Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps
Abstract Legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Yet, non-completely positive maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. Here, we in...
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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2024-12-01
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| Series: | npj Quantum Information |
| Online Access: | https://doi.org/10.1038/s41534-024-00949-z |
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| author | Fuchuan Wei Zhenhuan Liu Guoding Liu Zizhao Han Dong-Ling Deng Zhengwei Liu |
| author_facet | Fuchuan Wei Zhenhuan Liu Guoding Liu Zizhao Han Dong-Ling Deng Zhengwei Liu |
| author_sort | Fuchuan Wei |
| collection | DOAJ |
| description | Abstract Legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Yet, non-completely positive maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. Here, we introduce the Hermitian-preserving map exponentiation algorithm, which can effectively simulate the action of an arbitrary Hermitian-preserving map by exponentiating its output, $${\mathcal{N}}(\rho )$$ N ( ρ ) , into a quantum process, $${e}^{-i{\mathcal{N}}(\rho )t}$$ e − i N ( ρ ) t . We analyze the sample complexity of this algorithm and prove its optimality in certain cases. Utilizing positive but not completely positive maps, this algorithm provides exponential speedups in entanglement detection and quantification compared to protocols based on single-copy operations. In addition, it facilitates the encoding-free recovery of noiseless quantum states from multiple noisy ones by simulating the inverse map of the corresponding noise channel, providing a new approach to handling quantum noises. This algorithm acts as a building block of large-scale quantum algorithms and presents a pathway for exploring potential quantum speedups across a wide range of information-processing tasks. |
| format | Article |
| id | doaj-art-1b9ba2250708478fb26c1164a7abe584 |
| institution | Kabale University |
| issn | 2056-6387 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | npj Quantum Information |
| spelling | doaj-art-1b9ba2250708478fb26c1164a7abe5842025-01-05T12:42:32ZengNature Portfolionpj Quantum Information2056-63872024-12-0110111010.1038/s41534-024-00949-zSimulating non-completely positive actions via exponentiation of Hermitian-preserving mapsFuchuan Wei0Zhenhuan Liu1Guoding Liu2Zizhao Han3Dong-Ling Deng4Zhengwei Liu5Yau Mathematical Sciences Center and Department of Mathematics, Tsinghua UniversityCenter for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua UniversityCenter for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua UniversityCenter for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua UniversityCenter for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua UniversityYau Mathematical Sciences Center and Department of Mathematics, Tsinghua UniversityAbstract Legitimate quantum operations must adhere to principles of quantum mechanics, particularly the requirements of complete positivity and trace preservation. Yet, non-completely positive maps, especially Hermitian-preserving maps, play a crucial role in quantum information science. Here, we introduce the Hermitian-preserving map exponentiation algorithm, which can effectively simulate the action of an arbitrary Hermitian-preserving map by exponentiating its output, $${\mathcal{N}}(\rho )$$ N ( ρ ) , into a quantum process, $${e}^{-i{\mathcal{N}}(\rho )t}$$ e − i N ( ρ ) t . We analyze the sample complexity of this algorithm and prove its optimality in certain cases. Utilizing positive but not completely positive maps, this algorithm provides exponential speedups in entanglement detection and quantification compared to protocols based on single-copy operations. In addition, it facilitates the encoding-free recovery of noiseless quantum states from multiple noisy ones by simulating the inverse map of the corresponding noise channel, providing a new approach to handling quantum noises. This algorithm acts as a building block of large-scale quantum algorithms and presents a pathway for exploring potential quantum speedups across a wide range of information-processing tasks.https://doi.org/10.1038/s41534-024-00949-z |
| spellingShingle | Fuchuan Wei Zhenhuan Liu Guoding Liu Zizhao Han Dong-Ling Deng Zhengwei Liu Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps npj Quantum Information |
| title | Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps |
| title_full | Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps |
| title_fullStr | Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps |
| title_full_unstemmed | Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps |
| title_short | Simulating non-completely positive actions via exponentiation of Hermitian-preserving maps |
| title_sort | simulating non completely positive actions via exponentiation of hermitian preserving maps |
| url | https://doi.org/10.1038/s41534-024-00949-z |
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