Pilikia Hoʻokomo ʻia ʻo Connes

ʻO ka Connes Embedding Problemʻo ia kekahi o nā nīnau hohonu loa i ka makemakika hou, e noho ana ma ke kikowaena o nā algebras operator, quantum information theory, and computational complexity. Manaʻo ʻia e ka mea makemakika Palani ʻo Alain Connes i ka makahiki 1976 a ua hoʻoholo paʻa ʻia i ka makahiki 2020, ua hoʻololi hou kāna pane i ka hoʻomaopopo ʻana o ka poʻe makemakika a me nā physicists i ka hoʻoponopono quantum, nā hakahaka palena ʻole, a me ke ʻano o ka loina makemakika.

He aha ka Pilikia Hoʻokomo ʻana o Connes?

Ma kāna kumu, ua nīnau ka Connes Embedding Problem i kahi nīnau hoʻopunipuni maʻalahi: hiki anei i kēlā me kēia algebra von Neumann āpau me kahi kūlana tracial ke hoʻokomo ʻia i loko o ka mana nui o ka helu II₁ hyperfinite? Ma nā huaʻōlelo maʻalahi, ua hoʻāʻo ʻia inā hiki ke hoʻohālikelike ʻia nā ʻōnaehana kuantum palena ʻole "hana maikaʻi".

Ua kuhi mua ʻo Alain Connes i ka makahiki 1976 ʻo ka pane he ʻae — hiki mau kēia hoʻopili ʻana. No nā makahiki he ʻehā, ua paʻa ka pilikia, e kūʻē ana i nā hana a kekahi o nā mea makemakika akamai loa o ka honua. ʻAʻole hiki mai kāna hoʻoholo ʻana mai ke kumumanaʻo algebra operator maʻemaʻe, akā mai kahi ʻaoʻao i manaʻo ʻole ʻia: ka paʻakikī o ka helu helu o nā hōʻike pili kumu.

"ʻAʻole ʻo ka ʻike makemakika wale nō ka hōʻole ʻana o ka Connes Embedding Problem - e hōʻike ana ia i kahi ʻokoʻa ma waena o nā ʻōnaehana quantum e hana ai a me nā mea hoʻohālikelike kahiko e hiki ke hopu, me nā hopena e pili ana mai ka cryptography a hiki i ke kumu o ka physics."

Pehea i hoʻopau hope ai ʻo Quantum Computing i kahi pilikia makemakika 44 makahiki?

I ka makahiki 2020, ua paʻi nā mea noiʻi ʻo Ji, Natarajan, Vidick, Wright, a me Yuen i ka pepa ʻāina e hoʻokumu ana ʻo MIP* = RE, kahi e hōʻike ai ʻo MIP* i ka papa o nā pilikia e hoʻonā ʻia e kahi mea hōʻoia maʻamau e launa pū ana me ʻelua mau prover quantum entangled, a ʻo RE ka papa o nā ʻōlelo recursively enumerable. He mea hoʻopūʻiwa kēia hopena: ua hōʻike ʻia ka hāʻawi ʻana o ka quantum entanglement i mea hoʻoikaika kūʻokoʻa - ʻaʻole palena ʻole - i nā ʻōnaehana hōʻoia pili.

Ka pilina me Connes? Ua hōʻoia ka hui ʻo ka Connes Embedding Problem heequivalentme ka ʻōlelo MIP* = MIP (ka papa hana hōʻikeʻike multiprover maʻamau). No ka mea, ua ʻoi aku ka nui o ka MIP* ma mua o ka MIP - ʻoiaʻiʻo, e like me RE - he hewa ka Connes Embedding conjecture. ʻAʻole i hoʻopili ʻia kēlā me kēia helu helu von Neumann i loko o kahi mana nui o ka helu II₁ hyperfinite.

He aha nā Kumukānāwai Ma hope o ka pilikia?

No ka hoʻomaopopo ʻana i ka pilikia o ka hoʻokomo ʻana o Connes, pono ke kamaʻāina me nā ʻano hana makemakika nui:

• Nā Algebra Von Neumann: Nā Algebra o nā mea hoʻohana palena ʻia ma kahi ākea Hilbert i pani ʻia ma lalo o ka topology operator nāwaliwali, e hoʻomāhuahua ana i nā algebra matrix i nā ana palena ʻole.
• ʻO ka Hyperfinite II₁ Factor: He algebra kū hoʻokahi, canonical von Neumann, ʻo ia ka "palena" o nā algebra matrix palena - ʻo ka ʻōnaehana kuantum palena ʻole kūlohelohe.
• ʻAi Mokuʻāina: Nā hana laina ma nā algebra von Neumann e hana like me nā meheu maʻamau, e hāʻawi ana i ka manaʻo o ka "nui" a i ʻole "ana" no nā kuhi.
• Ultrapowers: He hana kumu hoʻohālike e hana ana i nā hale makemakika hou ma o ka lawe ʻana i nā palena o nā kaʻina algebra ma ke ʻano kikoʻī, ʻaʻole maʻamau.
• Kuantum Correlations: ʻO ka papa o ka hoʻoponopono ʻana i hiki ke hoʻokō ʻia e nā ʻaoʻao ʻelua e kaʻana like ana i nā mokuʻāina kuantum i hoʻopaʻa ʻia, ke kikowaena o ka manaʻo ʻike quantum a me ka hopena o ka pilikia.

He aha ke ʻano moʻolelo a me ka ulu ʻana o kēia pilikia?

Ke kumu o ka pilikia ma ka pepa a Connes i ka makahiki 1976 no na mea injective, he hana hoololi i na algebras operator. I nā makahiki i hala iho nei, ua ʻike ka poʻe makemakika ua like ka CEP me ka nui o nā pilikia pili ʻole ma waena o ka makemakika - mai ko Kirchberg's QWEP conjecture ma ka C*-algebra theory a hiki i ka pilikia ʻo Tsirelson i ka quantum information theory.

Ua lilo kēia pūnaewele o nā mea like i ka CEP i pilikia hoʻonohonoho koʻikoʻi, he "hub" e hoʻopili ana i nā kahua ʻokoʻa. I ka hāʻule ʻana i ka makahiki 2020, ua ʻike ʻia nā hopena ripple ma waena o ka makemakika, physics, a me ka ʻepekema kamepiula i ka manawa like. ʻO ka hōʻoiaʻiʻo o ka pilikia o Tsirelson he pane maikaʻi ʻole - i kuhikuhi pololei ʻia e MIP* = RE - ua hōʻoia ʻo ka quantum mechanics i loko o nā mea maʻalahi ma mua o ka manaʻo o nā physicists.

He aha nā ʻano o ka wā e hiki mai ana a me nā hopena kūpono o kēia ʻōlelo hoʻoholo?

Ke wehe nei ka hoʻonā ʻana o ka Connes Embedding Problem i nā palena noiʻi hou loa. I loko o ka quantum cryptography, hoʻonui ia i ko mākou ʻike i ke ʻano o nā correlations quantum e hiki ke ʻike kino ʻia me ka manaʻo makemakika wale nō. Ma ke kumumanaʻo paʻakikī, hōʻike ia he ʻoi aku ka mana o nā provers quantum entangled ma mua o ka hoʻohālike mua. Ma ke kahua o ka makemakika, ho'āla ia i nā nīnau hohonu e pili ana i ka pilina ma waena o ka palena palena a me nā mea makemakika palena ʻole.

No ka poʻe makemakika a me nā ʻenekini quantum, hōʻike ka hopena i ke koʻikoʻi o ke aʻo ʻana i ka ʻokoʻa ma waena o nā hoʻoponopono quantum "local" a me "commuting" — he āwaha me nā hopena pololei no ka cryptography quantum kūʻokoʻa me ka hoʻolālā ʻana o nā pūnaewele quantum.

Nīnau pinepine

He ʻoiaʻiʻo a he wahaheʻe paha ka Connes Embedding Conjecture?

Ua hōʻoiaʻiʻo ʻia ka manaʻo hewa ma 2020 e Ji, Natarajan, Vidick, Wright, a me Yuen. ʻO kā lākou hōʻoia, ʻo ka hoʻokumu ʻana i ka MIP* = RE, ua hōʻike i ke ola ʻana o nā algebras von Neumann ʻaʻole hiki ke hoʻokomo ʻia i loko o nā mana ultra o ka helu II₁ hyperfinite, e hōʻole pololei ana i ka manaʻo kumu mua o Connes.

No ke aha i pili ai ka pilikia o ka Connes Embedding ma waho o ka makemakika maʻemaʻe?

Hoʻopili pololei ka pilikia i ka quantum physics a me ka ʻepekema kamepiula. Ua hōʻoia ka ʻōlelo hoʻoholo e hiki i ka quantum entanglement ke hana i nā correlations ʻaʻole hiki ke hoʻohālikelike ʻia nā mea hoʻohālikelike kuantum-mechanical maʻamau. He hopena kēia no ka quantum cryptography, quantum computing architecture, a me nā kumu o ka quantum mechanics pono'ī.

He aha ka helu II₁ hyperfinite a no ke aha ke kumu o kēia pilikia?

ʻO ka helu II₁ hyperfinite, i hōʻike pinepine ʻia R, he ʻano helu helu von Neumann i kūkulu ʻia ma ke ʻano he palena o nā algebra matrix matrix. ʻO ia ka ʻōnaehana kuantum palena ʻole a maʻalahi loa. ʻO ka nīnau no ka hoʻokomo ʻia ʻana o nā algebra paʻakikī i loko o nā mana nui o Rke nīnau nei inā kaʻana like nā ʻōnaehana quantum a pau i kēia waiwai hoʻohālikelike - a ʻo ka pane, e like me ka hōʻike ʻana o ka hopena 2020, ʻaʻole.

Ke hōʻike ʻia nei nā hanana e like me ka hoʻonā ʻana o ka Connes Embedding Problem i ka mea e hiki mai ana ke hoʻomaopopo ʻia nā ʻōnaehana paʻakikī a pili i ko lākou pae hohonu - e hōʻike ana i nā pilina i manaʻo ʻole ʻia a wehe i nā mea hou loa. Ma Mewayz, ke manaʻoʻiʻo nei mākou pili ka loina like i ke kūkulu ʻana i kāu ʻoihana. Hāʻawi kā mākou ʻōnaehana hana pāʻoihana 207-module i nā mea hoʻohana 138,000 i nā mea hana e hoʻomaopopo, hoʻohui, a hoʻopaʻa pono i kēlā me kēia ʻāpana o kā lākou hana, mai ke kūʻai aku a me CRM a i ka analytics a ma waho aʻe - e hoʻomaka ana nā mea āpau ma $19/mahina wale nō.

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