Kvant optimal tashish masofasi \(W_{C_E,p}\) — belgilangan marginallarga ega kvant coupling holatlari ustida xarajat operatorining kutilgan qiymatini minimallashtirish orqali ikki zichlik matritsasi o‘rtasidagi munosabatni o‘lchashga urinadigan masofaga o‘xshash matematik kattalikdir. Tomasz Millerning ushbu tadqiqoti Friedland va hamkasblari 2022-yilda taklif qilgan maxsus kvant xarajat matritsasi uchun bu kattalik barcha o‘lchamlarda haqiqiy masofa bo‘lishi haqidagi conjecture-ni analitik qarshi misollar oilasi bilan tekshiradi va uchburchak tengsizligi uch o‘lchamli diagonal zichlik matritsalari uchun buzilishini ko‘rsatadi.

Tadqiqot markazidagi maxsus xarajat matritsasi antisymmetric subspace ustidagi projector bo‘lgan \(C^Q\) dir. Avvalgi ishda \(C^Q\) ishlatilganda aniqlangan quantum 2-Wasserstein semidistance Hilbert fazosining istalgan o‘lchamida triangle inequality-ni qanoatlantirishi conjecture qilingan edi. Miller esa muayyan \(\rho\), \(\sigma\) va \(\tau\) diagonal zichlik matritsalari oilasi uchun

\[ W(\rho,\tau) > W(\rho,\sigma)+W(\sigma,\tau) \]

ekanini analitik tarzda isbotlaydi. Masofa uchburchak tengsizligini qanoatlantirishi kerakligi sabab, ushbu bir yo‘nalishli qat’iy tengsizlikning o‘zi ham mazkur kattalik umumiy holatda metrik bo‘la olmasligini ko‘rsatadi.

Qarshi misollar faqat uch o‘lcham bilan cheklanmaydi. Manbada berilgan uch o‘lchamli zichlik matritsalari yuqori o‘lchamlarga nol bloklar qo‘shib joylashtirilishi mumkinligi sabab, ayni muammo barcha \(n>3\) o‘lchamlarga ko‘chirilishi mumkin. Shu bilan birga, tadqiqot ilgari isbotlangan maxsus holatlarni bekor qilmaydi: \(n=2\) holati va faqat pure states bilan cheklanganda, zarur \(p\ge2\) sharti ostida triangle inequality qanoatlantirilishi mumkin bo‘lgan maxsus holatlar saqlanib qoladi.

Tadqiqotning ikkinchi muhim hissasi diagonal, ya’ni bu kontekstda “classical” zichlik matritsalari orasidagi quantum transport muammosini klassik coupling-lar ustidagi minimallashtirishga keltiradigan umumiy formuladir. Proposition 1 orqali olingan ushbu formula quantum va classical transport o‘rtasidagi farq nima sababdan paydo bo‘lishini ham ko‘rsatadi: quantum coupling matritsasidagi off-diagonal hadlar xarajatga \(\sqrt{\gamma_{ij}\gamma_{ji}}\) ko‘rinishida kamaytiruvchi hissa qo‘shishi mumkin.

Bu yerdagi “quantum” fizik kvant kompyuterida hisoblash bajarilganini anglatmaydi. Tadqiqot nazariy kvant axboroti va matematik fizika darajasidadir; laboratoriya tajribasi, quantum processor yoki hardware ishlatilmagan. Natijalar zichlik matritsalari, tensor-product fazolar, partial traces, projectors, classical/quantum couplings va analitik tengsizliklar orqali chiqarilgan.

Kvant optimal tashish muammosi nima?

Kvant optimal tashish muammosi klassik optimal transport muammosidagi ikki ehtimollik taqsimotini o‘zaro moslashtirish g‘oyasini zichlik matritsalariga ko‘chiradigan va ikki kvant holatining belgilangan marginallarini saqlaydigan bipartite kvant holatlari orasida eng kichik xarajatli coupling-ni izlaydigan matematik optimallashtirish muammosidir.

Klassik optimal tashish g‘oyasi

Klassik optimal transport muammosida ikki probability distribution o‘rtasida “massa”ning bir taqsimotdan boshqasiga qanday ko‘chirilishi qaraladi. Coupling boshlang‘ich va maqsad taqsimotlarini marginal tarzda saqlaydigan qo‘shma ehtimollik taqsimotidir. Har bir \(i\rightarrow j\) ko‘chirishning \(E_{ij}\) xarajati bo‘lsa, optimallashtirish muammosi umumiy transport cost-ni eng kichik qiladigan coupling-ni izlaydi.

Wasserstein distance ushbu yondashuvning eng mashhur masofa oilalaridan biridir. Biroq bu maqolada klassik Wasserstein nazariyasining barchasi to‘g‘ridan-to‘g‘ri o‘rganilmaydi; asosiy masala Friedland va hamkasblari zichlik matritsalari uchun taklif qilgan kvant analogi haqiqatan metrikmi yoki yo‘qmi, shuni aniqlashdir.

Zichlik matritsasi nima?

Zichlik matritsasi \(\rho\) kvant holatini ifodalovchi musbat yarim aniqlangan va izi 1 ga teng operator hisoblanadi. Sof holatlar rank-1 projector ko‘rinishida ifodalanishi mumkin, mixed states esa bir nechta mumkin bo‘lgan holatlarning statistik aralashmasini o‘z ichiga oladi.

Ushbu maqoladagi qarshi misollarning muhim xususiyati ishlatilgan zichlik matritsalarining diagonal ekanidir. Masalan:

\[ \rho=\operatorname{diag}(1-s,s,0) \]

ifodasi computational basis da faqat diagonal entries ni o‘z ichiga olgan density matrix dir. Diagonal entries manfiy emas va ularning yig‘indisi 1 ga teng. Shu sabab ushbu entries ayni paytda probability vector sifatida ham o‘qilishi mumkin.

Manba bunday diagonal states ni “classical states” deb ataydi. Buning sababi tanlangan basis da off-diagonal quantum coherence hadlari mavjud emasligidir.

Semidistance bilan haqiqiy metrik o‘rtasidagi muhim farq nima?

Ushbu tadqiqot kontekstida \(W_{C_E,p}\) masofaga o‘xshash semidistance dir; haqiqiy metrik bo‘lishi uchun boshqa zarur xususiyatlar bilan birga uchburchak tengsizligini ham qanoatlantirishi kerak. Millerning qarshi misoli aynan uchburchak tengsizligi bajarilmasligini ko‘rsatgani uchun umumiy metrik da’vosini bekor qiladi.

Metrik \(d\) uchun uchta nuqta yoki holat \(A,B,C\) tanlanganda:

\[ d(A,C)\le d(A,B)+d(B,C) \]

bo‘lishi kerak.

Intuitiv ma’nosi shuki, \(A\) dan \(C\) gacha bo‘lgan to‘g‘ridan-to‘g‘ri masofa \(B\) orqali o‘tilgan yo‘l yig‘indisidan uzun bo‘la olmaydi.

Ushbu tadqiqotda esa muayyan \(\rho,\sigma,\tau\) uchun teskari yo‘nalishda qat’iy tengsizlik olinadi:

\[ W(\rho,\tau)> W(\rho,\sigma)+W(\sigma,\tau). \]

Demak, \(W\) mazkur holatlar fazosida haqiqiy metrik emas.

Kvant coupling nima?

Kvant coupling — ikki maqsad zichlik matritsasini partial trace orqali marginal tarzda qaytaradigan bipartite density matrix; maqolada \(\Gamma(\rho_1,\rho_2)\) \(\operatorname{Tr}_1\rho_{12}=\rho_2\) va \(\operatorname{Tr}_2\rho_{12}=\rho_1\) shartlarini qanoatlantiradigan barcha \(\rho_{12}\) holatlar to‘plamidir.

Ikki \(n\times n\) density matrix:

\[ \rho_1,\rho_2 \]

berilganda coupling:

\[ \rho_{12} \]

\(n^2\times n^2\) o‘lchamli bipartite density matrix dir.

Manba shartlari:

\[ \operatorname{Tr}_1\rho_{12}=\rho_2, \qquad \operatorname{Tr}_2\rho_{12}=\rho_1. \]

Bu yerda \(\operatorname{Tr}_1\) va \(\operatorname{Tr}_2\) tensor-product tizimining mos ravishda birinchi va ikkinchi quyi tizimini trace qilib yo‘qotadigan partial-trace amallaridir.

Bu tuzilma klassik coupling g‘oyasining kvant muqobilidir. Classical coupling da qo‘shma probability matrix ning satr va ustun yig‘indilari berilgan probability vectors ni qaytarsa, quantum coupling da xuddi shu vazifani partial traces bajaradi.

Antisymmetric subspace va SWAP operator ushbu tadqiqotda nega muhim?

Tadqiqot markazidagi maxsus quantum cost matrix \(C^Q\), ikki quyi tizimni almashtiradigan SWAP operatorning antisymmetric subspace ustidagi projector idir va avvalgi conjecture aynan ushbu maxsus xarajat uchun aniqlangan quantum 2-Wasserstein semidistance barcha o‘lchamlarda triangle inequality ni qanoatlantirishi haqidagi da’voga asoslanadi.

Antisymmetric basis state

Manba avval:

\[ |i\wedge j\rangle = \frac{1}{\sqrt{2}} \left( |ij\rangle-|ji\rangle \right) \]

holatini aniqlaydi.

Bu yerda:

• \(|i\rangle\) va \(|j\rangle\) belgilangan computational basis elementlaridir.
• \(|ij\rangle=|i\rangle\otimes|j\rangle\) tensor-product basis state dir.
• \(|i\wedge j\rangle\), \(i\) va \(j\) o‘rin almashganda ishorasi o‘zgaradigan normalized antisymmetric combination dir.
• \(1/\sqrt{2}\) koeffitsiyenti holat normasini 1 qilish uchun zarur.

Umumiy quantum cost matrix

Istalgan \(n\times n\) distance matrix \(E=(E_{ij})\) uchun manba Tenglama (1) da:

\[ C_E := \sum_{i

ta’rifidan foydalanadi.

\(|i\wedge j\rangle\langle i\wedge j|\) tegishli antisymmetric direction ustidagi rank-1 projector dir. \(E_{ij}\) koeffitsiyentlari ushbu directions uchun cost weights vazifasini bajaradi.

\(C_E\) fizik energiya Hamiltonian i bo‘lishi shart emas. Ushbu tadqiqotdagi vazifasi transport optimization doirasida xarajatni aniqlovchi Hermitian operator bo‘lishdir.

Quantum p-Wasserstein analogi

Manbaning Tenglama (2) si:

\[ W_{C_E,p}(\rho_1,\rho_2) := \min_{\rho_{12}\in\Gamma(\rho_1,\rho_2)} \left[ \operatorname{Tr} C_E^p\rho_{12} \right]^{1/p} \]

\[ = \min_{\rho_{12}\in\Gamma(\rho_1,\rho_2)} \left[ \sum_{i

Formulaning ma’nosi bosqichma-bosqich quyidagicha:

1. Ikki maqsad state \(\rho_1\) va \(\rho_2\) belgilab olinadi.
2. Ushbu states ni partial traces sifatida qaytaradigan barcha quantum couplings \(\rho_{12}\) ko‘rib chiqiladi.
3. Har bir coupling uchun \(C_E^p\) bilan aniqlangan cost hisoblanadi.
4. Barcha couplings orasidagi eng kichik qiymat tanlanadi.
5. Natija \(1/p\) darajaga ko‘tariladi.

Bu kattalik manbada quantum analogue of the \(p\)-Wasserstein distance sifatida qaraladi. Biroq maqolaning asosiy mavzusi ushbu analog qaysi sharoitlarda haqiqatan metrik xususiyatlarga ega bo‘lishidir.

Maxsus antisymmetric projector \(C^Q\)

Avvalgi conjecture uchun xarajat matritsasi:

\[ C^Q := \sum_{i

sifatida tanlanadi.

Bu yerda \(S\), SWAP operator dir:

\[ S|ij\rangle=|ji\rangle. \]

\(\mathbf{1}_{n^2}\), \(n^2\)-dimensional tensor-product space ustidagi identity operator dir.

Ushbu maxsus holatda quantum 2-Wasserstein semidistance:

\[ W_{C^Q,2}(\rho_1,\rho_2) = \min_{\rho_{12}\in\Gamma(\rho_1,\rho_2)} \sqrt{ \sum_{i

Avvalgi Conjecture I bu kattalik har bir dimension \(n\) uchun triangle inequality ni qanoatlantirishini ilgari surgan. Conjecture II esa ushbu xususiyat \(C^Q\) atrofidagi kichik cost-matrix perturbations ostida ham saqlanishini taxmin qilgan.

Proposition 1 nimani isbotlaydi?

Proposition 1 kirish states diagonal bo‘lganda quantum transport optimallashtirish to‘liq classical coupling matrix \(\gamma\) orqali yozilishi mumkinligini va quantum ta’sir \((\sqrt{\gamma_{ij}}-\sqrt{\gamma_{ji}})^2\) tuzilishida ko‘rinishini isbotlaydi.

Probability vectors:

\[ P=(p_1,\ldots,p_n), \qquad Q=(q_1,\ldots,q_n) \]

bo‘lsin.

Ularga mos diagonal density matrices:

\[ \operatorname{diag}(P), \qquad \operatorname{diag}(Q) \]

uchun Proposition 1:

\[ W_{C_E,p} \left( \operatorname{diag}(P), \operatorname{diag}(Q) \right) = \min_{\gamma\in\Gamma_{\mathrm{cl}}(P,Q)} \left[ \frac{1}{2} \sum_{i

natijasini beradi.

\(\Gamma_{\mathrm{cl}}(P,Q)\) — \(P\) va \(Q\) ning barcha classical couplings to‘plami. Coupling matrix \(\gamma=(\gamma_{ij})\) uchun:

\[ \sum_j\gamma_{ij}=p_i, \qquad \sum_i\gamma_{ij}=q_j, \qquad \gamma_{ij}\ge0. \]

Shunday qilib, satr yig‘indilari \(P\) ni, ustun yig‘indilari esa \(Q\) ni beradi.

Proposition 1 isboti nega uch bosqichga bo‘linadi?

Isbotning asosiy strategiyasi diagonal kirish states uchun keraksiz quantum phase information ni o‘rtachalash orqali yo‘qotish, coupling matrix ni kichik block larga keltirish va positivity constraint orqali optimal off-diagonal terms ni aniqlashdir.

Bosqich 1 — Diagonal unitary phase averaging

Manba:

\[ \theta=(\theta_1,\ldots,\theta_n) \in[0,2\pi)^n \]

uchun diagonal unitary operator ni aniqlaydi:

\[ U_\theta := \sum_{k=1}^{n} e^{i\theta_k} |k\rangle\langle k|. \]

Bir coupling \(\omega\in\Gamma(\rho,\sigma)\) uchun:

\[ \omega_\theta := \left( U_\theta^\dagger\otimes U_\theta^\dagger \right) \omega \left( U_\theta\otimes U_\theta \right). \]

So‘ng barcha phase angles bo‘yicha average olinadi:

\[ \widetilde{\omega} := \frac{1}{(2\pi)^n} \int_{[0,2\pi)^n} \omega_\theta\,d\theta. \tag{5} \]

Bu amal “phase twirling” ga o‘xshash symmetrization sifatida qaralishi mumkin. Maqsad coupling ni o‘zgartirish emas, diagonal marginals bilan bog‘liq bo‘lmagan phase-dependent matrix elements ni o‘rtachalash orqali yo‘qotishdir.

Matrix element o‘zgarishi:

\[ \langle ij| \omega_\theta |kl\rangle = e^{i(\theta_k+\theta_l-\theta_i-\theta_j)} \langle ij| \omega |kl\rangle. \tag{6} \]

Partial traces nega o‘zgarmaydi?

\(\rho\) diagonal bo‘lgani uchun:

\[ \begin{aligned} \langle i| \operatorname{Tr}_2\omega_\theta |k\rangle &= \sum_j \langle ij| \omega_\theta |kj\rangle\\ &= e^{i(\theta_k-\theta_i)} \sum_j \langle ij| \omega |kj\rangle\\ &= e^{i(\theta_k-\theta_i)} \langle i|\rho|k\rangle\\ &= \langle i|\rho|k\rangle. \end{aligned} \tag{7} \]

Oxirgi tenglik diagonal \(\rho\) uchun off-diagonal \(\langle i|\rho|k\rangle\) entries \(i\neq k\) bo‘lganda allaqachon nol ekaniga asoslanadi.

Xuddi shuningdek:

\[ \begin{aligned} \langle j| \operatorname{Tr}_1\omega_\theta |l\rangle &= \sum_i \langle ij| \omega_\theta |il\rangle\\ &= e^{i(\theta_l-\theta_j)} \sum_i \langle ij| \omega |il\rangle\\ &= e^{i(\theta_l-\theta_j)} \langle j|\sigma|l\rangle\\ &= \langle j|\sigma|l\rangle. \end{aligned} \tag{8} \]

Demak:

\[ \omega_\theta\in\Gamma(\rho,\sigma) \]

va phase average \(\widetilde{\omega}\) ham xuddi shu coupling set ichida qoladi.

Cost nega phase averaging ostida o‘zgarmaydi?

Antisymmetric state:

\[ |i\wedge j\rangle \]

diagonal unitary pair ostida faqat global phase oladi:

\[ (U_\theta\otimes U_\theta) |i\wedge j\rangle = e^{i(\theta_i+\theta_j)} |i\wedge j\rangle. \]

Global phase projector expectation value ni o‘zgartirmagani uchun:

\[ \sum_{i

Buning natijasi juda muhim: optimal coupling izlanayotganda barcha possible quantum couplings o‘rniga faqat phase-averaged couplings ustida minimization qilish yetarli.

Bosqich 2 — Phase averaging qaysi matrix elements ni qoldiradi?

Average matrix element:

\[ \langle ij| \widetilde{\omega} |kl\rangle = \frac{ \langle ij|\omega|kl\rangle }{ (2\pi)^n } \int_{[0,2\pi)^n} e^{i(\theta_k+\theta_l-\theta_i-\theta_j)} d\theta. \tag{10} \]

Bu integral faqat index multisets:

\[ \{i,j\}=\{k,l\} \]

bo‘lganda nonzero qolishi mumkin.

Shu sabab potentially nonzero matrix elements:

\[ \langle ii| \widetilde{\omega} |ii\rangle =:\gamma_{ii}, \]

\[ \langle ij| \widetilde{\omega} |ij\rangle =:\gamma_{ij}, \qquad \langle ji| \widetilde{\omega} |ji\rangle =:\gamma_{ji}, \]

va:

\[ \langle ij| \widetilde{\omega} |ji\rangle =:z_{ij}, \qquad \langle ji| \widetilde{\omega} |ij\rangle = \overline{z}_{ij}. \tag{11} \]

ko‘rinishidadir.

Hermiticity ikkinchi off-diagonal entry complex conjugate bo‘lishini talab qiladi.

Positive semidefiniteness esa har bir \(2\times2\) block uchun:

\[ \gamma_{ij}\ge0, \qquad \gamma_{ji}\ge0, \qquad |z_{ij}|^2 \le \gamma_{ij}\gamma_{ji} \]

constraint ni beradi.

\(\gamma\) nega haqiqatan classical coupling bo‘ladi?

Phase-averaged coupling ning diagonal entries i:

\[ \sum_j\gamma_{ij} = \sum_j \langle ij| \widetilde{\omega} |ij\rangle = \langle i|\rho|i\rangle = p_i \]

va:

\[ \sum_i\gamma_{ij} = \sum_i \langle ij| \widetilde{\omega} |ij\rangle = \langle j|\sigma|j\rangle = q_j \]

shartlarini qanoatlantiradi.

Shu sabab \((\gamma_{ij})\), \(P\) va \(Q\) orasidagi ordinary classical coupling matrix dir.

Bosqich 3 — Quantum cost qanday qilib classical coupling formulasiga aylanadi?

Phase-averaged coupling uchun cost:

\[ \sum_{i

Positive-semidefinite constraint:

\[ |z_{ij}| \le \sqrt{\gamma_{ij}\gamma_{ji}} \]

bo‘lgani uchun:

\[ \operatorname{Re}z_{ij} \le |z_{ij}| \le \sqrt{\gamma_{ij}\gamma_{ji}}. \]

Cost ichidagi had \(-2\operatorname{Re}z_{ij}\) bo‘lgani sabab cost ni kamaytirish uchun mumkin bo‘lgan eng katta \(\operatorname{Re}z_{ij}\) kerak. Quyi chegara:

\[ \begin{aligned} & \frac{1}{2} \sum_{i

Bu lower bound faqat formal bound bo‘lib qolmaydi. Manba:

\[ z_{ij} = \sqrt{\gamma_{ij}\gamma_{ji}} \]

tanlovi bilan bound ga attain qilish mumkinligini ko‘rsatadi.

Shunday qilib, Proposition 1 dagi Tenglama (4) aniq tenglik sifatida olinadi.

Quantum transport nega classical transport dan pastroq chiqishi mumkin?

Manbaning Remark 2 siga ko‘ra, diagonal states uchun quantum transport cost classical transport muqobilidan kichik bo‘lishining matematik manbai quantum coupling da ruxsat etilgan off-diagonal \(z_{ij}\) hadlari cost ichida \(-2\sqrt{\gamma_{ij}\gamma_{ji}}\) ko‘rinishida kamaytiruvchi interference hissasini hosil qila olishidir.

Tenglama (4) ishlatilganda:

\[ \begin{aligned} &W_{C_E,p} \left( \operatorname{diag}(P), \operatorname{diag}(Q) \right) \\ &= \min_{\gamma\in\Gamma_{\mathrm{cl}}(P,Q)} \left[ \frac{1}{2} \sum_{i

\(E\) haqiqiy distance matrix bo‘lganda eng o‘ngdagi ifoda classical \(p\)-Wasserstein distance ning \(2^{-1/p}\) koeffitsiyenti bilan masshtablangan ko‘rinishidir.

Bu yerdagi “arzonroq transport” iborasi fizik energiya sarfi degani emas. U faqat aniqlangan mathematical objective function optimal qiymati kichikroq bo‘lishini bildiradi.

Manbaning normalization taklifi

Miller ushbu \(2^{-1/p}\) factor sabab avvalgi ta’rifda:

\[ \min_{\rho_{12}\in\Gamma(\rho_1,\rho_2)} \left[ 2\operatorname{Tr} C_E^p\rho_{12} \right]^{1/p} \]

ko‘rinishida factor 2 ishlatilishi classical states va classical couplings bilan cheklanganda standard classical \(p\)-Wasserstein distance ni bevosita qayta olishni ta’minlashini aytadi.

Bu nuqta manbada muallifning matematik normalization bahosidir. Verianla uni yangi qabul qilingan standart sifatida taqdim etmaydi.

Ikki o‘lchamli maxsus holat nega muhim?

\(n=2\) holati muhim, chunki avvalgi ishda aynan shu maxsus o‘lchamda triangle inequality qanoatlantirilishi isbotlangan; Millerning Proposition 1 formulasi ham \(n=2\) uchun oldingi ma’lum natijani qayta hosil qilib, yangi formula uchun moslik tekshiruvini beradi.

Manba Remark 3 da:

\[ n=2, \qquad p=2, \qquad E= \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} \]

tanlovini ko‘rib chiqadi.

Manba matnida states:

\[ \rho=\operatorname{diag}(r,1-r), \qquad \sigma=\operatorname{diag}(s,1-s) \]

ko‘rinishida kiritiladi.

Notatsiya qaydi: Darhol keyingi coupling matrix va Tenglama (13) manbada \(r,s\) o‘rniga \(p,q\) belgilaridan foydalanadi. Quyida manba tenglamasining o‘z \(p,q\) notatsiyasi saqlangan.

Classical couplings bitta \(\lambda\) parametri bilan:

\[ \gamma= \begin{bmatrix} \lambda & p-\lambda\\ q-\lambda & 1-p-q+\lambda \end{bmatrix}, \]

\[ \max\{0,p+q-1\} \le \lambda \le \min\{p,q\}. \]

ko‘rinishidadir.

Tenglama (4) ishlatilganda:

\[ W_{C_E,2}(\rho,\sigma) = \frac{1}{\sqrt{2}} \min_{\lambda} \left| \sqrt{p-\lambda} - \sqrt{q-\lambda} \right|. \]

Minimum natijasida manba Tenglama (13) ni oladi:

\[ W_{C_E,2}(\rho,\sigma) = \begin{cases} \frac{1}{\sqrt{2}} \left| \sqrt{p}-\sqrt{q} \right|, & p+q\le1, \\[6pt] \frac{1}{\sqrt{2}} \left| \sqrt{1-p}-\sqrt{1-q} \right|, & p+q>1. \end{cases} \tag{13} \]

Ushbu expression shuningdek:

\[ \frac{1}{\sqrt{2}} \max \left\{ |\sqrt{p}-\sqrt{q}|, |\sqrt{1-p}-\sqrt{1-q}| \right\} \]

ko‘rinishiga ekvivalent.

Manba ekvivalentlikni:

\[ p+q\le1 \Longleftrightarrow |\sqrt{p}-\sqrt{q}| \ge |\sqrt{1-p}-\sqrt{1-q}| \]

munosabatiga bog‘laydi.

Tadqiqot Usuli va Natijalari

Uch o‘lchamli qarshi misollar oilasi qanday quriladi?

Qarshi misol uchta diagonal \(3\times3\) density matrix ning ikki parametrli oilasi bilan quriladi; \(s,t\) admissible region ichida tanlanganda direct \(\rho\rightarrow\tau\) quantum transport value, \(\sigma\) orqali o‘tadigan ikki transport value yig‘indisidan katta bo‘ladi va triangle inequality buziladi.

Uchta density matrix

Manba Tenglama (14) da:

\[ \rho := \operatorname{diag}(1-s,s,0), \]

\[ \sigma := \operatorname{diag}(s,1-s-t,t), \]

\[ \tau := \operatorname{diag}(t,1-s-t,s) \tag{14} \]

ta’riflaridan foydalanadi.

Parameter space:

\[ \Delta := \left\{ (s,t) \ \middle|\ 0\le t\le s\le\frac{1}{2}, \quad 2s+t\le1 \right\}. \]

Bu constraints tasodifiy emas. Barcha diagonal entries nonnegative bo‘lishi va states haqiqiy probability distributions/density matrices bo‘lib qolishi kerak.

Masalan, \(\rho\) uchun entries:

\[ 1-s,\ s,\ 0 \]

bo‘lib, yig‘indisi:

\[ (1-s)+s+0=1. \]

\(\sigma\) va \(\tau\) uchun ham entries 1 ga yig‘iladi.

Uchta pairwise transport value

Manba bu bo‘limda \(W_{C_E,2}\) ni qisqacha \(W\) bilan belgilaydi va avval quyidagi uchta closed-form natijani isbotlaydi:

\[ W(\rho,\sigma) = \sqrt{ \frac{1}{2} - \sqrt{s(1-s-t)} }. \tag{15} \]

\[ W(\rho,\tau) = \sqrt{ \frac{1}{2}(1+t-s) - \sqrt{t(1-2s)} }. \tag{16} \]

\[ W(\sigma,\tau) = \frac{1}{\sqrt{2}} \left( \sqrt{s}-\sqrt{t} \right). \tag{17} \]

\(\Delta\) ichida \(t\le s\) bo‘lgani uchun \(\sqrt{s}-\sqrt{t}\ge0\), demak Tenglama (17) da mutlaq qiymat talab qilmaydigan musbat tartib saqlanadi.

Tenglama (15) qanday chiqariladi?

\(\rho\) va \(\sigma\) ning probability vectors i:

\[ P=(1-s,s,0), \qquad Q=(s,1-s-t,t) \]

ko‘rinishida yoziladi.

Manba classical coupling family ni:

\[ \gamma= \begin{bmatrix} \lambda & 1-s-t-\lambda+\mu & t-\mu \\ s-\lambda & \lambda-\mu & \mu \\ 0&0&0 \end{bmatrix} \]

ko‘rinishida parametrizes qiladi.

Nonnegativity constraints:

\[ \lambda\in[0,s], \qquad \mu\in[0,\min\{\lambda,t\}] \]

ko‘rinishida beriladi.

Tenglama (4) qo‘llanganda:

\[ W^2(\rho,\sigma) = \frac{1}{2} \min_{\lambda,\mu} \left[ \left( \sqrt{1-s-t-\lambda+\mu} - \sqrt{s-\lambda} \right)^2 +t \right]. \]

Manba argumentida objective function avval \(\mu\) bo‘yicha nondecreasing bo‘lgani uchun minimum:

\[ \mu=0 \]

nuqtada olinadi.

So‘ng qolgan ifoda \(\lambda\) bo‘yicha ham nondecreasing bo‘lgani uchun:

\[ \lambda=0 \]

tanlanadi.

Shunday qilib:

\[ W^2(\rho,\sigma) = \frac{1}{2} \left[ \left( \sqrt{1-s-t} -\sqrt{s} \right)^2 +t \right]. \]

Kvadrat ochilganda:

\[ \left( \sqrt{1-s-t} -\sqrt{s} \right)^2 = 1-t-2s - 2\sqrt{s(1-s-t)} + 2s \]

va bunga \(t\) qo‘shilganda:

\[ 1-2\sqrt{s(1-s-t)} \]

hosil bo‘ladi.

Demak:

\[ W^2(\rho,\sigma) = \frac{1}{2} - \sqrt{s(1-s-t)}, \]

va musbat square root olinib Tenglama (15) hosil qilinadi.

Tenglama (16) qanday chiqariladi?

\(\rho\) va \(\tau\) uchun:

\[ P=(1-s,s,0), \qquad R=(t,1-s-t,s). \]

Manba coupling matrix ni:

\[ \gamma= \begin{bmatrix} \lambda & 1-2s-\lambda+\mu & s-\mu \\ t-\lambda & s-t+\lambda-\mu & \mu \\ 0&0&0 \end{bmatrix}. \]

Admissible parameters:

\[ \lambda\in[0,t], \qquad \mu\in[0,s-t+\lambda]. \]

Tenglama (4):

\[ W^2(\rho,\tau) = \frac{1}{2} \min_{\lambda,\mu} \left[ \left( \sqrt{1-2s-\lambda+\mu} - \sqrt{t-\lambda} \right)^2 +s \right]. \]

Manba yana objective function avval \(\mu\), so‘ng \(\lambda\) bo‘yicha nondecreasing ekanidan foydalanadi. Minimum:

\[ \mu=0, \qquad \lambda=0 \]

nuqtasida.

Shunday qilib:

\[ W^2(\rho,\tau) = \frac{1}{2} \left[ \left( \sqrt{1-2s} -\sqrt{t} \right)^2 +s \right]. \]

Bu ifoda tartibga keltirilganda:

\[ W^2(\rho,\tau) = \frac{1}{2}(1+t-s) - \sqrt{t(1-2s)} \]

va shu tariqa Tenglama (16) olinadi.

Tenglama (17) nega boshqacharoq isbotni talab qiladi?

\(\sigma\) va \(\tau\) orasidagi barcha classical couplings ni bevosita parametrizes qilish manbada to‘rtta parameter talab qilishi sabab, Miller buning o‘rniga objective function uchun lower bound tuzadi va keyin ushbu lower bound ga erishadigan aniq coupling ni ko‘rsatadi.

Ta’rif:

\[ \Phi(\gamma) := \sum_{1\le i

Avval nonnegative bo‘lgan bitta summand tashlab yuborilib:

\[ \Phi(\gamma) \ge \left( \sqrt{\gamma_{12}} -\sqrt{\gamma_{21}} \right)^2 + \left( \sqrt{\gamma_{13}} -\sqrt{\gamma_{31}} \right)^2. \]

Bu ikki had:

\[ \sum_{j=1}^{3} \left( \sqrt{\gamma_{1j}} - \sqrt{\gamma_{j1}} \right)^2 \]

ko‘rinishida yoziladi.

Cauchy–Schwarz inequality qo‘llanib:

\[ \sum_{j=1}^{3} \left( \sqrt{\gamma_{1j}} - \sqrt{\gamma_{j1}} \right)^2 \ge \left( \sqrt{ \sum_{j=1}^{3}\gamma_{1j} } - \sqrt{ \sum_{j=1}^{3}\gamma_{j1} } \right)^2. \]

Coupling marginals sabab:

\[ \sum_j\gamma_{1j}=s, \qquad \sum_j\gamma_{j1}=t. \]

Demak:

\[ \Phi(\gamma) \ge (\sqrt{s}-\sqrt{t})^2. \]

Tenglama (4) bilan:

\[ W^2(\sigma,\tau) = \frac{1}{2} \min_{\gamma\in\Gamma_{\mathrm{cl}}(Q,R)} \Phi(\gamma) \ge \frac{1}{2} (\sqrt{s}-\sqrt{t})^2. \]

Ushbu lower bound manbada berilgan:

\[ \gamma^\ast = \begin{bmatrix} 0&0&s\\ 0&1-s-t&0\\ t&0&0 \end{bmatrix} \]

coupling orqali attain qilinadi.

Shu sabab inequality tenglikka aylanadi:

\[ W^2(\sigma,\tau) = \frac{1}{2} (\sqrt{s}-\sqrt{t})^2 \]

va:

\[ W(\sigma,\tau) = \frac{1}{\sqrt{2}} (\sqrt{s}-\sqrt{t}). \]

Bu Tenglama (17) dir.

Uchburchak tengsizligi aynan qanday buziladi?

Manba admissible parameter region \(\Delta\) ning barcha ichki nuqtalarida \(W(\rho,\tau)-W(\rho,\sigma)-W(\sigma,\tau)>0\) ekanini ko‘rsatadi; shu sabab triangle inequality faqat bitta tasodifiy parameter tanlovida emas, balki ochiq ikki parametrli qarshi misollar oilasida bajarilmaydi.

Tenglama (15)–(17) triangle inequality ning zarur yo‘nalishi bilan solishtirilganda, manba quyidagi teskari tengsizlikni isbotlashni maqsad qiladi:

\[ W(\rho,\tau)> W(\rho,\sigma) + W(\sigma,\tau). \]

Manbaning Tenglama (18) dagi ekvivalent ko‘rinishi:

\[ \sqrt{ 1+t-s - 2\sqrt{t(1-2s)} } > \sqrt{ 1 - 2\sqrt{s(1-s-t)} } + \sqrt{s} - \sqrt{t}, \tag{18} \]

va validity region:

\[ 0

Ushbu shartlar aynan \(\Delta\) ning interior qismini ifodalaydi.

Nega yangi \(r\) va \(a\) o‘zgaruvchilar kiritiladi?

Manba proof ning oxirgi qismini boshqarish osonroq bo‘lishi uchun:

\[ r := \frac{\sqrt{t}}{\sqrt{s}} \]

va:

\[ a := \frac{ \sqrt{1-2s}-\sqrt{t} }{ \sqrt{s} } \]

ta’riflaridan foydalanadi.

Interior constraints sabab:

\[ 00. \]

Tenglama (18) ushbu variables bilan:

\[ \sqrt{1+a^2} +r-1 > \sqrt{ \left( \sqrt{1+a^2+2ar}-1 \right)^2 +r^2 }. \tag{19} \]

ko‘rinishiga keltiriladi.

Har ikki tomon musbat bo‘lgani sabab squares olish mumkin. Algebraic cancellation dan keyin talab qilinadigan inequality:

\[ (1-r) \sqrt{1+a^2} + r(1+a) < \sqrt{1+a^2+2ar}. \tag{20} \]

bo‘ladi.

Strict concavity oxirgi bosqichni nega hal qiladi?

Manba:

\[ f(r) = \sqrt{1+a^2+2ar} \]

function ni \([0,1]\) interval da ko‘rib chiqadi.

\(a>0\) uchun bu function strictly concave dir.

Strict concavity ning asosiy natijasi, \(0

\[ f(r)> (1-r)f(0)+rf(1). \]

Endpoint values:

\[ f(0)=\sqrt{1+a^2} \]

va:

\[ f(1) = \sqrt{1+a^2+2a} = \sqrt{(1+a)^2} = 1+a \]

bo‘lgani uchun:

\[ f(r)> (1-r)\sqrt{1+a^2} + r(1+a). \]

Bu aynan Tenglama (20) dir.

Shunday qilib:

\[ W(\rho,\tau)> W(\rho,\sigma)+W(\sigma,\tau) \]

butun interior \(\Delta^\circ\) da isbotlanadi.

Figure 1 ning matematik mazmuni

Manba Figure 1:

\[ g(s,t) := W(\rho,\tau) - W(\rho,\sigma) - W(\sigma,\tau) \]

function ni admissible parameter region \(\Delta\) ustida ko‘rsatadi.

Manba caption iga ko‘ra:

• \(g(s,t)=0\), boundary \(\partial\Delta\) ustida;
• \(g(s,t)>0\), interior \(\Delta^\circ\) ustida.

Demak, grafik analitik proof ning vizual muqobilidir; proof o‘rnini bosuvchi numerical evidence emas.

Qarshi misol nega barcha \(n>3\) o‘lchamlarga kengayadi?

Manba uch-dimensional counterexample states ni higher-dimensional space ichiga:

\[ \rho\oplus\mathbf{0}_{n-3}, \qquad \sigma\oplus\mathbf{0}_{n-3}, \qquad \tau\oplus\mathbf{0}_{n-3} \]

ko‘rinishida joylashtiradi.

\(\mathbf{0}_{n-3}\) qo‘shimcha dimensions ustidagi zero block dir.

Ushbu embedding dastlabki uch dimensions dagi probability structure ni o‘zgartirmaydi. Shu sabab \(n=3\) da mavjud triangle-inequality violation yuqori dimensions ga ham ko‘chadi.

Bu yerda to‘g‘ri xulosa:

Qarshi misol \(n=3\) dan boshlanadi va barcha \(n>3\) dimensions uchun ham counterexample hosil qiladi.

Noto‘g‘ri xulosa esa:

\(n=2\) uchun ham ayni failure isbotlandi.

bo‘lar edi. Manba aksincha \(n=2\) ni ma’lum maxsus metrik holatlardan biri sifatida eslatadi.

Conjecture I nega rad etiladi?

Conjecture I \(C^Q\) cost matrix bilan aniqlangan quantum 2-Wasserstein semidistance barcha dimensions uchun triangle inequality ni qanoatlantirishini ilgari surgan.

Millerning family si \(n=3\) uchun ochiq violation bergan zahoti “barcha \(n\)” da’vosi rad etiladi.

Bitta haqiqiy counterexample universal mathematical conjecture ni rad etish uchun yetarli. Bu tadqiqot esa bitta isolated point o‘rniga \(\Delta^\circ\) ichida keng ikki parametrli family ni ko‘rsatadi.

Conjecture II nega saqlanib qolmaydi?

Conjecture II ushbu metrik xatti-harakat \(C^Q\) ning kichik perturbations atrofi da ham saqlanishini taxmin qilgan. Tadqiqot \(C^Q\) ning o‘zi uchun asosiy triangle-inequality property bajarilmasligini ko‘rsatgani sabab, \(C^Q\) markazidagi ushbu conjectured metric neighborhood ning asosi ham yo‘qoladi. Manba abstract va introduction bo‘limlarida har ikki conjecture ham ushbu analytical counterexamples bilan disproved qilinganini aniq bildiradi.

Tadqiqot qo‘llab-quvvatlaydigan natijalar

• Umumiy \(W_{C_E,p}\) construction semidistance sifatida qaralishi mumkin, biroq triangle inequality umumiy holatda kafolatlanmaydi.
• Maxsus antisymmetric projector \(C^Q\) uchun aniqlangan quantum 2-Wasserstein quantity \(n=3\) da triangle inequality ni buzadigan diagonal-state counterexamples ga ega.
• Ushbu counterexamples zero-block embedding orqali barcha \(n>3\) o‘lchamlarga kengaytirilishi mumkin.
• Diagonal states uchun \(W_{C_E,p}\) Proposition 1 da berilgan classical-coupling minimization formulasiga keltirilishi mumkin.
• Quantum/classical transport farqida off-diagonal coupling terms hal qiluvchi matematik rol o‘ynaydi.
• Manbada aniqlangan \(\Delta^\circ\) sohasining barchasida triangle defect musbat.
• \(n=2\) va pure-state restrictions kabi avvaldan ma’lum maxsus natijalar ushbu counterexample bilan bekor qilinmaydi.

Tadqiqot qo‘llab-quvvatlamaydigan natijalar

• Tadqiqot “barcha quantum Wasserstein distances noto‘g‘ri” degan natijani ko‘rsatmaydi.
• Bu maqola yolg‘iz o‘zi barcha quantum optimal transport formulations metrik bo‘la olmasligini isbotlamaydi.
• Tadqiqot fizik quantum transport experiment bajarmaydi.
• Tadqiqot kvant kompyuterida algorithm benchmark qilmaydi.
• “Quantum transport cheaper” iborasi energiya, vaqt yoki iqtisodiy xarajat ustunligini anglatmaydi.
• Qarshi misol \(n=2\) holatini rad etmaydi.
• Manba \(p<2\), barcha possible cost operators yoki barcha muqobil quantum-transport definitions uchun yagona universal no-go theorem ni ilgari surmaydi.
• Figure 1 numerical simulation orqali teoremani isbotlamaydi; asosiy natija analytical proof dan keladi.

Belgi va tushunchalar jadvali

O‘lcham va birlik qaydi

Ushbu tadqiqot matematik quantum-information ishidir. Density-matrix entries, probabilities, coupling coefficients, \(s,t,r,a\) parameters va aniqlangan normalized transport quantities fizik SI birliklarida berilmaydi. Shu sabab metr, joul, soniya yoki boshqa dimensional units ushbu tenglamalarda mavjud emas.

Manbadagi notatsiya ogohlantirishi

Remark 3 da manba:

\[ \rho=\operatorname{diag}(r,1-r), \qquad \sigma=\operatorname{diag}(s,1-s) \]

ta’rifidan darhol keyin coupling matrix ichida:

\[ p,\ q \]

belgilariga o‘tadi.

Manba bu notation change uchun alohida izoh bermaydi. Verianla matematik kontekstdan \(p\) va \(q\) bu yerda tegishli two-state probability parameters rolida ishlatilishini ko‘rsatishi mumkin; biroq asl manbadagi harf almashishini ko‘rinmas tarzda qayta yozib, “manba shunday yozgan” degan taassurot yaratmaydi.

Acknowledgement ichidagi sun’iy intellekt ma’lumoti

Maqolaning Acknowledgements bo‘limida muallif qarshi misollarni topishda ChatGPT 5.4 Pro yordamidan foydalanganini bildiradi.

Bu ibora tadqiqotning matematik isbotlari avtomatik ravishda tekshirilganini anglatmaydi. E’lon qilingan preprintda analitik proof zanjiri ochiq yozilgan va ilmiy baholash ushbu proof ning matematik haqiqiyligi asosida amalga oshirilishi kerak.

Manba va Usul Haqida Izoh

Asl sarlavha: Comment on ‘Quantum Monge–Kantorovich Problem and Transport Distance between Density Matrices’

Muallif: Tomasz Miller

Mualliflar tartibi: Yakka muallifli tadqiqot.

Corresponding author: Tomasz Miller; manba PDF da yulduzcha belgisi bilan aloqa muallifi sifatida ko‘rsatilgan.

E-pochta: tomasz.miller@uj.edu.pl

Teng hissa: Qo‘llanilmaydi; tadqiqot yakka muallifli.

Muassasa: Copernicus Center for Interdisciplinary Studies, Jagiellonian University, Szczepańska 1/5, 31-011 Kraków, Poland.

Soha: Quantum Physics, Mathematical Physics, quantum optimal transport, density-matrix geometry.

Manba turi: Nazariy/matematik comment preprint.

Taqriz holati: Yuklangan manba arXiv preprintidir. Verianla ko‘rib chiqishi vaqtida tasdiqlangan rasmiy yozuv arXiv v1 yozuvidir; yuklangan faylda alohida taqrizlangan journal publication ma’lumoti mavjud emas.

arXiv: arXiv:2607.07764v1 [quant-ph].

DOI: 10.48550/arXiv.2607.07764.

arXiv topshirilgan sana: 8 Iyul 2026.

PDF dagi manuscript sanasi: 10 Iyul 2026.

Platforma: arXiv.

Asosiy mavzu toifasi: Quantum Physics (quant-ph).

Bog‘liq soha: Mathematical Physics.

Rasmiy havola: https://arxiv.org/abs/2607.07764

DOI havolasi: https://doi.org/10.48550/arXiv.2607.07764

Sahifalar soni: 7.

Shakllar soni: 1.

Jadval: Manbada alohida raqamlangan jadval mavjud emas.

Tajriba/simulyatsiya holati: Fizik tajriba yo‘q. Kvant uskunasi ishlatilmagan. Asosiy natija analytical counterexample va proof dir. Figure 1 analitik formulalardan aniqlangan triangle-defect function ning parameter-space ko‘rinishidir.

Asosiy muammo: Friedland, Eckstein, Cole va Życzkowski tomonidan aniqlangan quantum 2-Wasserstein semidistance maxsus \(C^Q\) cost matrix uchun barcha dimensions da haqiqiy metrikmi yoki yo‘qmi, shuni sinash.

Asosiy natija: \(n=3\) diagonal states uchun keng counterexample family da triangle inequality buziladi; family \(n>3\) ga zero-block embedding bilan kengaytirilishi mumkin.

Proposition 1: Diagonal density matrices orasidagi \(W_{C_E,p}\) classical couplings ustida \(\frac12\sum E_{ij}^p(\sqrt{\gamma_{ij}}-\sqrt{\gamma_{ji}})^2\) objective bilan ifodalanadi.

Muhim maxsus holat: \(n=2\) va pure-state restrictions uchun avvaldan ma’lum metrik natijalar ushbu tadqiqot bilan rad etilmaydi; manba ular uchun zarur \(p\ge2\) shartini ko‘rsatadi.

Manbadagi notatsiya muammosi: Remark 3 da two-level states \(r,s\) bilan aniqlanib, keyin coupling va Tenglama (13) da \(p,q\) belgilariga o‘tiladi. Manba buning uchun alohida izoh bermaydi.

Normalization izohi: Muallif classical Wasserstein limitini bevosita qayta olish nuqtai nazaridan Tenglama (2) da cost ni factor 2 bilan normallashtirish ma’qulroq bo‘lishi mumkinligini Remark 2 da taklif qiladi. Bu manbaning matematik izohi sifatida yetkazilgan.

Litsenziya: Yuklangan PDF da aniq Creative Commons litsenziya bayonoti ko‘rinmaydi. Shu sabab Verianla mazmunida manba matni yoki asl Figure 1 aynan qayta nashr etilmagan; ilmiy ma’lumot mustaqil o‘quv matni va original SVG tuzilmasi orqali yetkazilgan.

Moliyalashtirish: Manbada alohida funding statement berilmagan.

Manfaatlar to‘qnashuvi: Manbada alohida conflict-of-interest statement berilmagan.

Ma’lumotlar mavjudligi: Tadqiqot tajribaviy dataset ishlatmaydi; alohida data-availability statement berilmagan.

Muallif hissalari: Yakka muallifli ishda alohida CRediT breakdown berilmagan.

Acknowledgement: Muallif qarshi misollar ChatGPT 5.4 Pro yordamida topilganini bildirgan.

Manba tayangan asosiy ish: Shmuel Friedland, Michał Eckstein, Sam Cole va Karol Życzkowski, “Quantum Monge–Kantorovich problem and transport distance between density matrices”, Physical Review Letters 129, 110402 (2022).

Ikkinchi asosiy manba: Sam Cole, Michał Eckstein, Shmuel Friedland va Karol Życzkowski, “On quantum optimal transport”, Mathematical Physics, Analysis and Geometry 26, 14 (2023).

Uchinchi manba: Tomasz Miller va R. Bistroń, “Distances between pure quantum states induced by a distance matrix”, arXiv:2509.14727 [math-ph] (2025).