum Inf. Process. 12, 1307-1320 (2013) · <a href="/1264.81064" class="nowrap">Zbl 1264.81064</a> · <a href="https://doi.org/10.1007/s11128-012-0472-x" class="nowrap">doi:10.1007/s11128-012-0472-x</a></td> </tr><tr> <td>[41]</td> <td class="space">Fan, L.L., Xia, Y., Song, J.: Efficient entanglement concentration for arbitrary less-hyperentanglement multi-photon W states with linear optics. Quantum Inf. Process. 13, 1967-1978 (2014) · <a href="/1305.81022" class="nowrap">Zbl 1305.81022</a> · <a href="https://doi.org/10.1007/s11128-014-0789-8" class="nowrap">doi:10.1007/s11128-014-0789-8</a></td> </tr><tr> <td>[42]</td> <td class="space">Cao, C., Ding, H., Li, Y., Wang, T.J., Mi, S.C., Zhang, R., Wang, C.: Efficient multipartite entanglement concentration protocol for nitrogen-vacancy center and microresonator coupled systems. Quantum Inf. Process. 14, 1265-1277 (2015) · <a href="/1328.81030" class="nowrap">Zbl 1328.81030</a> · <a href="https://doi.org/10.1007/s11128-015-0924-1" class="nowrap">doi:10.1007/s11128-015-0924-1</a></td> </tr><tr> <td>[43]</td> <td class="space">Wang, C., Cao, C., He, L.Y., Zhang, C.L.: Hybrid entanglement concentration using quantum dot and microcavity coupled system. Quantum Inf. Process. 13, 1025-1034 (2014) · <a href="/1291.81061" class="nowrap">Zbl 1291.81061</a> · <a href="https://doi.org/10.1007/s11128-013-0707-5" class="nowrap">doi:10.1007/s11128-013-0707-5</a></td> </tr><tr> <td>[44]</td> <td class="space">Qu, C.C., Zhou, L., Sheng, Y.B.: Entanglement concentration for concatenated Greenberger-Horne-Zeilinger state. Quantum Inf. Process. 14, 4131-4146 (2015) · <a href="/1327.81062" class="nowrap">Zbl 1327.81062</a> · <a href="https://doi.org/10.1007/s11128-015-1113-y" class="nowrap">doi:10.1007/s11128-015-1113-y</a></td> </tr><tr> <td>[45]</td> <td class="space">Liu, J., Zhao, S.Y., Zhou, L., Sheng, Y.B.: Electronic cluster state entanglement concentration based on charge detection. Chin. Phys. B 23, 020313 (2014) · <a href="https://doi.org/10.1088/1674-1056/23/2/020313" class="nowrap">doi:10.1088/1674-1056/23/2/020313</a></td> </tr><tr> <td>[46]</td> <td class="space">Sheng, Y.B., Liu, J., Zhao, S.Y., Wang, L., Zhou, L.: Entanglement concentration for W-type entangled coherent states. Chin. Phys. B 23, 080305 (2014) · <a href="https://doi.org/10.1088/1674-1056/23/8/080305" class="nowrap">doi:10.1088/1674-1056/23/8/080305</a></td> </tr><tr> <td>[47]</td> <td class="space">Sheng, Y.B., Feng, Z.F., Ou-Yang, Y., Qu, C.C., Zhou, L.: Arbitrary partially entangled three-electron state concentration with controlled-not gates. Chin. Phys. Lett. 31, 050303 (2014) · <a href="https://doi.org/10.1088/0256-307X/31/5/050303" class="nowrap">doi:10.1088/0256-307X/31/5/050303</a></td> </tr><tr> <td>[48]</td> <td class="space">Sheng, Y.B., Zhou, L.: Efficient electronic entanglement concentration assisted by single mobile electrons. Chin. Phys. B 11, 110303 (2013) · <a href="https://doi.org/10.1088/1674-1056/22/11/110303" class="nowrap">doi:10.1088/1674-1056/22/11/110303</a></td> </tr><tr> <td>[49]</td> <td class="space">Fröwis, F., Dür, W.: Stable macroscopic quantum superpositions. Phys. Rev. Lett. 106, 110402 (2011) · <a href="https://doi.org/10.1103/PhysRevLett.106.110402" class="nowrap">doi:10.1103/PhysRevLett.106.110402</a></td> </tr><tr> <td>[50]</td> <td class="space">Fröwis, F., Dür, W.: Stability of encoded macroscopic quantum superpositions. Phys. Rev. A 85, 052329 (2012) · <a href="https://doi.org/10.1103/PhysRevA.85.052329" class="nowrap">doi:10.1103/PhysRevA.85.052329</a></td> </tr><tr> <td>[51]</td> <td class="space">Kesting, F., Fröwis, F., Dür, W.: Effective noise channels for encoded quantum systems. Phys. Rev. A 88, 042305 (2013) · <a href="https://doi.org/10.1103/PhysRevA.88.042305" class="nowrap">doi:10.1103/PhysRevA.88.042305</a></td> </tr><tr> <td>[52]</td> <td class="space">Ding, D., Yan, F.L., Gao, T.: Preparation of km-photon concatenated Greenberger-Horne-Zeilinger states for observing distinctive quantum effects at macroscopic scales. J. Opt. Soc. Am. B 30, 3075-3078 (2013) · <a href="https://doi.org/10.1364/JOSAB.30.003075" class="nowrap">doi:10.1364/JOSAB.30.003075</a></td> </tr><tr> <td>[53]</td> <td class="space">Lu, H., Chen, L.K., Liu, C., Xu, P., Yao, X.C., Li, L., Liu, N.L., Zhao, B., Chen, Y.A., Pan, J.W.: Experimental realization of a concatenated Greenberger-Horne-Zeilinger state for macroscopic quantum superpositions. Nat. Photonics 8, 364-368 (2014) · <a href="https://doi.org/10.1038/nphoton.2014.81" class="nowrap">doi:10.1038/nphoton.2014.81</a></td> </tr><tr> <td>[54]</td> <td class="space">Sheng, Y.B., Zhou, L.: Entanglement analysis for macroscopic Schrödinger’s Cat state. EPL 109, 40009 (2015) · <a href="https://doi.org/10.1209/0295-5075/109/40009" class="nowrap">doi:10.1209/0295-5075/109/40009</a></td> </tr><tr> <td>[55]</td> <td class="space">Sheng, Y.B., Zhou, L.: Two-step complete polarization logic Bell-state analysis. Sci. Rep. 5, 13453 (2015) · <a href="https://doi.org/10.1038/srep13453" class="nowrap">doi:10.1038/srep13453</a></td> </tr><tr> <td>[56]</td> <td class="space">Zhou, L., Sheng, Y.B.: Complete logic Bell-state analysis assisted with photonic Faraday rotation. Phys. Rev. A 92, 042314 (2015) · <a href="https://doi.org/10.1103/PhysRevA.92.042314" class="nowrap">doi:10.1103/PhysRevA.92.042314</a></td> </tr><tr> <td>[57]</td> <td class="space">Nemoto, K., Munro, W.J.: Nearly deterministic linear optical controlled-not gate. Phys. Rev. Lett. 93, 250502 (2004) · <a href="https://doi.org/10.1103/PhysRevLett.93.250502" class="nowrap">doi:10.1103/PhysRevLett.93.250502</a></td> </tr><tr> <td>[58]</td> <td class="space">Dong, L., Wang, J.X., Shen, H.Z., Li, D., Xiu, X.M., Gao, Y.J., Yi, X.X.: Deterministic transmission of an arbitrary single-photon polarization state through bit-flip error channel. Quantum Inf. Process. 13, 1413-1424 (2014) · <a href="/1303.81057" class="nowrap">Zbl 1303.81057</a> · <a href="https://doi.org/10.1007/s11128-014-0736-8" class="nowrap">doi:10.1007/s11128-014-0736-8</a></td> </tr><tr> <td>[59]</td> <td class="space">Dong, L., Xiu, X.M., Gao, Y.J., Yi, X.X.: A nearly deterministic scheme for generating \[\chi\] χ-type entangled states with weak cross-Kerr nonlinearities. Quantum Inf. Process. 12, 1787-1795 (2013) · <a href="/1267.81061" class="nowrap">Zbl 1267.81061</a> · <a href="https://doi.org/10.1007/s11128-012-0481-9" class="nowrap">doi:10.1007/s11128-012-0481-9</a></td> </tr><tr> <td>[60]</td> <td class="space">Sheng, Y.B., Guo, R., Pan, J., Zhou, L., Wang, X.F.: Two-step measurement of the concurrence for hyperentangled state. Quantum Inf. Process. 14, 963-978 (2015) · <a href="/1311.81040" class="nowrap">Zbl 1311.81040</a> · <a href="https://doi.org/10.1007/s11128-015-0916-1" class="nowrap">doi:10.1007/s11128-015-0916-1</a></td> </tr><tr> <td>[61]</td> <td class="space">Lin, Q., He, B.: Highly efficient processing of multi-photon states. Sci. Rep. 5, 12792 (2015) · <a href="https://doi.org/10.1038/srep12792" class="nowrap">doi:10.1038/srep12792</a></td> </tr><tr> <td>[62]</td> <td class="space">Zhou, L., Sheng, Y.B.: Recyclable amplification protocol for the single-photon entangled state. Laser Phys. Lett. 12, 045203 (2015) · <a href="https://doi.org/10.1088/1612-2011/12/4/045203" class="nowrap">doi:10.1088/1612-2011/12/4/045203</a></td> </tr><tr> <td>[63]</td> <td class="space">Hu, J.R., Lin, Q.: W state generation by adding independent single photons. Quantum Inf. Process. 14, 2847-2860 (2015) · <a href="/1327.81044" class="nowrap">Zbl 1327.81044</a> · <a href="https://doi.org/10.1007/s11128-015-1030-0" class="nowrap">doi:10.1007/s11128-015-1030-0</a></td> </tr><tr> <td>[64]</td> <td class="space">Lin, Q.: Optical parity gate and a wide range of entangled states generation. Sci. Chin. Phys. Mech. Astro. 58, 1-10 (2015) · <a href="https://doi.org/10.1007/s11433-014-5620-0" class="nowrap">doi:10.1007/s11433-014-5620-0</a></td> </tr><tr> <td>[65]</td> <td class="space">Ding, D., Yan, F.L., Gao, T.: Entangler and analyzer for multiphoton Greenberger-Horne-Zeilinger states using weak nonlinearities. Sci. Chin. Phys. Mech. Astro. 57, 2098-2103 (2014) · <a href="https://doi.org/10.1007/s11433-014-5498-x" class="nowrap">doi:10.1007/s11433-014-5498-x</a></td> </tr><tr> <td>[66]</td> <td class="space">He, Y.Q., Ding, D., Yan, F.L., Gao, T.: Exploration of photon-number entangled states using weak nonlinearities. Opt. Express 23, 21671-21677 (2015) · <a href="https://doi.org/10.1364/OE.23.021671" class="nowrap">doi:10.1364/OE.23.021671</a></td> </tr><tr> <td>[67]</td> <td class="space">Han, X., Hu, S., Guo, Q., Wang, H.F., Zhang, S.: Effective scheme for W-state fusion with weak cross-Kerr nonlinearities. Quantum Inf. Process. 14, 1919-1932 (2015) · <a href="/1317.81028" class="nowrap">Zbl 1317.81028</a> · <a href="https://doi.org/10.1007/s11128-015-0960-x" class="nowrap">doi:10.1007/s11128-015-0960-x</a></td> </tr><tr> <td>[68]</td> <td class="space">He, B., Lin, Q., Simon, C.: Cross-Kerr nonlinearity between continuous-mode coherent states and single photons. Phys. Rev. A 83, 053826 (2011) · <a href="https://doi.org/10.1103/PhysRevA.83.053826" class="nowrap">doi:10.1103/PhysRevA.83.053826</a></td> </tr><tr> <td>[69]</td> <td class="space">Dong, L., Xiu, X.M., Shen, H.Z., Gao, Y.J., Yi, X.X.: Quantum Fourier transform of polarization photons mediated by weak cross-Kerr nonlinearity. J. Opt. Soc. Am. B 30, 2765-2773 (2013) · <a href="https://doi.org/10.1364/JOSAB.30.002765" class="nowrap">doi:10.1364/JOSAB.30.002765</a></td> </tr><tr> <td>[70]</td> <td class="space">Xiu, X.M., Dong, L., Shen, H.Z., Gao, Y.J., Yi, X.X.: Construction scheme of a two-photon polarization controlled arbitrary phase gate mediated by weak cross-phase modulation. J. Opt. Soc. Am. B 30, 589-597 (2013) · <a href="https://doi.org/10.1364/JOSAB.30.000589" class="nowrap">doi:10.1364/JOSAB.30.000589</a></td> </tr><tr> <td>[71]</td> <td class="space">Xiu, X.M., Dong, L., Shen, H.Z., Gao, Y.J., Yi, X.X.: Preparing, linking, and unlinking cluster-type polarization-entangled states by integrating modules. Prog. Theor. Exp. Phys 9, 0903A01 (2013)</td> </tr><tr> <td>[72]</td> <td class="space">Dong, L., Xiu, X.M., Shen, H.Z., Gao, Y.J., Yi, X.X.: Perfect distribution of four-photon \[\chi\] χ-type entangled states over an arbitrary collective noise channel by spatial degree of freedom. Opt. Commun. 308, 304-308 (2013) · <a href="https://doi.org/10.1016/j.optcom.2013.07.047" class="nowrap">doi:10.1016/j.optcom.2013.07.047</a></td> </tr><tr> <td>[73]</td> <td class="space">Xiu, X.M., Dong, L., Shen, H.Z., Gao, Y.J., Yi, X.X.: Two-party quantum privacy comparison with polarization-entangled Bell states and the coherent states. Quantum Inf. Comput. 14, 236-254 (2014)</td> </tr><tr> <td>[74]</td> <td class="space">Xiu, X.M., Li, Q.Y., Dong, L., Shen, H.Z., Li, D., Gao, Y.J., Yi, X.X.: Distributing a multi-photon polarization-entangled state with unitary fidelity via arbitrary collective noise channels. Quantum Inf. Process. 14, 361-372 (2015) · <a href="/1311.81043" class="nowrap">Zbl 1311.81043</a> · <a href="https://doi.org/10.1007/s11128-014-0844-5" class="nowrap">doi:10.1007/s11128-014-0844-5</a></td> </tr><tr> <td>[75]</td> <td class="space">Wang, H.F., Zhu, A.D., Zhang, S.: Physical optimization of quantum error correction circuits with spatially separated quantum dot spins. Opt. Express 21, 12484-12494 (2013) · <a href="https://doi.org/10.1364/OE.21.012484" class="nowrap">doi:10.1364/OE.21.012484</a></td> </tr><tr> <td>[76]</td> <td class="space">Nielsen, A.E.B., Muschik, C.A., Giedke, G., Vollbrecht, K.G.H.: Quantum state engineering, purification, and number-resolved photon detection with high-finesse optical cavities. Phys. Rev. A 81, 043832 (2010) · <a href="https://doi.org/10.1103/PhysRevA.81.043832" class="nowrap">doi:10.1103/PhysRevA.81.043832</a></td> </tr><tr> <td>[77]</td> <td class="space">Shapiro, J.H.: Single-photon Kerr nonlinearities do not help quantum computation. Phys. Rev. A 73, 062305 (2006) · <a href="https://doi.org/10.1103/PhysRevA.73.062305" class="nowrap">doi:10.1103/PhysRevA.73.062305</a></td> </tr><tr> <td>[78]</td> <td class="space">Shapiro, J.H., Razavi, M.: Continuous-time cross-phase modulation and quantum computation. New J. Phys. 9, 16 (2007) · <a href="https://doi.org/10.1088/1367-2630/9/1/016" class="nowrap">doi:10.1088/1367-2630/9/1/016</a></td> </tr><tr> <td>[79]</td> <td class="space">Munro, W.J., Nemoto, K., Spiller, T.P.: Weak nonlinearities: a new route to optical quantum computation. New J. Phys. 7, 137 (2005) · <a href="https://doi.org/10.1088/1367-2630/7/1/137" class="nowrap">doi:10.1088/1367-2630/7/1/137</a></td> </tr><tr> <td>[80]</td> <td class="space">Barrett, S.D., Milburn, G.J.: Quantum-information processing via a lossy bus. Phys. Rev. A 74, 060302(R) (2006) · <a href="https://doi.org/10.1103/PhysRevA.74.060302" class="nowrap">doi:10.1103/PhysRevA.74.060302</a></td> </tr><tr> <td>[81]</td> <td class="space">Lin, Q., Li, J.: Quantum control gates with weak cross-Kerr nonlinearity. Phys. Rev. A 79, 022301 (2009) · <a href="https://doi.org/10.1103/PhysRevA.79.022301" class="nowrap">doi:10.1103/PhysRevA.79.022301</a></td> </tr><tr> <td>[82]</td> <td class="space">Jeong, H.: Quantum computation using weak nonlinearities: robustness against decoherence. Phys. Rev. A 73, 052320 (2006) · <a href="https://doi.org/10.1103/PhysRevA.73.052320" class="nowrap">doi:10.1103/PhysRevA.73.052320</a></td> </tr><tr> <td>[83]</td> <td class="space">Feizpour, A., Xing, X.X., Steinberg, A.M.: Amplifying single-photon nonlinearity using weak measurements. Phys. Rev. Lett. 107, 133603 (2011) · <a href="https://doi.org/10.1103/PhysRevLett.107.133603" class="nowrap">doi:10.1103/PhysRevLett.107.133603</a></td> </tr><tr> <td>[84]</td> <td class="space">Hofmann, H.F., Kojima, K., Takeuchi, S., Sasaki, K.: Optimized phase switching using a single-atom nonlinearity. J. Opt. B 5, 218-221 (2003) · <a href="https://doi.org/10.1088/1464-4266/5/3/304" class="nowrap">doi:10.1088/1464-4266/5/3/304</a></td> </tr><tr> <td>[85]</td> <td class="space">Hoi, I.C., Kockum, A.F., Palomaki, T., Stace, T.M., Fan, B., Tornberg, L., Sathyamoorthy, S.R., Johansson, G., Delsing, P., Wilson, C.M.: Giant cross-Kerr effect for propagating microwaves induced by an artificial atom. Phys. Rev. Lett. 111, 053601 (2013) · <a href="https://doi.org/10.1103/PhysRevLett.111.053601" class="nowrap">doi:10.1103/PhysRevLett.111.053601</a></td> </tr><tr> <td>[86]</td> <td class="space">He, B., Sharypov, A.V., Sheng, J., Simon, C., Xiao, M.: Two-photon dynamics in coherent Rydberg atomic ensemble. Phys. Rev. Lett. 112, 133606 (2014) · <a href="https://doi.org/10.1103/PhysRevLett.112.133606" class="nowrap">doi:10.1103/PhysRevLett.112.133606</a></td> </tr><tr> <td>[87]</td> <td class="space">Feizpour, A., Hallaji, M., Dmochowski, G., Steinberg, A.M.: Observation of the nonlinear phase shift due to single post-selected photons. Nat. Phys. 11, 905-909 (2015) · <a href="https://doi.org/10.1038/nphys3433" class="nowrap">doi:10.1038/nphys3433</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 1338.81104 - zbMATH Open