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Operator theory

Relatively Random Unitary Operators

Karol Życzkowski

Published Jan 1, 1994• 2 citations
Publisher

Abstract

Consider unitary operators Û1 and Û2 represented by matrices U 1 and U 2 of size N. We shall call operators Û1 and Û2 relatively random. if (1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey % ypa0ZaaSaaaeaaciGGsbGaaiyzamaadmaabaGaamivaiaadkhacaGG % OaGaamyvamaaDaaaleaacaaIXaaabaGaaiiiGaaakiaadwfadaqhaa % WcbaGaaGOmaaqaaiaaccciaaGccaWGvbWaaSbaaSqaaiaaigdaaeqa % aOGaamyvamaaBaaaleaacaaIYaaabeaakiaacMcaaiaawUfacaGLDb % aaaeaacaWGobaaaiablQMi9iaaigdacaGGUaaaaa!4BEE! $$ \mu = \frac{{\operatorname{Re} \left[ {Tr(U_1^\dag U_2^\dag {U_1}{U_2})} \right]}}{N} \ll 1. $$

DOI
10.1007/978-3-0348-8545-4_33

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Feller [1] and Kato [2] proved the existence of a unique minimal semigroup associated with the classical Fokker-Planck equation: (1.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaeaada % WcaaqaaiaadsgacaWGWbWaaSbaaSqaaiaadMgacaWGQbaabeaaaOqa % aiaadsgacaWG0baaaiabg2da9maaqafabaGaamiCamaaBaaaleaaca % WGPbGaam4AaaqabaGccqqHPoWvdaWgaaWcbaGaam4AaiaadQgaaeqa % aOGaaiilaiaaywW7caWG0bGaeyyzImRaaGimaiaaywW7caWG3bGaam % yAaiaadshacaWGObGaaGPaVlaadMgacaWGUbGaamyAaiaadshacaWG % PbGaamyyaiaadYgacaaMc8Uaam4yaiaad+gacaWGUbGaamizaiaadM % gacaWG0bGaamyAaiaad+gacaWGUbGaaGPaVlaadchadaWgaaWcbaGa % amyAaiaadQgaaeqaaOGaaiikaiaaicdacaGGPaGaeyypa0JaeqiTdq % 2aaSbaaSqaaiaadMgacaWGQbaabeaaaeaacaWGRbaabeqdcqGHris5 % aaGccaGL9baaaaa!703E! dpijdt=∑kpikΩkj,t⩾0with initial condition pij(0)=δij} \left. {\frac{{d{p_{ij}}}}{{dt}} = \sum\limits_k {{p_{ik}}{\Omega _{kj}},\quad t \geqslant 0\quad with\,initial\,condition\,{p_{ij}}(0) = {\delta _{ij}}} } \right\} dtdpij​​=k∑​pik​Ωkj​,t⩾0withinitialconditionpij​(0)=δij​} and subject to the Markov condition: (1.2) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaqUbcqqHPo % WvdaWgaaWcbaGaam4AaiaadQgaaeqaaOGaeyyzImRaaGimaiaadAga % caWGVbGaamOCaiaaykW7caWGRbGaeyiyIKRaamOAaiaaykW7caWGHb % GaamOBaiaadsgacaaMc8UaeyOhIu6aaabuaeaacqqHPoWvdaWgaaWc % baGaam4AaiaadQgaaeqaaOGaeyypa0JaeyOeI0IaeuyQdC1aaSbaaS % qaaiaadUgacaWGRbaabeaatCvAUfeBSn0BKvguHDwzZbqegeezVjwz % GyuyUD2CV52zGmfDKbIuaGqbaOGaa8hpaiabg6HiLcWcbaGaamOAai % abgcMi5kaadUgaaeqaniabggHiLdGccaGGUaaaaa!6B15! Ωkj⩾0for k≠j and ∞∑j≠kΩkj=−Ωkk<∞. {\Omega _{kj}} \geqslant 0for\,k \ne j\,and\,\infty \sum\limits_{j \ne k} {{\Omega _{kj}} = - {\Omega _{kk}} < \infty } . Ωkj​⩾0fork=jand∞j=k∑​Ωkj​=−Ωkk​<∞.
In Analogie zu den 3-Wellen-Mischprozessen ist es nicht mehr schwer, die Typologie für 4-Wellen-Phänomene zusammenzustellen. Drei der vier Wellen erzeugen eine Polarisation % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGqbWdamaaBaaaleaapeGaamyAaaWdaeqaaOWdbiaacIcapaGa % eqyYdC3dbiaacMcacqGH9aqpcqaH1oqzpaWaaSbaaSqaa8qacaaIWa % aapaqabaGcpeGaamiEa8aadaqhaaWcbaWdbiaadMgacaWGQbGaam4A % aiaadYgaa8aabaWdbiaacIcacaaIZaGaaiykaaaak8aacqaHjpWDpe % Gaai4oa8aacqaHjpWDdaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGG % SaWdaiabeM8a3naaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaacYcapa % GaeqyYdC3aaSbaaSqaa8qacaaIZaaapaqabaGcpeGaaiykaiaadwea % paWaaSbaaSqaa8qacaWGQbaapaqabaGcpeGaaiika8aacqaHjpWDda % WgaaWcbaWdbiaaigdaa8aabeaak8qacaGGPaGaamyra8aadaWgaaWc % baWdbiaadUgaa8aabeaak8qacaGGOaWdaiabeM8a3naaBaaaleaape % GaaGOmaaWdaeqaaOWdbiaacMcacaWGfbWdamaaBaaaleaapeGaamiB % aaWdaeqaaOWdbiaacIcapaGaeqyYdC3aaSbaaSqaa8qacaaIZaaapa % qabaGcpeGaaiykaaaa!67E5! Pi(ω)=ε0xijkl(3)ω;ω1,ω2,ω3)Ej(ω1)Ek(ω2)El(ω3)P_i (\omega ) = \varepsilon _0 x_{ijkl}^{(3)} \omega ;\omega _1 ,\omega _2 ,\omega _3 )E_j (\omega _1 )E_k (\omega _2 )E_l (\omega _3 )Pi​(ω)=ε0​xijkl(3)​ω;ω1​,ω2​,ω3​)Ej​(ω1​)Ek​(ω2​)El​(ω3​)
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