{"title":"混合态量子异常与多部纠缠","authors":"Leonardo A. Lessa, Meng Cheng, Chong Wang","doi":"10.1103/physrevx.15.011069","DOIUrl":null,"url":null,"abstract":"Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here, we explore a surprising connection between mixed-state entanglement and ’t Hooft anomaly. More specifically, we consider lattice systems in d</a:mi></a:math> space dimensions with anomalous symmetry <c:math xmlns:c=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><c:mi>G</c:mi></c:math> where the anomaly is characterized by an invariant in the group cohomology <e:math xmlns:e=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><e:msup><e:mi>H</e:mi><e:mrow><e:mi>d</e:mi><e:mo>+</e:mo><e:mn>2</e:mn></e:mrow></e:msup><e:mo stretchy=\"false\">[</e:mo><e:mi>G</e:mi><e:mo>,</e:mo><e:mi>U</e:mi><e:mo stretchy=\"false\">(</e:mo><e:mn>1</e:mn><e:mo stretchy=\"false\">)</e:mo><e:mo stretchy=\"false\">]</e:mo></e:math>. We show that any mixed state <k:math xmlns:k=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><k:mi>ρ</k:mi></k:math> that is strongly symmetric under <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:mi>G</m:mi></m:math>, in the sense that <o:math xmlns:o=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><o:mi>G</o:mi><o:mi>ρ</o:mi><o:mo>∝</o:mo><o:mi>ρ</o:mi></o:math> is necessarily (<q:math xmlns:q=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><q:mrow><q:mi>d</q:mi><q:mo>+</q:mo><q:mn>2</q:mn></q:mrow></q:math>)-nonseparable, i.e., is not the mixture of tensor products of <s:math xmlns:s=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><s:mi>d</s:mi><s:mo>+</s:mo><s:mn>2</s:mn></s:math> states in the Hilbert space. Furthermore, such states cannot be prepared from any (<u:math xmlns:u=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><u:mrow><u:mi>d</u:mi><u:mo>+</u:mo><u:mn>2</u:mn></u:mrow></u:math>)-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in <w:math xmlns:w=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><w:mi>d</w:mi><w:mo>≤</w:mo><w:mn>1</w:mn></w:math> and plausibility arguments in <y:math xmlns:y=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><y:mi>d</y:mi><y:mo>></y:mo><y:mn>1</y:mn></y:math>. The anomaly-nonseparability connection, thus, allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in d</ab:mi>=</ab:mo>1</ab:mn></ab:math> we find an example of quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze a mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly, in general, does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20161,"journal":{"name":"Physical Review X","volume":"209 1","pages":""},"PeriodicalIF":15.7000,"publicationDate":"2025-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Mixed-State Quantum Anomaly and Multipartite Entanglement\",\"authors\":\"Leonardo A. Lessa, Meng Cheng, Chong Wang\",\"doi\":\"10.1103/physrevx.15.011069\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here, we explore a surprising connection between mixed-state entanglement and ’t Hooft anomaly. More specifically, we consider lattice systems in d</a:mi></a:math> space dimensions with anomalous symmetry <c:math xmlns:c=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><c:mi>G</c:mi></c:math> where the anomaly is characterized by an invariant in the group cohomology <e:math xmlns:e=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><e:msup><e:mi>H</e:mi><e:mrow><e:mi>d</e:mi><e:mo>+</e:mo><e:mn>2</e:mn></e:mrow></e:msup><e:mo stretchy=\\\"false\\\">[</e:mo><e:mi>G</e:mi><e:mo>,</e:mo><e:mi>U</e:mi><e:mo stretchy=\\\"false\\\">(</e:mo><e:mn>1</e:mn><e:mo stretchy=\\\"false\\\">)</e:mo><e:mo stretchy=\\\"false\\\">]</e:mo></e:math>. We show that any mixed state <k:math xmlns:k=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><k:mi>ρ</k:mi></k:math> that is strongly symmetric under <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><m:mi>G</m:mi></m:math>, in the sense that <o:math xmlns:o=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><o:mi>G</o:mi><o:mi>ρ</o:mi><o:mo>∝</o:mo><o:mi>ρ</o:mi></o:math> is necessarily (<q:math xmlns:q=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><q:mrow><q:mi>d</q:mi><q:mo>+</q:mo><q:mn>2</q:mn></q:mrow></q:math>)-nonseparable, i.e., is not the mixture of tensor products of <s:math xmlns:s=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><s:mi>d</s:mi><s:mo>+</s:mo><s:mn>2</s:mn></s:math> states in the Hilbert space. Furthermore, such states cannot be prepared from any (<u:math xmlns:u=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><u:mrow><u:mi>d</u:mi><u:mo>+</u:mo><u:mn>2</u:mn></u:mrow></u:math>)-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in <w:math xmlns:w=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><w:mi>d</w:mi><w:mo>≤</w:mo><w:mn>1</w:mn></w:math> and plausibility arguments in <y:math xmlns:y=\\\"http://www.w3.org/1998/Math/MathML\\\" display=\\\"inline\\\"><y:mi>d</y:mi><y:mo>></y:mo><y:mn>1</y:mn></y:math>. The anomaly-nonseparability connection, thus, allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in d</ab:mi>=</ab:mo>1</ab:mn></ab:math> we find an example of quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze a mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly, in general, does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>\",\"PeriodicalId\":20161,\"journal\":{\"name\":\"Physical Review X\",\"volume\":\"209 1\",\"pages\":\"\"},\"PeriodicalIF\":15.7000,\"publicationDate\":\"2025-03-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Physical Review X\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://doi.org/10.1103/physrevx.15.011069\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review X","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevx.15.011069","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
Mixed-State Quantum Anomaly and Multipartite Entanglement
Quantum entanglement measures of many-body states have been increasingly useful to characterize phases of matter. Here, we explore a surprising connection between mixed-state entanglement and ’t Hooft anomaly. More specifically, we consider lattice systems in d space dimensions with anomalous symmetry G where the anomaly is characterized by an invariant in the group cohomology Hd+2[G,U(1)]. We show that any mixed state ρ that is strongly symmetric under G, in the sense that Gρ∝ρ is necessarily (d+2)-nonseparable, i.e., is not the mixture of tensor products of d+2 states in the Hilbert space. Furthermore, such states cannot be prepared from any (d+2)-separable states using finite-depth local quantum channels, so the nonseparability is long-ranged in nature. We provide proof of these results in d≤1 and plausibility arguments in d>1. The anomaly-nonseparability connection, thus, allows us to generate simple examples of mixed states with nontrivial long-ranged multipartite entanglement. In particular, in d=1 we find an example of quantum phase, in the sense that states in this phase cannot be two-way connected to any pure state through finite-depth local quantum channels. We also analyze a mixed anomaly involving both strong and weak symmetries, including systems constrained by the Lieb-Schultz-Mattis type of anomaly. We find that, while strong-weak mixed anomaly, in general, does not constrain quantum entanglement, it does constrain long-range correlations of mixed states in nontrivial ways. Namely, such states are not symmetrically invertible and not gapped Markovian, generalizing familiar properties of anomalous pure states. Published by the American Physical Society2025
期刊介绍:
Physical Review X (PRX) stands as an exclusively online, fully open-access journal, emphasizing innovation, quality, and enduring impact in the scientific content it disseminates. Devoted to showcasing a curated selection of papers from pure, applied, and interdisciplinary physics, PRX aims to feature work with the potential to shape current and future research while leaving a lasting and profound impact in their respective fields. Encompassing the entire spectrum of physics subject areas, PRX places a special focus on groundbreaking interdisciplinary research with broad-reaching influence.