The compositional structure of multipartite quantum entanglement
While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a long-standing open problem in quantum computer science. In this paper we expose the...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Springer
2010
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| _version_ | 1826305535762235392 |
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| author | Coecke, B Kissinger, A |
| author_facet | Coecke, B Kissinger, A |
| author_sort | Coecke, B |
| collection | OXFORD |
| description | While multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a long-standing open problem in quantum computer science. In this paper we expose the algebraic and graphical structure of the GHZ-state and the W-state, as well as a purely graphical distinction that characterises the behaviours of these states. In turn, this structure yields a compositional graphical model for expressing general multipartite states. We identify those states, named Frobenius states, which canonically induce an algebraic structure, namely the structure of a commutative Frobenius algebra (CFA). We show that all SLOCC-maximal tripartite qubit states are locally equivalent to Frobenius states. Those that are SLOCC-equivalent to the GHZ-state induce special commutative Frobenius algebras, while those that are SLOCC-equivalent to the W-state induce what we call anti-special commutative Frobenius algebras. From the SLOCC-classification of tripartite qubit states follows a representation theorem for two dimensional CFAs. Together, a GHZ and a W Frobenius state form the primitives of a graphical calculus. This calculus is expressive enough to generate and reason about arbitrary multipartite states, which are obtained by "composing" the GHZ- and W-states, giving rise to a rich graphical paradigm for general multipartite entanglement. |
| first_indexed | 2024-03-07T06:34:19Z |
| format | Journal article |
| id | oxford-uuid:f713ea67-18cc-49ae-a315-a7738ea76663 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:34:19Z |
| publishDate | 2010 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:f713ea67-18cc-49ae-a315-a7738ea766632022-03-27T12:39:58ZThe compositional structure of multipartite quantum entanglementJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f713ea67-18cc-49ae-a315-a7738ea76663EnglishSymplectic Elements at OxfordSpringer2010Coecke, BKissinger, AWhile multipartite quantum states constitute a (if not the) key resource for quantum computations and protocols, obtaining a high-level, structural understanding of entanglement involving arbitrarily many qubits is a long-standing open problem in quantum computer science. In this paper we expose the algebraic and graphical structure of the GHZ-state and the W-state, as well as a purely graphical distinction that characterises the behaviours of these states. In turn, this structure yields a compositional graphical model for expressing general multipartite states. We identify those states, named Frobenius states, which canonically induce an algebraic structure, namely the structure of a commutative Frobenius algebra (CFA). We show that all SLOCC-maximal tripartite qubit states are locally equivalent to Frobenius states. Those that are SLOCC-equivalent to the GHZ-state induce special commutative Frobenius algebras, while those that are SLOCC-equivalent to the W-state induce what we call anti-special commutative Frobenius algebras. From the SLOCC-classification of tripartite qubit states follows a representation theorem for two dimensional CFAs. Together, a GHZ and a W Frobenius state form the primitives of a graphical calculus. This calculus is expressive enough to generate and reason about arbitrary multipartite states, which are obtained by "composing" the GHZ- and W-states, giving rise to a rich graphical paradigm for general multipartite entanglement. |
| spellingShingle | Coecke, B Kissinger, A The compositional structure of multipartite quantum entanglement |
| title | The compositional structure of multipartite quantum entanglement |
| title_full | The compositional structure of multipartite quantum entanglement |
| title_fullStr | The compositional structure of multipartite quantum entanglement |
| title_full_unstemmed | The compositional structure of multipartite quantum entanglement |
| title_short | The compositional structure of multipartite quantum entanglement |
| title_sort | compositional structure of multipartite quantum entanglement |
| work_keys_str_mv | AT coeckeb thecompositionalstructureofmultipartitequantumentanglement AT kissingera thecompositionalstructureofmultipartitequantumentanglement AT coeckeb compositionalstructureofmultipartitequantumentanglement AT kissingera compositionalstructureofmultipartitequantumentanglement |