Ingxaki yokuNgenisa iiConnes

I-Ingxaki yokuNgeniswa kweConnes yenye yeyona mibuzo inzulu kwizibalo zangoku, ehleli kwisiphambuka se-algebras, ithiyori yolwazi lwe-quantum, kunye nobunzima bokubala. Icetywe yingcali yezibalo yaseFransi u-Alain Connes ngo-1976 kwaye yasonjululwa ngokuqinisekileyo ngo-2020, impendulo yayo yalungisa kwakhona indlela iingcali zezibalo kunye neengcali zefiziksi eziqonda ngayo ulungelelwaniso lobungakanani, izithuba ezinobude obungenasiphelo, kunye nelaphu lengqiqo yemathematika.

Yintoni kanye kanye iNgxaki yokuNgenisa iiConnes?

Embindini wayo, iConnes Embedding Problem ibuze umbuzo olula ngokukhohlisayo: ingaba yonke ifinite von Neumann algebra ene-tracial state inokuzinziswa kumandla amakhulu wehyperfinite II₁ factor? Ngokucacileyo, iphonononge ukuba ingaba zonke iisistim "eziziphethe kakuhle" ezinomlinganiselo ongenamkhawulo we-quantum systems zinokuqikelelwa ngobuchule bemathematika obunokuphela, obuphathekayo.

U-Alain Connes ekuqaleni wayecingela ngo-1976 ukuba impendulo nguewe - ukuba oku kuhlonyelwa kwakusoloko kunokwenzeka. Ngaphezu kweminyaka engamashumi amane, le ngxaki yahlala ivulekile, ixhathisa imigudu yezinye zezibalo ezikrelekrele zehlabathi. Isisombululo sayo asizukuvela kwithiyori yealgebra esulungekileyo, kodwa kwicala ebelingalindelwanga kwaphela: ukuntsonkotha kobungqina bokusetyenziswa kobuninzi.

"Ukuchaswa kweNgxaki yokuNgeniswa kweConnes ayingomdla nje wezibalo - iveza umsantsa osisiseko phakathi kwezinto ezinokwenziwa ziinkqubo ze-quantum kunye nokuba yintoni uqikelelo lwakudala olunokuthi lubanjwe, kunye neempembelelo ezisuka kwi-cryptography ukuya kwiziseko zefiziksi."

Iyenze njani i-Quantum Computing ekugqibeleni iSombulula iNgxaki yeMathematika eneminyaka engama-44?

Ngo-2020, abaphandi uJi, uNatarajan, uVidick, uWright, kunye noYuen bapapasha iphepha elibalulekileyo elibonisa ukuba MIP* = RE, apho i-MIP* ibonisa udidi lweengxaki ezisonjululwa ngumqinisekisi weklasikhi osebenzisana neeprover ezimbini ezibambeneyo zequantum, kwaye RE yiklasi yeelwimi eziphindaphindwayo. Esi siphumo sasothusa: sabonisa ukuba ukufakwa kwe-quantum kunika isibonelelo esingaqhelekanga - ngokusisiseko esingenasiphelo - ukomeleza kwiinkqubo zobungqina obusebenzisanayo.

Uqhagamshelwano kwiiConnes? Iqela libonise ukuba i-Connes Embedding Ingxakiilinganakwingxelo ye-MIP* = MIP (iklasi ye-multiprover interactive proof proof). Ekubeni i-MIP* yajika yaba nkulu kakhulu kune-MIP - eneneni, ilingana ne-RE - i-Connes Embedding conjecture yayibubuxoki. Ayizizo zonke i-algebra ze-von Neumann ezizinzisayo kumandla aphezulu we-hyperfinite II₁ factor.

Yeyiphi iMigaqo eSisiseko emva kweNgxaki?

Ukuqonda iNgxaki yokuHlelela iiConnes kufuna ukuqhelana nezakheko ezininzi eziphambili zemathematika:

• Von Neumann Algebras: Ii-Algebra zabasebenzisi ababambeneyo kwisithuba se-Hilbert abavalwa phantsi kwe-topology yabaqhubi ababuthathaka, ukwenza ngokubanzi ii-algebra ze-matrix ukuya kwimilinganiselo engapheliyo.
• I-Hyperfinite II₁ Factor: Eyodwa, i-canonical ye-von Neumann algebra "engumda" we-algebras ye-matrix enesiphelo — eyona nkqubo yendalo engenasiphelo-dimensional quantum system.
• Amazwe oMthombo: Imisebenzi yomgca kwi-von Neumann algebras eziphatha njengetrayisi eqhelekileyo, ibonelela ngombono "wobungakanani" okanye "umlinganiselo" woqikelelo.
• Amandla amakhulu: Ulwakhiwo lwe-model-theoretic oluvelisa izakhiwo ezintsha zemathematika ngokuthatha imida yolandelelwano lwealgebra ngendlela ethile, engekho mgangathweni.
• Unxulumano lwe-Quantum: Udidi lolungelelwaniso olufikelelwa ngamaqela amabini abelane ngeemeko ze-quantum ezibambeneyo, ezingundoqo kwithiyori yolwazi lomyinge kunye nesisombululo sengxaki ekugqibeleni.

Yintoni uMxholo weMbali kunye nokuvela kwale Ngxaki?

Imvelaphi yengxaki ilandelela kwiphepha lika-Connes lika-1976 malunga nezinto zokutofa, umsebenzi oguqulayo kwiialgebra zokusebenza. Kumashumi eminyaka eyalandelayo, iingcali zemathematika zafumanisa ukuba iCEP yayilingana neengxaki ezininzi ezibonakala zinganxulumananga kwimathematika yonke - ukusuka kwingqikelelo yeQWEP yaseKirchberg kwithiyori yeC *-algebra ukuya kwingxaki kaTsirelson kwithiyori yolwazi lwequantum, eyabuza ukuba ngaba ulungelelwaniso lwequantum oluveliswa ngabasebenzisi bemveliso abalishumi.

Le web of equivalences yenza i-CEP yaba yingxaki enkulu yolungiselelo, "i-hub" edibanisa imimandla eyahlukeneyo. Xa yawa ngo-2020, iziphumo ze-ripple zavakala kwimathematika, ifiziksi, kunye nesayensi yekhompyuter ngaxeshanye. Ubungqina bokuba ingxaki kaTsirelson yayinempendulo engalunganga - ichazwa ngokuthe ngqo yi-MIP * = RE - iqinisekisile ukuba i-quantum mechanics igcina izinto ezifihlakeleyo ezinzulu ngakumbi kunokuba iingcali zefiziksi bezicinga.

Zithini iiNdlela zexesha elizayo kunye neziphumo ezisebenzayo zesi sigqibo?

Isisombululo seNgxaki yokuNgenisa iiConnes sivula imida yophando emitsha. Kwi-cryptography ye-quantum, ilola ukuqonda kwethu ukuba zeziphi iintlobo zolungelelwaniso lwe-quantum olunokwenzeka ngokwasemzimbeni xa kuthelekiswa nokucingeleka ngokwemathematika. Kwithiyori entsonkothileyo, icebisa ukuba amandla eprover yequantum ebambeneyo aqhelekile kakhulu kunendlela ebiyimodeli ngaphambili. Kwiziseko zemathematika, iphakamisa imibuzo enzulu malunga nobudlelwane phakathi kokusondelelana okunomda kunye nezinto ezingenasiphelo zemathematika.

Kumathematika asetyenzisiweyo kunye neenjineli zequantum, isiphumo sigxininisa ukubaluleka kokufunda umsantsa phakathi kolungelelwaniso lwe-"local" kunye "nohambo" lwe-quantum - umsantsa oneziphumo ezithe ngqo kwisixhobo esizimeleyo se-cryptography ye-quantum kunye noyilo lothungelwano lwe-quantum.

Imibuzo Ebuzwa Rhoqo

Ingaba iConnes Embedding Conjecture ingqinwe iyinyani okanye bubuxoki?

Uqikelelo lungqinwe bubuxoki ngo-2020 nguJi, uNatarajan, uVidick, uWright, kunye noYuen. Ubungqina babo, ukuseka i-MIP* = RE, kubonise ubukho be-algebra ye-von Neumann engenakho ukumiliselwa kumandla amakhulu e-hyperfinite II₁ factor, ephikisa ngokuthe ngqo intelekelelo yokuqala kaConnes.

Kutheni iNgxaki yokuFakela iiConnes ibalulekile ngaphandle kwezibalo ezisulungekileyo?

Ingxaki idibanisa ngokuthe ngqo kwi-quantum physics kunye ne-computer yesayensi. Isisombululo sayo siqinisekisile ukuba ukunyanzeliswa kwe-quantum kunokuvelisa ulungelelwaniso lwe-classical kunye ne-standard quantum-mechanical approximations ayikwazi ukuphindaphinda. Oku kuneempembelelo kwi-quantum cryptography, i-quantum computing architecture, kunye neziseko ze-quantum mechanics ngokwayo.

Yintoni i-hyperfinite II₁ factor kwaye kutheni ingundoqo kule ngxaki?

I-hyperfinite II₁ factor, edla ngokuchazwa R, yinto eyodwa ye-von Neumann algebra eyakhiwe njengomda we-finite-dimensional matrix algebras. Yeyona nkqubo ilula neyona "iqikelelwayo" engapheliyo-dimensional quantum system. Umbuzo wokuba ingaba ii-algebra ezintsonkothileyo ezizinziswe kumagunya amakhulu eRubuza ukuba ngaba zonke iinkqubo ze-quantum zabelana ngale propati eqikelelweyo eqikelelwayo - kwaye impendulo, njengoko iziphumo zika-2020 zibonisa, hayi.

Ukuphumelela okufana nesisombululo seNgxaki yokuNgeniswa kweConnes kubonisa okwenzekayo xa iisistim ezintsonkothileyo, ezidityanisiweyo ziqondwa kwinqanaba labo elinzulu - ukuveza uqhagamshelo olungalindelekanga kunye nokuvula amathuba amatsha ngokupheleleyo. KwaMewayz, sikholelwa ukuba umgaqo ofanayo usebenza ekwakheni ishishini lakho. Inkqubo yethu yokusebenza yeemodyuli ezingama-207 inika abasebenzisi abangaphezu kwe-138,000 izixhobo zokuqonda, ukudibanisa, kunye nokwandisa yonke imilinganiselo yemisebenzi yabo, ukusuka kwintengiso kunye neCRM ukuya kuhlalutyo nangaphaya - zonke ziqala kwi-$ 19 kuphela ngenyanga.

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