Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric Extensions
Symmetric extensions are essential in quantum mechanics, providing a lens through which to investigate the correlations of entangled quantum systems and to address challenges like the quantum marginal problem. Though semi-definite programming (SDP) is a recognized method for handling symmetric exten...
| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/10/1425 |
| _version_ | 1797573927623458816 |
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| author | Youning Li Chao Zhang Shi-Yao Hou Zipeng Wu Xuanran Zhu Bei Zeng |
| author_facet | Youning Li Chao Zhang Shi-Yao Hou Zipeng Wu Xuanran Zhu Bei Zeng |
| author_sort | Youning Li |
| collection | DOAJ |
| description | Symmetric extensions are essential in quantum mechanics, providing a lens through which to investigate the correlations of entangled quantum systems and to address challenges like the quantum marginal problem. Though semi-definite programming (SDP) is a recognized method for handling symmetric extensions, it struggles with computational constraints, especially due to the large real parameters in generalized qudit systems. In this study, we introduce an approach that adeptly leverages permutation symmetry. By fine-tuning the SDP problem for detecting <i>k</i>-symmetric extensions, our method markedly diminishes the searching space dimensionality and trims the number of parameters essential for positive-definiteness tests. This leads to an algorithmic enhancement, reducing the complexity from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>d</mi><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>k</mi><msup><mi>d</mi><mn>2</mn></msup></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in the qudit <i>k</i>-symmetric extension scenario. Additionally, our approach streamlines the process of verifying the positive definiteness of the results. These advancements pave the way for deeper insights into quantum correlations, highlighting potential avenues for refined research and innovations in quantum information theory. |
| first_indexed | 2024-03-10T21:16:57Z |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T21:16:57Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-c1f3489bb0424edd95d11b5eef57d0672023-11-19T16:24:42ZengMDPI AGEntropy1099-43002023-10-012510142510.3390/e25101425Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric ExtensionsYouning Li0Chao Zhang1Shi-Yao Hou2Zipeng Wu3Xuanran Zhu4Bei Zeng5College of Science, China Agricultural University, Beijing 100080, ChinaDepartment of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, ChinaCollege of Physics and Electronic Engineering, Center for Computational Sciences, Sichuan Normal University, Chengdu 610068, ChinaDepartment of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, ChinaDepartment of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, ChinaDepartment of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, ChinaSymmetric extensions are essential in quantum mechanics, providing a lens through which to investigate the correlations of entangled quantum systems and to address challenges like the quantum marginal problem. Though semi-definite programming (SDP) is a recognized method for handling symmetric extensions, it struggles with computational constraints, especially due to the large real parameters in generalized qudit systems. In this study, we introduce an approach that adeptly leverages permutation symmetry. By fine-tuning the SDP problem for detecting <i>k</i>-symmetric extensions, our method markedly diminishes the searching space dimensionality and trims the number of parameters essential for positive-definiteness tests. This leads to an algorithmic enhancement, reducing the complexity from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>d</mi><mrow><mn>2</mn><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>k</mi><msup><mi>d</mi><mn>2</mn></msup></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> in the qudit <i>k</i>-symmetric extension scenario. Additionally, our approach streamlines the process of verifying the positive definiteness of the results. These advancements pave the way for deeper insights into quantum correlations, highlighting potential avenues for refined research and innovations in quantum information theory.https://www.mdpi.com/1099-4300/25/10/1425symmetric extensionirreducible representation of <i>su</i>(<i>n</i>)permutation symmetrycomputational complexity |
| spellingShingle | Youning Li Chao Zhang Shi-Yao Hou Zipeng Wu Xuanran Zhu Bei Zeng Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric Extensions Entropy symmetric extension irreducible representation of <i>su</i>(<i>n</i>) permutation symmetry computational complexity |
| title | Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric Extensions |
| title_full | Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric Extensions |
| title_fullStr | Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric Extensions |
| title_full_unstemmed | Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric Extensions |
| title_short | Tapping into Permutation Symmetry for Improved Detection of <i>k</i>-Symmetric Extensions |
| title_sort | tapping into permutation symmetry for improved detection of i k i symmetric extensions |
| topic | symmetric extension irreducible representation of <i>su</i>(<i>n</i>) permutation symmetry computational complexity |
| url | https://www.mdpi.com/1099-4300/25/10/1425 |
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