Klassik QSL$$ T\ge\frac{\pi}{2\Delta H}, \quad \Delta H=\sqrt{\langle H^2\rangle-\langle H\rangle^2}. $$$$ T\ge\frac{\pi}{2\bar H}, \quad \bar H=\langle H\rangle-E_g. $$$$ T_{\mathrm{QSL}}=\max\{T_{\mathrm{MT}},T_{\mathrm{ML}}\}. $$Instantaneous incoherent set$$ H_t|n_t\rangle=E_{n,t}|n_t\rangle, $$$$ \mathcal I_t=\{\sigma_t:[\sigma_t,H_t]=0\}, $$$$ C(\rho_t)=\min_{\sigma_t\in\mathcal I_t}D(\rho_t,\sigma_t). $$Schatten QSL$$ - \frac{dF_{\mathrm{RP}}(\rho_0,\rho_t)}{dt} \le \frac{C_p(\rho_t)\|[H_t,\rho_0]\|_q} {\operatorname{Tr}(\rho_0^2)}. \tag{1} $$$$ C_p(\rho_t)= \min_{\sigma_t\in\mathcal I_t} \|\rho_t-\sigma_t\|_p. \tag{2} $$$$ \frac{1-F_{\mathrm{RP}}(\rho_0,\rho_T)}{T} \le \frac1T\int_0^Tdt\, \frac{C_p(\rho_t)\|[H_t,\rho_0]\|_q} {\operatorname{Tr}(\rho_0^2)}. \tag{3} $$$$ T\ge T_S(p,q) = \frac{[1-F_{\mathrm{RP}}]\operatorname{Tr}(\rho_0^2)} {\frac1T\int_0^Tdt\,C_p(\rho_t)\|[H_t,\rho_0]\|_q}. \tag{4} $$$$ T_{S,\mathrm{Pure}}(1,\infty) = \frac{1-F} {\frac1T\int_0^Tdt\,C_1(|\psi_t\rangle)\Delta H_t(|\psi_0\rangle)}. \tag{5} $$$$ \widetilde T_{S,\mathrm{Pure}}(1,\infty) = \frac{1-F} {C_1(|\psi_0\rangle)\Delta H(|\psi_0\rangle)}. \tag{6} $$$$ T_S(2,2) = \frac{[1-F_{\mathrm{RP}}]\operatorname{Tr}(\rho_0^2)} {\frac{\sqrt2}{T}\int_0^Tdt\,C_2(\rho_t) \sqrt{\operatorname{Tr}[\rho_0^2H_t^2-(\rho_0H_t)^2]}}. \tag{7} $$$$ T_{S,\mathrm{Pure}}(2,2) = \frac{1-F} {\frac{\sqrt2}{T}\int_0^Tdt\,C_2(|\psi_t\rangle)\Delta H_t(|\psi_0\rangle)}. \tag{8} $$$$ \widetilde T_{S,\mathrm{Pure}}(2,2) = \frac{1-F} {\sqrt2 C_2(|\psi_0\rangle)\Delta H(|\psi_0\rangle)}. \tag{9} $$Hellinger QSL$$ D_H(\rho,\sigma)=2-2F_A(\rho,\sigma), \quad F_A=\operatorname{Tr}(\sqrt\rho\sqrt\sigma). $$$$ T\ge T_H(p,q) = \frac{1-F_A(\rho_0,\rho_T)} {\frac1T\int_0^Tdt\, \widetilde C_p(\rho_t) \|[H_t,\sqrt{\rho_0}]\|_q}. \tag{10} $$$$ C_H(\rho_t)= \min_{\sigma_t\in\mathcal I_t} D_H(\rho_t,\sigma_t). \tag{11} $$$$ T_H(2,2) = \frac{1-F_A(\rho_0,\rho_T)} {\frac{\sqrt2}{T}\int_0^Tdt\, \sqrt{C_H(\rho_t)} \sqrt{I(\rho_0,H_t)}}. \tag{12} $$Landau–Zener$$ H_{\mathrm{LZ}}(t)= \Delta\sigma^x+J_t\sigma^z, \qquad J_t=J(-1+2t/\tau). \tag{13} $$$$ C_2(\rho_t) = \sqrt{2P_{\mathrm{exc}}(1-P_{\mathrm{exc}})}. $$$$ \varrho_c= \frac12 \begin{pmatrix} D_c&O_c\ -O_c&-D_c \end{pmatrix}. \tag{14} $$STA$$ H_{\mathrm{STA}}=H_{\mathrm{LZ}}+V, $$$$ V(t) = -\frac{dJ_t}{dt} \frac{\Delta}{2(\Delta^2+J_t^2)} \sigma^y. \tag{15} $$Fig. 1(b) da $\Delta\tau=10,20,50$: evolyutsiya tezlashganda coherence peak oshadi.Ikki-qubit mixed model$$ H_{\mathrm{TQ}}(t) = \Delta(\sigma_1^x+\sigma_2^x) + J_t\sigma_1^z\sigma_2^z. \tag{16} $$$$ \varrho_0(\eta) = \eta|\phi_0\rangle\langle\phi_0| + (1-\eta)|\phi_1\rangle\langle\phi_1|. $$Fig. 1(c) uchun $\eta=3/5$. $T_S(2,2)$ va $T_H(2,2)$ comparison boundsdan tighter va adiabatic limitda saturable.Appendix A–D$$ - \frac{dF_{\mathrm{RP}}}{dt} = -i\frac{\operatorname{Tr}(\rho_0[\rho_t,H_t])}{\operatorname{Tr}(\rho_0^2)}. \tag{A1} $$$$ -i\operatorname{Tr}((\rho_t-\sigma_t)[H_t,\rho_0]) \le \|\rho_t-\sigma_t\|_p \|[H_t,\rho_0]\|_q. \tag{A2} $$$$ - \frac{dF_{\mathrm{RP}}}{dt} \le \frac{C_p(\rho_t)\|[H_t,\rho_0]\|_q}{\operatorname{Tr}(\rho_0^2)}. \tag{A3} $$$$ - \frac{dF_A}{dt} = -i\operatorname{Tr}(\sqrt{\rho_0}[\sqrt{\rho_t},H_t]). \tag{A4} $$$$ -i\operatorname{Tr}(\sqrt{\rho_0}[\sqrt{\rho_t},H_t]) \le \|\sqrt{\rho_t}-\sqrt{\sigma_t}\|_p \|[H_t,\sqrt{\rho_0}]\|_q. \tag{A5} $$$$ H_t|\psi_0\rangle= \bar H_t|\psi_0\rangle+ \Delta H_t|\psi_0^\perp\rangle. \tag{B1} $$$$ [H_t,\rho_0] = \Delta H_t( |\psi_0^\perp\rangle\langle\psi_0| - |\psi_0\rangle\langle\psi_0^\perp|). \tag{B2} $$$$ G_t^\dagger G_t = [\Delta H_t]^2 ( |\psi_0\rangle\langle\psi_0| + |\psi_0^\perp\rangle\langle\psi_0^\perp|). \tag{B3} $$$$ \|[H_t,\rho_0]\|_p = 2^{1/p}\Delta H_t. \tag{B4} $$$$ \|\rho_t-\sigma\|_p = \|\rho_0-\sigma\|_p. \tag{C1} $$$$ \rho_t= U_t\rho_0U_t^\dagger = \sum_n\lambda_n|u_t^n\rangle\langle u_t^n|. \tag{D1} $$$$ \frac{d\sqrt{\rho_t}}{dt} = i[\sqrt{\rho_t},H_t]. $$O‘zbekiston kontekstiIsh O‘zbekistonda bajarilgan eksperiment emas. Lekin quantum control, simulation, adiabatic computing va state engineering uchun coherence’ni energetik drivingdan alohida resurs sifatida hisoblash g‘oyasini beradi. Real platformada decoherence, control limits va calibration alohida o‘lchanishi kerak.Nimani ko‘rsatadi?

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Ikki infinite coherent-QSL family umumiy unitary dynamics uchun matematik jihatdan chiqariladi. LZ va two-qubit misollarida ular comparison boundsdan tighter. Bu universal “har bir sistemada eng yaxshi QSL” da’vosi emas va open-system dynamicsni qamramaydi.

Family	Distance	Coherence	Generator
Schatten	\(F_{RP}\)	\(C_p\)	\([H_t,\rho_0]\)
Hellinger	\(F_A\)	\(C_H\)	WYSI

Bosqich	Jarayon	Natija	Source
1	Energy eigenbasis	\(\mathcal I_t\)	Main text
2	Liouville	Evolution rate	Appendix A
3	Hölder	Coherence × energy	Eq. 1
4	Integration	QSL families	Eq. 4,10
5	LZ + STA	Saturation	Fig. 1a-b
6	Mixed two-qubit	Tighter bounds	Fig. 1c

Kogerent Kvant Tezlik Chegaralari: Kvant Evolyutsiyasi Tezligida Kogerentlikning Fundamental Roli

Klassik QSL

Instantaneous incoherent set

Schatten QSL

Hellinger QSL

Landau–Zener

STA

Ikki-qubit mixed model

Appendix A–D

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