Shrinking of operators in quantum error correction and AdS/CFT
Abstract We first show that a class of operators acting on a given bipartite pure state on ℋ A ⊗ ℋ B can shrink its supports on ℋ A ⊗ ℋ B to only ℋ A or ℋ B while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes again...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-12-01
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| Series: | Journal of High Energy Physics |
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| Online Access: | https://doi.org/10.1007/JHEP12(2019)128 |
| _version_ | 1819078172914745344 |
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| author | Hayato Hirai |
| author_facet | Hayato Hirai |
| author_sort | Hayato Hirai |
| collection | DOAJ |
| description | Abstract We first show that a class of operators acting on a given bipartite pure state on ℋ A ⊗ ℋ B can shrink its supports on ℋ A ⊗ ℋ B to only ℋ A or ℋ B while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The “sub- algebra code with complementary recovery” introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators. |
| first_indexed | 2024-12-21T19:08:53Z |
| format | Article |
| id | doaj.art-e733e27d37174a09b0ec572717e9c039 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-12-21T19:08:53Z |
| publishDate | 2019-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-e733e27d37174a09b0ec572717e9c0392022-12-21T18:53:16ZengSpringerOpenJournal of High Energy Physics1029-84792019-12-0120191213810.1007/JHEP12(2019)128Shrinking of operators in quantum error correction and AdS/CFTHayato Hirai0Department of Physics, Osaka UniversityAbstract We first show that a class of operators acting on a given bipartite pure state on ℋ A ⊗ ℋ B can shrink its supports on ℋ A ⊗ ℋ B to only ℋ A or ℋ B while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The “sub- algebra code with complementary recovery” introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators.https://doi.org/10.1007/JHEP12(2019)128AdS-CFT CorrespondenceGauge-gravity correspondence |
| spellingShingle | Hayato Hirai Shrinking of operators in quantum error correction and AdS/CFT Journal of High Energy Physics AdS-CFT Correspondence Gauge-gravity correspondence |
| title | Shrinking of operators in quantum error correction and AdS/CFT |
| title_full | Shrinking of operators in quantum error correction and AdS/CFT |
| title_fullStr | Shrinking of operators in quantum error correction and AdS/CFT |
| title_full_unstemmed | Shrinking of operators in quantum error correction and AdS/CFT |
| title_short | Shrinking of operators in quantum error correction and AdS/CFT |
| title_sort | shrinking of operators in quantum error correction and ads cft |
| topic | AdS-CFT Correspondence Gauge-gravity correspondence |
| url | https://doi.org/10.1007/JHEP12(2019)128 |
| work_keys_str_mv | AT hayatohirai shrinkingofoperatorsinquantumerrorcorrectionandadscft |