Kvant Qurilmasiga Ishonmasdan Tekshirish: Har Qanday Kvant Holati va O‘lchovi uchun Universal Self-Testing Sxemasi

Kvant hisoblash va kvant aloqa qurilmalari rivojlangan sari ularning ichki tuzilishiga ishonmasdan to‘g‘ri ishlayotganini tekshirish muhim bo‘lib bormoqda. Device-independent sertifikatlash aynan shuni maqsad qiladi. Self-testing esa kuzatilgan statistik korrelyatsiyalarning o‘zidan noma’lum kvant holati va o‘lchovlarini, mahalliy unitary ekvivalentlik va yordamchi erkinlik darajalarigacha, aniqlashga harakat qiladi.

Ushbu ish \(N\) ta tashqi tomon va bitta markaziy Eve dan iborat star quantum networkdan foydalanadi. Har bir tashqi tomon Eve bilan alohida va statistik mustaqil kvant manbasini bo‘lishadi. Avval Pauli o‘lchovlari, maksimal entangled ikki-qubit manbalar va Eve ning GHZ-basis birinchi o‘lchovi self-test qilinadi. Keyin shu sertifikatlangan tuzilma arbitrary extremal POVM ni tekshirish uchun tomografik tayanch sifatida ishlatiladi.

Projective measurements extremal bo‘lgani uchun ular to‘liq qamrab olinadi. Non-extremal POVM extremal POVMlarning convex mixture shaklida ifodalanib bilvosita self-test qilinadi. Certified remote state preparation orqali istalgan pure state va tegishli \(3d\)-outcome extremal POVM konstruksiyasi orqali istalgan mixed state ham sertifikatlanadi.

Natija matematik jihatdan juda umumiy, lekin tajribaviy xarajat bepul emas. Source independence sharti muhim, dimension oshishi bilan correlation conditions soni tez o‘sadi va GHZ-basis measurement optical platformlarda murakkab.

Star network

\[ \rho_{AE} = \bigotimes_{i=1}^{N}\rho_{A_iE_i}. \tag{1} \]

Har bir \(A_i\) uchta binary measurementga ega:

\[ x_i\in\{0,1,2\}, \qquad a_i\in\{0,1\}. \]

Eve ikki measurement inputdan birini tanlaydi:

\[ e\in\{0,1\}. \]

\(e=0\) — \(2^N\)-outcome certification measurement, \(e=1\) — \(K\le2^N\)-outcome target measurement.

Observed probabilities

\[ p(al|xe)= \operatorname{Tr} \left[ \rho_{AE} \left( \bigotimes_{i=1}^{N}M^{a_i}_{i|x_i} \right) \otimes R_{l|e} \right]. \tag{2} \]

\[ A_{i,x_i}=M^0_{i|x_i}-M^1_{i|x_i}. \]

\[ \left\langle A_{1,x_1}\cdots A_{N,x_N}R_{l|e} \right\rangle = \sum_{a_1,\ldots,a_N} (-1)^{\sum_i a_i}p(al|xe). \tag{3} \]

Self-testing equivalence

\[ (U_{A_i}\otimes U_{E_i}) |\psi_{A_iE_i}\rangle = |\psi'_{A'_iE'_i}\rangle \otimes|\xi_{A''_iE''_i}\rangle. \tag{4} \]

\[ U_{A_i}A_{i,x_i}U_{A_i}^{\dagger} = A'_{i,x_i}\otimes\mathbb I, \qquad U_ER_{l|e}U_E^\dagger = R'_{l|e}\otimes\mathbb I. \tag{5} \]

\[ U_A\rho_A^{l|e}U_A^\dagger = \widetilde\rho_{A'}^{l|e} \otimes\varrho_{A''}^{l|e}. \tag{6} \]

Oxirgi tenglama device-independent certified remote preparationni ifodalaydi.

Bell inequalities

\[ \begin{aligned} I_l=(-1)^{l_1}\Bigg[ &(N-1) \left\langle \widetilde A_{1,1} \prod_{i=2}^{N}A_{i,1} \right\rangle\\ &+\sum_{i=2}^{N}(-1)^{l_i} \left\langle \widetilde A_{1,0}A_{i,0} \right\rangle\\ &-(-1)^{l_1} \sum_{i=2}^{N}(-1)^{l_i} \left\langle A_{1,2}A_{i,2} \prod_{\substack{j=2\\j\neq i}}^{N}A_{j,1} \right\rangle \Bigg]. \end{aligned} \tag{7} \]

\[ \widetilde A_{1,0} = \frac{A_{1,0}-A_{1,1}}{\sqrt2}, \qquad \widetilde A_{1,1} = \frac{A_{1,0}+A_{1,1}}{\sqrt2}. \tag{8} \]

Classical bound:

\[ \beta_C=(\sqrt2+1)(N-1). \]

Quantum bound:

\[ \beta_Q=3(N-1). \]

Reference observables

\[ A_{1,0}=\frac{X+Z}{\sqrt2}, \quad A_{1,1}=\frac{X-Z}{\sqrt2}, \quad A_{1,2}=Y, \]

\[ A_{i,0}=Z,\quad A_{i,1}=X,\quad A_{i,2}=Y. \tag{9} \]

GHZ-like states

\[ |\phi_l\rangle = \frac1{\sqrt2} \left( |l_1\ldots l_N\rangle+ (-1)^{l_1} |\bar l_1\ldots\bar l_N\rangle \right), \qquad \bar l_i=1-l_i. \tag{10} \]

Theorem 1

Agar barcha Bell inequalities maksimal quantum valuega yetsa va

\[ P(l|0)=\frac1{2^N}, \]

bo‘lsa, tashqi Pauli measurements, har bir source dagi maximal two-qubit entangled state va Eve ning GHZ-basis measurementi self-test qilinadi.

\[ (U_{A_i}\otimes U_{E_i})|\psi_{A_iE_i}\rangle = |\phi^+\rangle\otimes|\xi\rangle. \]

SOS proof structure

Sum-of-squares decomposition maksimal Bell violation vaqtida barcha positive square termlarni nolga majbur qiladi:

\[ (-1)^{l_1}\widetilde A_{1,1}|\psi\rangle = \bigotimes_{i=2}^{N}A_{i,1}|\psi\rangle, \]

\[ (-1)^{l_1+l_i}\widetilde A_{1,0}|\psi\rangle = A_{i,0}|\psi\rangle. \]

Shundan \(X\)- va \(Z\)-type observables anticommutation relations olinadi.

Y sign ambiguity

\[ U_{A_i}A_{i,2}U_{A_i}^{\dagger} = \pm Y\otimes\mathbb I. \]

Bu global sign complex-conjugation ambiguity bilan bog‘liq.

Arbitrary extremal POVM

\[ \left\langle \widetilde A_{1,i_1} \otimes \bigotimes_{k=2}^{N}A_{k,i_k} \otimes R_{l|1} \right\rangle = f^{\,l}_{i_1,\ldots,i_N}. \tag{11} \]

\(f\) coefficients reference POVMning Pauli-tensor decompositionidan olinadi. Extremality auxiliary degrees of freedom ichida turli POVMlarning bir xil average statisticsni berishini yo‘qqa chiqaradi.

\[ U_ER_{l|1}U_E^\dagger = (R'_{l|1})^*\otimes\mathbb I \]

yoki mos sign tanlovida

\[ U_ER_{l|1}U_E^\dagger = R'_{l|1}\otimes\mathbb I. \]

Finite dimension embedding

\[ D\le2^N, \qquad \sum_lR'_{l|1}+M_\perp=\mathbb I_{2^N}. \]

Shu orqali arbitrary finite-dimensional extremal measurement N-qubit frameworkga joylashtiriladi.

Non-extremal measurements

\[ \mathcal E = \sum_ep_e\mathcal E_e^{\mathrm{ex}}. \]

Extremal components alohida self-test qilinib, mixture bilvosita sertifikatlanadi.

Pure states

\[ R'_{0|1}=|\psi\rangle\langle\psi|. \]

Bu projective element Eve outcome orqali istalgan pure state ni tashqi tomonlarda sertifikatlangan holda tayyorlaydi.

Mixed states

\[ \rho=\sum_kp_k|\psi_k\rangle\langle\psi_k|. \tag{12} \]

\[ M_{k,1}=p_k|\psi_k\rangle\langle\psi_k|, \]

\[ M_{k,2}= \frac{2-p_k}{2} |\tau_{k,2}\rangle\langle\tau_{k,2}|, \qquad M_{k,3}= \frac{2-p_k}{2} |\tau_{k,3}\rangle\langle\tau_{k,3}|. \]

\[ |\tau_{k,2}\rangle = \sqrt{\frac{1-p_k}{2-p_k}}|\psi_k\rangle + \sqrt{\frac1{2-p_k}}|\varphi_k\rangle, \]

\[ |\tau_{k,3}\rangle = -\sqrt{\frac{1-p_k}{2-p_k}}|\psi_k\rangle + \sqrt{\frac1{2-p_k}}|\varphi_k\rangle. \]

Natijadagi \(3d\)-outcome rank-one POVM extremal bo‘ladi. Kerakli \((k,1)\) outcomes post-select qilinganda

\[ U_A\rho_AU_A^\dagger = \rho_{A'}\otimes\widetilde\rho_{A''} \tag{13} \]

yoki complex conjugate olinadi. Success probability \(1/2^N\).

Verianla Live: Universal self-testing jarayoni

Bosqich	Kirish	Sertifikatlanadi	Shart
1	N independent source	Maximal Bell pairs	Source independence
2	2^N Bell expressions	Pauli measurements	I_l=3(N−1)
3	Eve e=0	GHZ-basis	P(l\|0)=1/2^N
4	Pauli correlations	Extremal POVM	Eq. (11)
5	POVM outcome	Arbitrary pure state	Remote preparation
6	3d POVM	Arbitrary mixed state	Post-selection
7	Convex mixture	Non-extremal POVM	E=Σp_eE_e^ex

Robustness

\[ \langle\widehat I_l\rangle \ge\beta_Q-\epsilon, \qquad \left|P(l)-\frac1{2^N}\right|\le\epsilon. \]

\[ \left\| |\widetilde\psi_l\rangle- |\phi_l\rangle|\xi_l\rangle \right\| \le 17(N^2-1)\sqrt{2\epsilon}. \]

\[ \left\| |\psi_{A_iE_i}\rangle- |\phi^+\rangle|\xi\rangle \right\| \le [17N(N^2-1)]^2 \sqrt{\frac\epsilon2}. \]

\[ \left\| U_iA_{i,2}U_i^\dagger|\widetilde\psi_l\rangle - Y\otimes K|\widetilde\psi_l\rangle \right\| \le 226N^2\sqrt{2\epsilon}. \]

Bu bounds universal analytical upper bounds; practical experimental optimum emas.

O‘zbekiston konteksti

Tadqiqot O‘zbekistonda eksperimental tekshirilmagan. Ammo quantum network verification, quantum communication va hardware trust muammolari uchun nazariy asos beradi. Mahalliy qo‘llash uchun photon source fidelity, detector efficiency, independent sources va multi-party measurements alohida tajriba bilan tekshirilishi kerak.

Tadqiqot Usuli va Natijalari

Bu theoretical quantum information paper. Dataset yoki statistical p-value yo‘q; natijalar theoremlar va operator identities.

Theorem 1: Bell maximum + uniform Eve outcome → Pauli measurements, Bell sources va GHZ measurement.

Theorem 2: Additional Pauli-tensor statistics → arbitrary extremal POVM.

Corollary 1: Post-measurement states device-independently certified.

Mixed-state construction: \(3d\)-outcome extremal POVM orqali arbitrary mixed state.

Robustness: Deviation bounds \(\sqrt\epsilon\) scaling.

Limitations: Independent sources, high entanglement, GHZ measurement, rapidly increasing complexity.

Manba va Usul Bo‘yicha Izoh

Manba: Shubhayan Sarkar, Alexandre C. Orthey Jr., Remigiusz Augusiak, Nature Physics 22, 446–451 (2026).

DOI: 10.1038/s41567-026-03181-y.

Peer review: Nature Physics hakamli maqolasi.

Uploaded version: arXiv:2312.04405v4 [quant-ph], 8-iyun 2026, 30 sahifa.

ArXiv DOI: 10.48550/arXiv.2312.04405.

ArXiv license: CC BY 4.0.

Funding: QuantERA II VERIqTAS, Horizon 2020 va Polish National Science Center grants.

Competing interests: Mualliflar competing interest yo‘qligini bildirgan.

Appendices: GHZ/source proof, extremal POVM proof, arbitrary-state construction va robustness.

Axborot va tuzatish eslatmasi

Ushbu mazmun tegishli akademik tadqiqot asosida sun’iy intellekt yordamidagi vositalardan foydalanib tayyorlangan. Maqsad tadqiqotning asosiy mavzusi, usuli va natijalarini yanada tushunarli hamda qulay shaklda yetkazishdir.

Mazmun tayyorlashda manba tadqiqotga sodiq qolishga e’tibor berilgan bo‘lsa-da, sun’iy intellekt yordamida mazmun yaratish jarayonida ma’no, ifoda, tarjima yoki talqin bilan bog‘liq xatolar yuz berishi mumkin. Ilmiy, texnik yoki kontekstual xatoni aniqlasangiz, bizga info@verianla.com.tr manzili orqali xabar berishingiz mumkin. Bildirilgan masalalar ko‘rib chiqilib, zarur tuzatishlar kiritiladi.

Bundan tashqari, Verianla saytida tadqiqoti yoritilgan maqolaning muallifi, huquq egasi yoki vakolatli vakili tegishli mazmunni nashrdan olib tashlashni talab qilishi mumkin. Olib tashlash talabi uchun maqola havolasi va talabning qisqa izohini info@verianla.com.tr manziliga yuborish kifoya. Talab ko‘rib chiqilgach, zarur choralar ko‘riladi.

Verianla akademik tadqiqotlarning o‘rnini egallashni emas, ilmiy mazmunni kengroq auditoriya tushunishiga hissa qo‘shishni maqsad qiladi. Akademik yoki professional baholashlarda har doim tadqiqotning asl manbasiga tayanish tavsiya etiladi.

Ulashish: