Connes Embedding ƒe Kuxi

Connes Embedding Problem nye nyabiase goglotɔwo dometɔ ɖeka le egbegbe akɔntabubu me, si bɔbɔ nɔ operator algebras, quantum information theory, kple computational complexity ƒe nutome. Franseawo ƒe akɔntanyala Alain Connes ye doe ɖa le ƒe 1976 me eye wokpɔ egbɔ keŋkeŋ le ƒe 2020 me, eƒe ŋuɖoɖoa trɔ asi le alesi akɔntanyala kple dzɔdzɔmeŋusẽŋununyalawo sea quantum kadodowo, seɖoƒemanɔsitɔ-dzidzeme teƒewo, kple akɔntabubu ƒe susuŋudɔwɔwɔ ƒe avɔ ŋutɔŋutɔ gɔmee.

Nuka Tututue Nye Connes Embedding Kuxia?

Le eƒe gɔmedzedze la, Connes Embedding Problem bia nya bɔbɔe aɖe si blea ame: ɖe woate ŋu atsɔ von Neumann ƒe akɔntabubu ɖesiaɖe si seɖoƒe li na si si tracial nɔnɔme le la ade ultrapower si nye hyperfinite II1 factor mea? Le nya siwo me kɔ me la, eku nu me kpɔ be woate ŋu atsɔ akɔntabubu ƒe xɔtuɖoɖo siwo seɖoƒe li na, siwo woate ŋu akpɔ la asɔ kple quantum-ɖoɖo siwo katã "wɔa nu nyuie" siwo ƒe didime nye seɖoƒemanɔsitɔ.

Alain Connes susui gbã le ƒe 1976 me be ŋuɖoɖoa nye ɛ̃ — be nusiwo wotsɔ de eme sia ate ŋu adzɔ ɣesiaɣi. Ƒe blaene kple edzivɔ la, kuxia gakpɔtɔ nɔ ʋuʋu ɖi, eye wòtsi tre ɖe xexeame ƒe akɔntanyala bibi aɖewo ƒe agbagbadzedzewo ŋu. Eƒe nyametsotso matso pure operator algebra nufiafia me o, ke boŋ tso mɔfiame si womele mɔ kpɔm na kura o gbɔ: akɔntabubu ƒe sesẽ si le quantum interactive proofs me.

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"Connes Embedding Problem ƒe gbegbe menye akɔntabubu ƒe didi be yeanya nu dzro aɖe ko o — eɖe dometsotso vevi aɖe si le nusi quantum systems ateŋu awɔ kple nusi classical approximations ateŋu alé dome, kple gɔmesese siwo keke tso cryptography dzi va ɖo physics ƒe gɔmeɖoanyiwo dzi."

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Aleke Quantum Computing Kpɔ Akɔntabubu ƒe Kuxi Si Xɔ Ƒe 44 Gbɔ Mlɔeba?

| Nusi do tso eme sia wɔ nuku ŋutɔ: eɖee fia be quantum entanglement naa ŋusẽ tɔxɛ aɖe — le nyateƒe me la, seɖoƒe meli na o — na kpeɖodziɖoɖo siwo wɔa dɔ aduadu.

Kadodo si le Connes ŋua? Agbalẽtaƒea ɖo kpe edzi be Connes Embedding Problem la sɔ kple nyagbɔgblɔ MIP* = MIP (the classical multiprover interactive proof class). Esi wònye be MIP* va lolo ŋutɔ wu MIP — le nyateƒe me la, esɔ kple RE — ta la, Connes Embedding ƒe nususugblɔa nye alakpa. Menye von Neumann ƒe akɔntabubu ɖesiaɖe si seɖoƒe li na lae gena ɖe ŋusẽ gã aɖe si nye hyperfinite II1 factor me o.

Nukae Nye Gɔmeɖose Vevi Siwo Le Kuxia Me?

Connes Embedding Problem gɔmesese bia be nànya akɔntabubu ƒe ɖoɖo vevi geɖewo nyuie:

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• Von Neumann Algebras: Algebras of bounded operators le Hilbert space si wotu le operator topology gbɔdzɔ te, generalizing matrix algebras to infinite dimensions.
• The Hyperfinite II1 Factor: Von Neumann ƒe akɔntabubu tɔxɛ aɖe, si le se nu si nye "seɖoƒe" na matriki akɔntabubu siwo seɖoƒe li na — dzɔdzɔme seɖoƒemanɔsitɔ-dzidzeme quantum ɖoɖo.
• Tracial States: Linear functionals le von Neumann algebras dzi siwo wɔa nu abe normalized traces ene, si naa susu si nye "lolome" alo "dzidzeme" na projections.
• Ŋusẽ gãwo: Kpɔɖeŋu-nukpɔsusu ƒe xɔtutu si wɔa akɔntabubu ƒe wɔwɔme yeyewo to seɖoƒewo xɔxɔ na akɔntabubu ƒe ɖoɖo siwo le ɖoɖo nu le mɔ tɔxɛ aɖe nu, si mele ɖoɖo nu o nu.
• Quantum Correlations: Kadodowo ƒe hatsotso si akpa eve siwo ma quantum nɔnɔme siwo xaxa, si le vevie le quantum nyatakaka nufiafia kple kuxia gbɔ kpɔkpɔ mlɔeba me ate ŋu awɔ.

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Nukae Nye Ŋutinya me Nɔnɔme Kple Kuxi sia ƒe Nɔnɔmetɔtrɔ?

Kuxia ƒe gɔmedzedze tso Connes ƒe agbalẽ si wòŋlɔ le ƒe 1976 me ku ɖe injective factors ŋu, si nye dɔ si trɔa ame le operator algebras me. Le ƒe bla nane siwo kplɔe ɖo me la, akɔntanyalagãwo kpɔe be CEP sɔ kple kuxi gbogbo aɖewo siwo dze abe ɖe womedo ƒome o ene le akɔntabubuwo katã me — tso Kirchberg ƒe QWEP nususugblɔ le C*-algebra nufiafia me dzi va ɖo Tsirelson ƒe kuxi si le quantum nyatakaka nufiafia me, si bia be ɖe quantum kadodo siwo commuting operators hena vɛ la sɔ kple esiwo tensor product hena vɛ hã dɔwɔlawo.

Nu siwo sɔ ƒe nyatakakadzraɖoƒe sia na CEP nye ɖoɖowɔwɔ ƒe kuxi vevi aɖe, "hub" si doa ka kple agble siwo to vovo. Esi wòɖiɖi le ƒe 2020 me la, wose ƒutsotsoeawo ƒe ŋusẽkpɔɖeamedzi le akɔntabubu, dzɔdzɔmeŋusẽŋununya, kple kɔmpiutaŋutinunya me le ɣeyiɣi ɖeka me. Kpeɖodzi si nye be ŋuɖoɖo manyomanyo aɖe nɔ Tsirelson ƒe kuxia ŋu — si MIP* = RE fia tẽ — ɖo kpe edzi be quantum mechanics dzraa nusiwo menya kpɔna dzea sii bɔbɔe o wu alesi dzɔdzɔmeŋusẽŋununyalawo susui gɔ̃ hã ɖo.

Nukae Nye Etsɔme Nɔnɔmewo Kple Nyametsotso Sia Ƒe Gɔmesese Nyuiwo?

Connes Embedding Problem ƒe egbɔkpɔkpɔ ʋu numekuku ƒe liƒo yeyewo kura. Le quantum cryptography me la, enaa míaƒe gɔmesese le quantum ƒe kadodo ƒomevi siwo woate ŋu akpɔ le ŋutilã me tsɔ wu esiwo woate ŋu abu le akɔntabubu dzro ko me ŋu la nu sẽna ɖe edzi. Le nusiwo gɔme sese sesẽ ƒe nufiafia me la, efia be ŋusẽ si le quantum provers siwo xaxa me la to vovo kura wu alesi wowɔ kpɔɖeŋui tsã. Le akɔntabubu ƒe gɔmeɖoanyiwo me la, efɔa nyabiase goglowo ɖe te ku ɖe ƒomedodo si le akɔntabubu ƒe nusiwo seɖoƒe li na kple esiwo seɖoƒe meli na o dome ŋu.

Le akɔntanyala siwo wozãna kple quantum engineers gome la, emetsonua te gbe ɖe vovototo si le "nutoa me" kple "commuting" quantum correlations dome sɔsrɔ̃ ƒe vevienyenye dzi — dometsotso si me tsonu tẽ le na device-independent quantum cryptography kple quantum networks ƒe ɖoɖowɔwɔ.

Nyabiase Siwo Wobiana Enuenu

Ðe woɖo kpe Connes Embedding Conjecture dzi be enye nyateƒe loo alo alakpae?

Woɖo kpe nususugblɔa dzi aʋatso le ƒe 2020 me to Ji, Natarajan, Vidick, Wright, kple Yuen dzi. Woƒe kpeɖodzi, si ɖo MIP* = RE anyi, ɖe von Neumann ƒe akɔntabubu siwo womate ŋu atsɔ ade ultrapowers of the hyperfinite II1 factor me o la ƒe anyinɔnɔ fia, si tsi tre ɖe Connes ƒe nususugblɔ gbãtɔ ŋu tẽ.

Nukatae Connes Embedding Problem le vevie le akɔntabubu dzadzɛ godo?

Kuxia do ƒome kple quantum physics kple kɔmpiutaŋutinunya tẽ. Eƒe nyametsotso ɖo kpe edzi be quantum entanglement ate ŋu ahe kadodo siwo quantum-mechanical approximations xoxowo kple esiwo wozãna ɖaa gɔ̃ hã mate ŋu agbugbɔ awɔ o. Esia le ŋusẽ kpɔm ɖe quantum cryptography, quantum computing architecture, kple quantum mechanics ŋutɔ ƒe gɔmeɖoanyiwo dzi.

Nukae nye hyperfinite II1 factor eye nukatae wònye nu vevitɔ le kuxi sia me?

Hyperfinite II1 factor, si woyɔna zi geɖe be R, nye von Neumann ƒe akɔntabubu tɔxɛ aɖe si wotu abe seɖoƒe na seɖoƒe-dzidzeme matriki akɔntabubuwo ene. Enye quantum-ɖoɖo si ƒe didime seɖoƒe meli na o si le bɔbɔe wu eye "woate ŋu asɔ kple wo nɔewo" wu. Nyabiase si nye be ɖe algebra siwo sesẽ wu la gena ɖe ultrapowers of R me la le biabiam vevie be ɖe quantum systems katã kpɔa gome le seɖoƒe ƒe nɔnɔme sia si nye be woate ŋu asɔ kple wo nɔewo — eye ŋuɖoɖoa, abe alesi ƒe 2020 ƒe emetsonua ɖee fia ene la, enye ao.

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Ŋgɔyiyiwo abe Connes Embedding Kuxia gbɔkpɔkpɔ ene ɖea nusi dzɔna ne wose ɖoɖo sesẽ siwo do ƒome kple wo nɔewo gɔme le woƒe gogloƒe kekeake — si ɖea kadodo siwo womele mɔ kpɔm na o fiana eye wòʋua mɔnukpɔkpɔ yeye kura. Le Mewayz la, míexɔe se be gɔmeɖose ma ke ku ɖe wò dɔwɔƒea tutuɖo ŋu. Míaƒe 207-module asitsadɔwɔɖoɖoa naa dɔwɔnu siwo wu 138,000 zãla be woase woƒe dɔwɔnawo ƒe akpa ɖesiaɖe gɔme, aɖo kadodo me, eye woawɔe wòanyo wu, tso asitsatsa kple CRM dzi va ɖo numekuku kple edzivɔ dzi — wo katã wodzea egɔme tso $19/ɣleti ko dzi.

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