Connes Embedding Ɔhaw no

Connes Embedding Problem yɛ nsɛmmisa a emu dɔ sen biara wɔ nnɛyi akontabuo mu no mu baako, ɛte operator algebras, quantum information theory, ne computational complexity ntam. Franseni akontabuo ho nimdefoɔ Alain Connes na ɔhyɛɛ ho nyansa wɔ afe 1976 mu na wɔsiesiee no koraa wɔ afe 2020 mu, ne mmuaeɛ no sesaa sɛdeɛ akontabuo ho nimdefoɔ ne abɔdeɛ mu nneɛma ho animdefoɔ te quantum nkitahodiɛ, ahunmu a ɛnni ano-afã, ne akontabuo mu nteaseɛ ntama ankasa ase.

Dɛn Pɛpɛɛpɛ ne Connes Embedding Ɔhaw no?

Wɔ ne titiriw mu no, Connes Embedding Problem no bisaa asɛmmisa bi a ɛyɛ nnaadaa a ɛnyɛ den: so wobetumi de von Neumann algebra biara a ɛwɔ anohyeto a ɛwɔ tracial tebea ahyɛ ultrapower a ɛwɔ hyperfinite II1 factor no mu? Wɔ nsɛm a emu da hɔ mu no, ɛhwehwɛɛ mu sɛ ebia wobetumi de akontaabu nhyehyɛe ahorow a ɛwɔ anohyeto, a wotumi di ho dwuma abɛn quantum nhyehyɛe ahorow a "ɛyɛ wɔn ade yiye" a enni ano.

Alain Connes mfiase no susuwii wɔ 1976 mu sɛ mmuae no yɛ yiw — sɛ saa embedding yi betumi aba bere nyinaa. Bɛboro mfe aduanan no, ɔhaw no kɔɔ so buei, na ɛsɔre tiaa wiase no mu akontaabufo a wɔn ho akokwaw sen biara no bi mmɔdenbɔ. Ne gyinaesi no remfi pure operator algebra theory mu, na mmom efi akwankyerɛ bi a wɔnhwɛ kwan koraa mu: akontaabu mu nsɛnnennen a ɛwɔ quantum nkitahodi adanse ahorow mu.

a wɔde ahyɛ mu

"Connes Embedding Problem no a wɔpow no nyɛ akontaabu mu anigye ara kwa — ɛda nsonsonoe titiriw bi adi wɔ nea quantum nhyehyɛe ahorow betumi ayɛ ne nea tete approximations betumi akyere ntam, a nea ɛkyerɛ fi cryptography so kosi physics fapem so."

Ɛbɛyɛ dɛn na Quantum Computing Awiei koraa no Di Nkontaabu Ho Ɔhaw a Adi Mfe 44 Adi?

Nkitahodi a ɛda Connes ntam? Kuw no daa no adi sɛ Connes Embedding Problem no equivalent ne asɛm MIP* = MIP (class multiprover interactive proof class) no yɛ pɛ. Esiane sɛ MIP* bɛdaa adi sɛ ɛsõ sen MIP kɛse — nokwarem no, ɛne RE yɛ pɛ — nti na Connes Embedding nsusuwii no yɛ atoro. Ɛnyɛ von Neumann algebra biara a ɛwɔ anohyeto na ɛde ne ho hyɛ tumi kɛse bi a ɛwɔ hyperfinite II1 factor no mu.

Nnyinasosɛm Titiriw Bɛn na Ɛwɔ Ɔhaw no Akyi?

Connes Embedding Problem no nteaseɛ hwehwɛ sɛ wobɛhunu akontabuo nhyehyɛeɛ titire pii:

• Von Neumann Algebras: Algebras a ɛkyerɛ bounded operators wɔ Hilbert space a wɔato mu wɔ operator topology a ɛyɛ mmerɛw ase, generalizing matrix algebras to infinite dimensions.
• Hyperfinite II1 Factor: Von Neumann algebra soronko, canonical a ɛyɛ "anohyeto" a ɛwɔ finite matrix algebras — abɔdeɛ mu infinite-dimensional quantum nhyehyɛeɛ a ɛsen biara.
• Tracial States: Linear functionals wɔ von Neumann algebras so a ɛyɛ wɔn ade te sɛ normalized traces, a ɛma adwene a ɛfa "size" anaa "dimension" ho ma projections.
• Ultrapowers: Model-theoretic construction a ɛma akontabuo nhyehyeɛ foforɔ denam anohyetoɔ a ɛfa algebra ahodoɔ a ɛtoatoa soɔ wɔ ɔkwan pɔtee bi a ɛnyɛ gyinapɛn so.
• Quantum Nkitahodi: Nkitahodi kuw a nnipa baanu a wɔkyɛ quantum tebea horow a ɛyɛ den, a ɛyɛ ade titiriw wɔ quantum nsɛm ho nsusuwii ne ɔhaw no ano aduru awiei koraa no mu.

Dɛn Ne Abakɔsɛm mu Nsɛm ne Nkɔsoɔ a Ɛfa Ɔhaw Yi Ho?

Ɔhaw no mfiase fi Connes krataa a ɔkyerɛwee wɔ 1976 mu a ɛfa injective factors ho, adwuma a ɛsakra wɔ operator algebras mu. Wɔ mfe du du a edii hɔ no mu no, akontaabufo hui sɛ CEP no ne ɔhaw ahorow du du pii a ɛte sɛ nea enni abusuabɔ wɔ akontaabu nyinaa mu yɛ pɛ — efi Kirchberg QWEP nsusuwii a ɛwɔ C*-algebra nsusuwii mu kosi Tsirelson haw a ɛwɔ quantum information nsusuwii mu, a ɛbisae sɛ ebia quantum nkitahodi ahorow a commuting operators de ba no ne nea tensor product de ba no yɛ pɛ anaa adwumayɛfoɔ.

Saa web a ɛyɛ pɛ yi maa CEP yɛɛ nhyehyɛe ho haw titiriw, "hub" a ɛka afuw a ɛsono emu biara bom. Bere a ɛhwee ase wɔ afe 2020 mu no, wɔtee asorɔkye nkɛntɛnso no nka wɔ akontaabu, abɔde mu nneɛma, ne kɔmputa ho nimdeɛ nyinaa mu bere koro mu. Adanse a ɛkyerɛ sɛ na Tsirelson haw no wɔ mmuae a enye — a MIP* = RE kyerɛ tẽẽ — sii so dua sɛ quantum mfiridwuma kura nneɛma a ɛyɛ anifere a emu dɔ mpo sen sɛnea abɔde mu nneɛma ho animdefo susuwii.

Dɛn ne Daakye Nkɔsoɔ ne Nkyerɛkyerɛmu a Ɛyɛ Mfasoɔ a Ɛfiri Saa Gyinaesi Yi Mu?

Connes Embedding Problem no ano aduru bue nhwehwɛmu ahye foforo koraa. Wɔ quantum cryptography mu no, ɛma ntease a yɛwɔ wɔ quantum nkitahodi ahorow a wotumi hu wɔ honam fam ne nea wotumi susuw ho wɔ akontaabu mu ara kwa ho no yɛ nnam. Wɔ nkyerɛkyerɛ a ɛfa nneɛma a ɛyɛ den ho mu no, ɛkyerɛ sɛ tumi a entangled quantum provers wɔ no yɛ nwonwa koraa sen sɛnea na wɔayɛ ho nhwɛso kan no. Wɔ akontabuo fapem mu no, ɛma nsemmisa a emu dɔ sɔre wɔ abusuabɔ a ɛda bɛyɛ a ɛwɔ anohyeto ne akontabuo nneɛma a enni ano ntam.

Nsɛmmisa a Wɔtaa Bisa

So wɔdaa no adi sɛ Connes Embedding Conjecture no yɛ nokware anaasɛ ɛyɛ atoro?

Wɔdaa no adi sɛ nsusuwii no yɛ atoro wɔ afe 2020 mu denam Ji, Natarajan, Vidick, Wright, ne Yuen so. Wɔn adanse a ɛde MIP* = RE sii hɔ no kyerɛɛ sɛ von Neumann algebras a wontumi mfa nhyɛ tumi a ɛboro so a ɛwɔ hyperfinite II1 factor no mu wɔ hɔ, na ɛbɔ Connes mfitiase nsusuwii no gu tẽẽ.

Dɛn nti na Connes Embedding Problem no ho hia wɔ akontabuo kronkron akyi?

Ɔhaw no ne quantum physics ne kɔmputa ho nyansahu di nkitaho tẽẽ. Ne gyinaesi no sii so dua sɛ quantum entanglement betumi ama nkitahodi ahorow a classical ne mpo standard quantum-mechanical approximations ntumi nyɛ bio. Eyi wɔ nkyerɛkyerɛmu ma quantum cryptography, quantum kɔmputa nhyehyɛe, ne quantum mfiridwuma ankasa fapem.

Dɛn ne hyperfinite II1 factor na adɛn nti na ɛyɛ ɔhaw yi titire?

Hyperfinite II1 factor a wɔtaa kyerɛ no R no yɛ von Neumann algebra soronko a wɔayɛ sɛ anohyeto a ɛwɔ finite-dimensional matrix algebras mu. Ɛyɛ quantum nhyehyɛe a ɛyɛ mmerɛw na "ɛbɛyɛ sɛ" a enni ano-dimensional. Asɛmmisa a ɛfa sɛ ebia algebras a ɛyɛ den kɛse no hyɛ ultrapowers a ɛwɔ R mu no titiriw rebisa sɛ ebia quantum nhyehyɛe nyinaa kyɛ saa finite approximability property yi — na mmuae no, sɛnea 2020 aba no kyerɛ no, yɛ dabi.

Nkɔsoɔ te sɛ Connes Embedding Ɔhaw no ano aduru kyerɛ deɛ ɛba berɛ a wɔte nhyehyɛeɛ a ɛyɛ den, a ɛka bom ase wɔ wɔn gyinabea a emu dɔ — ɛda nkitahodiɛ a wɔnhwɛ kwan adi na ɛbue nneɛma foforɔ a ɛbɛtumi aba koraa. Wɔ Mewayz no, yɛgye di sɛ nnyinasosɛm koro no ara fa w’adwuma a wobɛkyekyere ho. Yɛn 207-module adwumayɛ dwumadie nhyehyɛeɛ no ma bɛboro 138,000 a wɔde di dwuma no nnwinnadeɛ a wɔde bɛte aseɛ, abɔ, na wɔayɛ wɔn dwumadie no fã biara yie, ɛfiri aguadi ne CRM so kɔsi nhwehwɛmu ne nea ɛboro saa — ne nyinaa firi aseɛ firi $19/ɔsram pɛ.

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