Sang-Jun Park, Yeong-Gwang Jung, Jeongeun Park, Sang-Gyun Youn
{"title":"群对称下纠缠检测的通用框架","authors":"Sang-Jun Park, Yeong-Gwang Jung, Jeongeun Park, Sang-Gyun Youn","doi":"10.1088/1751-8121/ad6413","DOIUrl":null,"url":null,"abstract":"One of the fundamental questions in quantum information theory is determining entanglement of quantum states, which is generally an NP-hard problem. In this paper, we prove that all PPT <inline-formula>\n<tex-math><?CDATA $(\\overline{\\pi}_A\\otimes \\pi_B)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mover><mml:mi>π</mml:mi><mml:mo accent=\"true\">―</mml:mo></mml:mover><mml:mi>A</mml:mi></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ieqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>-invariant quantum states are separable if and only if all extremal unital positive <inline-formula>\n<tex-math><?CDATA $(\\pi_B,\\pi_A)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ieqn2.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>-covariant maps are decomposable where <inline-formula>\n<tex-math><?CDATA $\\pi_A,\\pi_B$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:msub><mml:mi>π</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ieqn3.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> are unitary representations of a compact group and <italic toggle=\"yes\">π</italic>\n<sub>\n<italic toggle=\"yes\">A</italic>\n</sub> is irreducible. Moreover, an extremal unital positive <inline-formula>\n<tex-math><?CDATA $(\\pi_B,\\pi_A)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ieqn4.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>-covariant map <inline-formula>\n<tex-math><?CDATA $\\mathcal{L}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ieqn5.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is decomposable if and only if <inline-formula>\n<tex-math><?CDATA $\\mathcal{L}$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ieqn6.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form <inline-formula>\n<tex-math><?CDATA $ \\Phi\\left(\\rho\\right) = a\\frac{\\textrm{Tr}\\left(\\rho\\right)}{d}\\textrm{Id}_d+ b\\rho+c\\rho^T+\\left(1-a-b-c\\right)\\textrm{diag}\\left(\\rho\\right)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mtable columnalign=\"left\" displaystyle=\"true\"><mml:mtr><mml:mtd><mml:mi mathvariant=\"normal\">Φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mfrac><mml:mrow><mml:mtext>Tr</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>d</mml:mi></mml:mfrac><mml:msub><mml:mtext>Id</mml:mtext><mml:mi>d</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mi>ρ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mi>ρ</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>a</mml:mi><mml:mo>−</mml:mo><mml:mi>b</mml:mi><mml:mo>−</mml:mo><mml:mi>c</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>diag</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ueqn1.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula> are entanglement-breaking, and that there is no <italic toggle=\"yes\">A</italic>-<italic toggle=\"yes\">BC</italic> PPT-entangled <inline-formula>\n<tex-math><?CDATA $(U\\otimes \\overline{U}\\otimes U)$?></tex-math>\n<mml:math overflow=\"scroll\"><mml:mrow><mml:mo stretchy=\"false\">(</mml:mo><mml:mi>U</mml:mi><mml:mo>⊗</mml:mo><mml:mover><mml:mi>U</mml:mi><mml:mo accent=\"true\">―</mml:mo></mml:mover><mml:mo>⊗</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy=\"false\">)</mml:mo></mml:mrow></mml:math>\n<inline-graphic xlink:href=\"aad6413ieqn7.gif\" xlink:type=\"simple\"></inline-graphic>\n</inline-formula>-invariant tripartite quantum state. The former strengthens some conclusions in (Vollbrecht and Werner 2001 <italic toggle=\"yes\">Phys. Rev.</italic> A <bold>64</bold> 062307; Kopszak <italic toggle=\"yes\">et al</italic> 2020 <italic toggle=\"yes\">J. Phys. A: Math. Theor.</italic>\n<bold>53</bold> 395306), and the latter resolves some open questions raised in (Collins <italic toggle=\"yes\">et al</italic> 2018 <italic toggle=\"yes\">Linear Algebra Appl.</italic>\n<bold>555</bold> 398–411).","PeriodicalId":16763,"journal":{"name":"Journal of Physics A: Mathematical and Theoretical","volume":"51 1","pages":""},"PeriodicalIF":2.0000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A universal framework for entanglement detection under group symmetry\",\"authors\":\"Sang-Jun Park, Yeong-Gwang Jung, Jeongeun Park, Sang-Gyun Youn\",\"doi\":\"10.1088/1751-8121/ad6413\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"One of the fundamental questions in quantum information theory is determining entanglement of quantum states, which is generally an NP-hard problem. In this paper, we prove that all PPT <inline-formula>\\n<tex-math><?CDATA $(\\\\overline{\\\\pi}_A\\\\otimes \\\\pi_B)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:msub><mml:mover><mml:mi>π</mml:mi><mml:mo accent=\\\"true\\\">―</mml:mo></mml:mover><mml:mi>A</mml:mi></mml:msub><mml:mo>⊗</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ieqn1.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>-invariant quantum states are separable if and only if all extremal unital positive <inline-formula>\\n<tex-math><?CDATA $(\\\\pi_B,\\\\pi_A)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ieqn2.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>-covariant maps are decomposable where <inline-formula>\\n<tex-math><?CDATA $\\\\pi_A,\\\\pi_B$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:msub><mml:mi>π</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ieqn3.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> are unitary representations of a compact group and <italic toggle=\\\"yes\\\">π</italic>\\n<sub>\\n<italic toggle=\\\"yes\\\">A</italic>\\n</sub> is irreducible. Moreover, an extremal unital positive <inline-formula>\\n<tex-math><?CDATA $(\\\\pi_B,\\\\pi_A)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>B</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>π</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ieqn4.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>-covariant map <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{L}$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ieqn5.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> is decomposable if and only if <inline-formula>\\n<tex-math><?CDATA $\\\\mathcal{L}$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mrow><mml:mi>L</mml:mi></mml:mrow></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ieqn6.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form <inline-formula>\\n<tex-math><?CDATA $ \\\\Phi\\\\left(\\\\rho\\\\right) = a\\\\frac{\\\\textrm{Tr}\\\\left(\\\\rho\\\\right)}{d}\\\\textrm{Id}_d+ b\\\\rho+c\\\\rho^T+\\\\left(1-a-b-c\\\\right)\\\\textrm{diag}\\\\left(\\\\rho\\\\right)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mtable columnalign=\\\"left\\\" displaystyle=\\\"true\\\"><mml:mtr><mml:mtd><mml:mi mathvariant=\\\"normal\\\">Φ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:mfrac><mml:mrow><mml:mtext>Tr</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mi>d</mml:mi></mml:mfrac><mml:msub><mml:mtext>Id</mml:mtext><mml:mi>d</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mi>ρ</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi><mml:msup><mml:mi>ρ</mml:mi><mml:mi>T</mml:mi></mml:msup><mml:mo>+</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>a</mml:mi><mml:mo>−</mml:mo><mml:mi>b</mml:mi><mml:mo>−</mml:mo><mml:mi>c</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mtext>diag</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mi>ρ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ueqn1.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula> are entanglement-breaking, and that there is no <italic toggle=\\\"yes\\\">A</italic>-<italic toggle=\\\"yes\\\">BC</italic> PPT-entangled <inline-formula>\\n<tex-math><?CDATA $(U\\\\otimes \\\\overline{U}\\\\otimes U)$?></tex-math>\\n<mml:math overflow=\\\"scroll\\\"><mml:mrow><mml:mo stretchy=\\\"false\\\">(</mml:mo><mml:mi>U</mml:mi><mml:mo>⊗</mml:mo><mml:mover><mml:mi>U</mml:mi><mml:mo accent=\\\"true\\\">―</mml:mo></mml:mover><mml:mo>⊗</mml:mo><mml:mi>U</mml:mi><mml:mo stretchy=\\\"false\\\">)</mml:mo></mml:mrow></mml:math>\\n<inline-graphic xlink:href=\\\"aad6413ieqn7.gif\\\" xlink:type=\\\"simple\\\"></inline-graphic>\\n</inline-formula>-invariant tripartite quantum state. 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引用次数: 0
摘要
量子信息论的基本问题之一是确定量子态的纠缠，这通常是一个 NP 难问题。在本文中，我们证明了所有 PPT (π-A⊗πB)-invariant 量子态都是可分的，当且仅当所有极值单元正 (πB,πA)-covariant 映射都是可分解的，其中 πA,πB 是紧凑群的单元表示，且 πA 是不可还原的。此外，当且仅当 L 是完全正或完全共正时，极值单元正 (πB,πA)- 变映射 L 是可分解的。然后，我们应用这些结果证明所有形式为Φ(ρ)=aTr(ρ)dIdd+bρ+cρT+(1-a-b-c)diag(ρ)的 PPT 量子通道都是纠缠断裂的，并且不存在 A-BC PPT-entangled (U⊗U-⊗U)-invariant 三方量子态。前者加强了（Vollbrecht 和 Werner 2001 Phys. Rev. A 64 062307; Kopszak et al 2020 J. Phys.Theor.53 395306）中的一些结论，后者则解决了（Collins et al 2018 Linear Algebra Appl.555 398-411）中提出的一些悬而未决的问题。
A universal framework for entanglement detection under group symmetry
One of the fundamental questions in quantum information theory is determining entanglement of quantum states, which is generally an NP-hard problem. In this paper, we prove that all PPT (π―A⊗πB)-invariant quantum states are separable if and only if all extremal unital positive (πB,πA)-covariant maps are decomposable where πA,πB are unitary representations of a compact group and πA is irreducible. Moreover, an extremal unital positive (πB,πA)-covariant map L is decomposable if and only if L is completely positive or completely copositive. We then apply these results to prove that all PPT quantum channels of the form Φ(ρ)=aTr(ρ)dIdd+bρ+cρT+(1−a−b−c)diag(ρ) are entanglement-breaking, and that there is no A-BC PPT-entangled (U⊗U―⊗U)-invariant tripartite quantum state. The former strengthens some conclusions in (Vollbrecht and Werner 2001 Phys. Rev. A 64 062307; Kopszak et al 2020 J. Phys. A: Math. Theor.53 395306), and the latter resolves some open questions raised in (Collins et al 2018 Linear Algebra Appl.555 398–411).
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Publishing 50 issues a year, Journal of Physics A: Mathematical and Theoretical is a major journal of theoretical physics reporting research on the mathematical structures that describe fundamental processes of the physical world and on the analytical, computational and numerical methods for exploring these structures.
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