Projective ring line encompassing two-qubits View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2008-06

AUTHORS

M. Saniga, M. Planat, P. Pracna

ABSTRACT

We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications. More... »

PAGES

905

References to SciGraph publications

  • 2005-03. On distant-isomorphisms of projective lines in AEQUATIONES MATHEMATICAE
  • 2001-09. Generalized Flatland in THE MATHEMATICAL INTELLIGENCER
  • 2007-04. Projective line over the finite quotient ring GF(2)[x]/〈x3 − x〉 and quantum entanglement: Theoretical background in THEORETICAL AND MATHEMATICAL PHYSICS
  • 2000-12. Projective representations i. projective lines over rings in ABHANDLUNGEN AUS DEM MATHEMATISCHEN SEMINAR DER UNIVERSITÄT HAMBURG
  • 2007-05. Projective line over the finite quotient ring GF(2)[x]/〈x3 ™ x〉 and quantum entanglement: The Mermin “magic” square/pentagram in THEORETICAL AND MATHEMATICAL PHYSICS
  • 1998. A Geometrical Picture Book in NONE
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s11232-008-0076-x

    DOI

    http://dx.doi.org/10.1007/s11232-008-0076-x

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1041700739


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Astronomical Institute, Slovak Academy of Sciences, Tatransk\u00e1 Lomnica, Slovak Republic", 
              "id": "http://www.grid.ac/institutes/grid.493212.f", 
              "name": [
                "Astronomical Institute, Slovak Academy of Sciences, Tatransk\u00e1 Lomnica, Slovak Republic"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Saniga", 
            "givenName": "M.", 
            "id": "sg:person.015610617470.96", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015610617470.96"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Institut FEMTO-ST, CNRS, D\u00e9partement LPMO, Besan\u00e7on, France", 
              "id": "http://www.grid.ac/institutes/grid.462068.e", 
              "name": [
                "Institut FEMTO-ST, CNRS, D\u00e9partement LPMO, Besan\u00e7on, France"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Planat", 
            "givenName": "M.", 
            "id": "sg:person.016076407625.27", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016076407625.27"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Heyrovsk\u00fd Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic", 
              "id": "http://www.grid.ac/institutes/grid.418095.1", 
              "name": [
                "Heyrovsk\u00fd Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Pracna", 
            "givenName": "P.", 
            "id": "sg:person.012473732565.40", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012473732565.40"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/s11232-007-0035-y", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1015916706", 
              "https://doi.org/10.1007/s11232-007-0035-y"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s11232-007-0049-5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028110868", 
              "https://doi.org/10.1007/s11232-007-0049-5"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf02940921", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1037915995", 
              "https://doi.org/10.1007/bf02940921"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00010-004-2745-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1013666784", 
              "https://doi.org/10.1007/s00010-004-2745-7"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-1-4419-8526-2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028682858", 
              "https://doi.org/10.1007/978-1-4419-8526-2"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf03024601", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039097670", 
              "https://doi.org/10.1007/bf03024601"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2008-06", 
        "datePublishedReg": "2008-06-01", 
        "description": "We find that the projective line over the (noncommutative) ring of 2\u00d72 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) \u00d7 GF(2) (the \u201cMermin\u201d part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s11232-008-0076-x", 
        "inLanguage": "en", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1327888", 
            "issn": [
              "0040-5779", 
              "2305-3135"
            ], 
            "name": "Theoretical and Mathematical Physics", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "3", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "155"
          }
        ], 
        "keywords": [
          "projective line", 
          "finite-dimensional quantum systems", 
          "algebra of observables", 
          "geometric notions", 
          "commutation relations", 
          "order two", 
          "generalized quadrangles", 
          "quantum systems", 
          "two-qubit system", 
          "operators", 
          "algebra", 
          "operational notion", 
          "remarkable configuration", 
          "disjoint union", 
          "numerous applications", 
          "ring line", 
          "two-qubit", 
          "factorization", 
          "unexpected possibilities", 
          "subconfigurations", 
          "observables", 
          "point", 
          "distant points", 
          "notion", 
          "modeling", 
          "system", 
          "geometry", 
          "neighbors", 
          "quadrangle", 
          "set", 
          "matrix", 
          "applications", 
          "coefficient", 
          "ovoids", 
          "different ways", 
          "two", 
          "subset", 
          "configuration", 
          "way", 
          "lines", 
          "relation", 
          "ring", 
          "possibility", 
          "new prospects", 
          "Union", 
          "prospects", 
          "group", 
          "findings", 
          "relevant subconfiguration", 
          "ring geometric notions", 
          "respective operational notions", 
          "algebro-geometric modeling", 
          "Projective ring line"
        ], 
        "name": "Projective ring line encompassing two-qubits", 
        "pagination": "905", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1041700739"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s11232-008-0076-x"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s11232-008-0076-x", 
          "https://app.dimensions.ai/details/publication/pub.1041700739"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2021-12-01T19:20", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20211201/entities/gbq_results/article/article_456.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s11232-008-0076-x"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s11232-008-0076-x'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s11232-008-0076-x'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s11232-008-0076-x'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s11232-008-0076-x'


     

    This table displays all metadata directly associated to this object as RDF triples.

    155 TRIPLES      22 PREDICATES      85 URIs      71 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s11232-008-0076-x schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N0d3c2a0304794916aa30a96f99df39ca
    4 schema:citation sg:pub.10.1007/978-1-4419-8526-2
    5 sg:pub.10.1007/bf02940921
    6 sg:pub.10.1007/bf03024601
    7 sg:pub.10.1007/s00010-004-2745-7
    8 sg:pub.10.1007/s11232-007-0035-y
    9 sg:pub.10.1007/s11232-007-0049-5
    10 schema:datePublished 2008-06
    11 schema:datePublishedReg 2008-06-01
    12 schema:description We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications.
    13 schema:genre article
    14 schema:inLanguage en
    15 schema:isAccessibleForFree true
    16 schema:isPartOf N44d25a3911854e44ba2f845da854beda
    17 Na034205e60b84081a778d5ae7b84a54c
    18 sg:journal.1327888
    19 schema:keywords Projective ring line
    20 Union
    21 algebra
    22 algebra of observables
    23 algebro-geometric modeling
    24 applications
    25 coefficient
    26 commutation relations
    27 configuration
    28 different ways
    29 disjoint union
    30 distant points
    31 factorization
    32 findings
    33 finite-dimensional quantum systems
    34 generalized quadrangles
    35 geometric notions
    36 geometry
    37 group
    38 lines
    39 matrix
    40 modeling
    41 neighbors
    42 new prospects
    43 notion
    44 numerous applications
    45 observables
    46 operational notion
    47 operators
    48 order two
    49 ovoids
    50 point
    51 possibility
    52 projective line
    53 prospects
    54 quadrangle
    55 quantum systems
    56 relation
    57 relevant subconfiguration
    58 remarkable configuration
    59 respective operational notions
    60 ring
    61 ring geometric notions
    62 ring line
    63 set
    64 subconfigurations
    65 subset
    66 system
    67 two
    68 two-qubit
    69 two-qubit system
    70 unexpected possibilities
    71 way
    72 schema:name Projective ring line encompassing two-qubits
    73 schema:pagination 905
    74 schema:productId N45a77c14ba6e4999b983ce42b284f678
    75 Nee4438ecef204bf5a4b4393cf0d5171a
    76 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041700739
    77 https://doi.org/10.1007/s11232-008-0076-x
    78 schema:sdDatePublished 2021-12-01T19:20
    79 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    80 schema:sdPublisher Nc602553e8bf74e59a36dd812e5eb1afa
    81 schema:url https://doi.org/10.1007/s11232-008-0076-x
    82 sgo:license sg:explorer/license/
    83 sgo:sdDataset articles
    84 rdf:type schema:ScholarlyArticle
    85 N0d3c2a0304794916aa30a96f99df39ca rdf:first sg:person.015610617470.96
    86 rdf:rest Nf665253034694ff986d2847f78ea4fce
    87 N0ebc2d958f604408a4af07e58ca1c984 rdf:first sg:person.012473732565.40
    88 rdf:rest rdf:nil
    89 N44d25a3911854e44ba2f845da854beda schema:issueNumber 3
    90 rdf:type schema:PublicationIssue
    91 N45a77c14ba6e4999b983ce42b284f678 schema:name doi
    92 schema:value 10.1007/s11232-008-0076-x
    93 rdf:type schema:PropertyValue
    94 Na034205e60b84081a778d5ae7b84a54c schema:volumeNumber 155
    95 rdf:type schema:PublicationVolume
    96 Nc602553e8bf74e59a36dd812e5eb1afa schema:name Springer Nature - SN SciGraph project
    97 rdf:type schema:Organization
    98 Nee4438ecef204bf5a4b4393cf0d5171a schema:name dimensions_id
    99 schema:value pub.1041700739
    100 rdf:type schema:PropertyValue
    101 Nf665253034694ff986d2847f78ea4fce rdf:first sg:person.016076407625.27
    102 rdf:rest N0ebc2d958f604408a4af07e58ca1c984
    103 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    104 schema:name Mathematical Sciences
    105 rdf:type schema:DefinedTerm
    106 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    107 schema:name Pure Mathematics
    108 rdf:type schema:DefinedTerm
    109 sg:journal.1327888 schema:issn 0040-5779
    110 2305-3135
    111 schema:name Theoretical and Mathematical Physics
    112 schema:publisher Springer Nature
    113 rdf:type schema:Periodical
    114 sg:person.012473732565.40 schema:affiliation grid-institutes:grid.418095.1
    115 schema:familyName Pracna
    116 schema:givenName P.
    117 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012473732565.40
    118 rdf:type schema:Person
    119 sg:person.015610617470.96 schema:affiliation grid-institutes:grid.493212.f
    120 schema:familyName Saniga
    121 schema:givenName M.
    122 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015610617470.96
    123 rdf:type schema:Person
    124 sg:person.016076407625.27 schema:affiliation grid-institutes:grid.462068.e
    125 schema:familyName Planat
    126 schema:givenName M.
    127 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016076407625.27
    128 rdf:type schema:Person
    129 sg:pub.10.1007/978-1-4419-8526-2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028682858
    130 https://doi.org/10.1007/978-1-4419-8526-2
    131 rdf:type schema:CreativeWork
    132 sg:pub.10.1007/bf02940921 schema:sameAs https://app.dimensions.ai/details/publication/pub.1037915995
    133 https://doi.org/10.1007/bf02940921
    134 rdf:type schema:CreativeWork
    135 sg:pub.10.1007/bf03024601 schema:sameAs https://app.dimensions.ai/details/publication/pub.1039097670
    136 https://doi.org/10.1007/bf03024601
    137 rdf:type schema:CreativeWork
    138 sg:pub.10.1007/s00010-004-2745-7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1013666784
    139 https://doi.org/10.1007/s00010-004-2745-7
    140 rdf:type schema:CreativeWork
    141 sg:pub.10.1007/s11232-007-0035-y schema:sameAs https://app.dimensions.ai/details/publication/pub.1015916706
    142 https://doi.org/10.1007/s11232-007-0035-y
    143 rdf:type schema:CreativeWork
    144 sg:pub.10.1007/s11232-007-0049-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028110868
    145 https://doi.org/10.1007/s11232-007-0049-5
    146 rdf:type schema:CreativeWork
    147 grid-institutes:grid.418095.1 schema:alternateName Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic
    148 schema:name Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Prague, Czech Republic
    149 rdf:type schema:Organization
    150 grid-institutes:grid.462068.e schema:alternateName Institut FEMTO-ST, CNRS, Département LPMO, Besançon, France
    151 schema:name Institut FEMTO-ST, CNRS, Département LPMO, Besançon, France
    152 rdf:type schema:Organization
    153 grid-institutes:grid.493212.f schema:alternateName Astronomical Institute, Slovak Academy of Sciences, Tatranská Lomnica, Slovak Republic
    154 schema:name Astronomical Institute, Slovak Academy of Sciences, Tatranská Lomnica, Slovak Republic
    155 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...