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   Master Gleichung Eq: 25
TeX (as stored in database):
\begin{align}
  & {{d}_{t}}\tilde{\rho }={{d}_{t}}\left( U_{0}^{\dagger }\rho {{U}_{0}} \right) \\
 & =\mathfrak{i}{{H}_{0}}U_{0}^{\dagger }\rho {{U}_{0}}-iU_{0}^{\dagger }\rho {{H}_{0}}{{U}_{0}}+U_{0}^{\dagger }{{d}_{t}}\left( \rho  \right){{U}_{0}} \\
 & =\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}U_{0}^{\dagger }\left[ H,\rho  \right]{{U}_{0}} \\
 & =\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}U_{0}^{\dagger }\left[ {{H}_{0}}+{{H}_{I}},\rho  \right]{{U}_{0}} \\
 & =\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}\left[ {{H}_{0}},\tilde{\rho } \right]-\mathfrak{i}U_{0}^{\dagger }\left[ {{H}_{I}},\rho  \right]{{U}_{0}} \\
 & =-\mathfrak{i}\left[ {{{\tilde{H}}}_{I}},\tilde{\rho } \right] \\
\end{align}
MathML (38.898 KB / 4.462 KB) :
dtρ~=dt(U0ρU0)subscriptdtnormal-~ρsubscriptdtsuperscriptsubscriptU0normal-†ρsubscriptU0\displaystyle{{d}_{{t}}}\tilde{\rho}={{d}_{{t}}}\left(U_{{0}}^{{\dagger}}\rho{% {U}_{{0}}}\right)
=𝔦H0U0ρU0-iU0ρH0U0+U0dt(ρ)U0absent𝔦subscriptH0superscriptsubscriptU0normal-†ρsubscriptU0isuperscriptsubscriptU0normal-†ρsubscriptH0subscriptU0superscriptsubscriptU0normal-†subscriptdtρsubscriptU0\displaystyle=\mathfrak{i}{{H}_{{0}}}U_{{0}}^{{\dagger}}\rho{{U}_{{0}}}-iU_{{0% }}^{{\dagger}}\rho{{H}_{{0}}}{{U}_{{0}}}+U_{{0}}^{{\dagger}}{{d}_{{t}}}\left(% \rho\right){{U}_{{0}}}
=𝔦[H0,ρ~]-𝔦U0[H,ρ]U0absent𝔦subscriptH0normal-~ρ𝔦superscriptsubscriptU0normal-†HρsubscriptU0\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_% {{0}}^{{\dagger}}\left[H,\rho\right]{{U}_{{0}}}
=𝔦[H0,ρ~]-𝔦U0[H0+HI,ρ]U0absent𝔦subscriptH0normal-~ρ𝔦superscriptsubscriptU0normal-†subscriptH0subscriptHIρsubscriptU0\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_% {{0}}^{{\dagger}}\left[{{H}_{{0}}}+{{H}_{{I}}},\rho\right]{{U}_{{0}}}
=𝔦[H0,ρ~]-𝔦[H0,ρ~]-𝔦U0[HI,ρ]U0absent𝔦subscriptH0normal-~ρ𝔦subscriptH0normal-~ρ𝔦superscriptsubscriptU0normal-†subscriptHIρsubscriptU0\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}% \left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_{{0}}^{{\dagger}}\left[{{H% }_{{I}}},\rho\right]{{U}_{{0}}}
=-𝔦[H~I,ρ~]absent𝔦subscriptnormal-~HInormal-~ρ\displaystyle=-\mathfrak{i}\left[{{{\tilde{H}}}_{{I}}},\tilde{\rho}\right]

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MathML


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xref="S0.Ex2.m2.1.21"/><apply xml:id="S0.Ex2.m2.1.32.2.1.cmml" xref="S0.Ex2.m2.1.32.2.1"><minus xml:id="S0.Ex2.m2.1.11.cmml" xref="S0.Ex2.m2.1.11"/><apply xml:id="S0.Ex2.m2.1.32.2.1.1.cmml" xref="S0.Ex2.m2.1.32.2.1.1"><times xml:id="S0.Ex2.m2.1.32.2.1.1.1.cmml" xref="S0.Ex2.m2.1.32.2.1.1.1"/><ci xml:id="S0.Ex2.m2.1.2.cmml" xref="S0.Ex2.m2.1.2">𝔦</ci><apply xml:id="S0.Ex2.m2.1.32.2.1.1.2.cmml" xref="S0.Ex2.m2.1.32.2.1.1.2"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.1.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.3.cmml" xref="S0.Ex2.m2.1.3">H</ci><cn type="integer" xml:id="S0.Ex2.m2.1.4.1.cmml" xref="S0.Ex2.m2.1.4.1">0</cn></apply><apply xml:id="S0.Ex2.m2.1.32.2.1.1.3.cmml" xref="S0.Ex2.m2.1.32.2.1.1.3"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.1.3.1.cmml">superscript</csymbol><apply xml:id="S0.Ex2.m2.1.32.2.1.1.3.2.cmml"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.1.3.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.5.cmml" xref="S0.Ex2.m2.1.5">U</ci><cn type="integer" xml:id="S0.Ex2.m2.1.6.1.cmml" xref="S0.Ex2.m2.1.6.1">0</cn></apply><ci xml:id="S0.Ex2.m2.1.7.1.cmml" xref="S0.Ex2.m2.1.7.1">normal-†</ci></apply><ci xml:id="S0.Ex2.m2.1.8.cmml" xref="S0.Ex2.m2.1.8">ρ</ci><apply xml:id="S0.Ex2.m2.1.32.2.1.1.4.cmml" xref="S0.Ex2.m2.1.32.2.1.1.4"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.1.4.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.9.cmml" xref="S0.Ex2.m2.1.9">U</ci><cn type="integer" xml:id="S0.Ex2.m2.1.10.1.cmml" xref="S0.Ex2.m2.1.10.1">0</cn></apply></apply><apply xml:id="S0.Ex2.m2.1.32.2.1.2.cmml" xref="S0.Ex2.m2.1.32.2.1.2"><times xml:id="S0.Ex2.m2.1.32.2.1.2.1.cmml" xref="S0.Ex2.m2.1.32.2.1.2.1"/><ci xml:id="S0.Ex2.m2.1.12.cmml" xref="S0.Ex2.m2.1.12">i</ci><apply xml:id="S0.Ex2.m2.1.32.2.1.2.2.cmml" xref="S0.Ex2.m2.1.32.2.1.2.2"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.2.2.1.cmml">superscript</csymbol><apply xml:id="S0.Ex2.m2.1.32.2.1.2.2.2.cmml"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.2.2.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.13.cmml" xref="S0.Ex2.m2.1.13">U</ci><cn type="integer" xml:id="S0.Ex2.m2.1.14.1.cmml" xref="S0.Ex2.m2.1.14.1">0</cn></apply><ci xml:id="S0.Ex2.m2.1.15.1.cmml" xref="S0.Ex2.m2.1.15.1">normal-†</ci></apply><ci xml:id="S0.Ex2.m2.1.16.cmml" xref="S0.Ex2.m2.1.16">ρ</ci><apply xml:id="S0.Ex2.m2.1.32.2.1.2.3.cmml" xref="S0.Ex2.m2.1.32.2.1.2.3"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.2.3.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.17.cmml" xref="S0.Ex2.m2.1.17">H</ci><cn type="integer" xml:id="S0.Ex2.m2.1.18.1.cmml" xref="S0.Ex2.m2.1.18.1">0</cn></apply><apply xml:id="S0.Ex2.m2.1.32.2.1.2.4.cmml" xref="S0.Ex2.m2.1.32.2.1.2.4"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.1.2.4.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.19.cmml" xref="S0.Ex2.m2.1.19">U</ci><cn type="integer" xml:id="S0.Ex2.m2.1.20.1.cmml" xref="S0.Ex2.m2.1.20.1">0</cn></apply></apply></apply><apply xml:id="S0.Ex2.m2.1.32.2.2.cmml" xref="S0.Ex2.m2.1.32.2.2"><times xml:id="S0.Ex2.m2.1.32.2.2.1.cmml" xref="S0.Ex2.m2.1.32.2.2.1"/><apply xml:id="S0.Ex2.m2.1.32.2.2.2.cmml" xref="S0.Ex2.m2.1.32.2.2.2"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.2.2.1.cmml">superscript</csymbol><apply xml:id="S0.Ex2.m2.1.32.2.2.2.2.cmml"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.2.2.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.22.cmml" xref="S0.Ex2.m2.1.22">U</ci><cn type="integer" xml:id="S0.Ex2.m2.1.23.1.cmml" xref="S0.Ex2.m2.1.23.1">0</cn></apply><ci xml:id="S0.Ex2.m2.1.24.1.cmml" xref="S0.Ex2.m2.1.24.1">normal-†</ci></apply><apply xml:id="S0.Ex2.m2.1.32.2.2.3.cmml" xref="S0.Ex2.m2.1.32.2.2.3"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.2.3.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.25.cmml" xref="S0.Ex2.m2.1.25">d</ci><ci xml:id="S0.Ex2.m2.1.26.1.cmml" xref="S0.Ex2.m2.1.26.1">t</ci></apply><ci xml:id="S0.Ex2.m2.1.28.cmml" xref="S0.Ex2.m2.1.28">ρ</ci><apply xml:id="S0.Ex2.m2.1.32.2.2.4.cmml" xref="S0.Ex2.m2.1.32.2.2.4"><csymbol cd="ambiguous" xml:id="S0.Ex2.m2.1.32.2.2.4.1.cmml">subscript</csymbol><ci xml:id="S0.Ex2.m2.1.30.cmml" xref="S0.Ex2.m2.1.30">U</ci><cn type="integer" xml:id="S0.Ex2.m2.1.31.1.cmml" xref="S0.Ex2.m2.1.31.1">0</cn></apply></apply></apply></apply></annotation-xml><annotation xml:id="S0.Ex2.m2.1b" encoding="application/x-tex" xref="S0.Ex2.m2.1.cmml">\displaystyle=\mathfrak{i}{{H}_{{0}}}U_{{0}}^{{\dagger}}\rho{{U}_{{0}}}-iU_{{0%
}}^{{\dagger}}\rho{{H}_{{0}}}{{U}_{{0}}}+U_{{0}}^{{\dagger}}{{d}_{{t}}}\left(%
\rho\right){{U}_{{0}}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/></tr>
<tr id="S0.Ex3" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_td ltx_align_right"/>
<td class="ltx_td ltx_align_left"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex3.m2" class="ltx_Math" alttext="\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_%&#10;{{0}}^{{\dagger}}\left[H,\rho\right]{{U}_{{0}}}" xml:id="S0.Ex3.m2.1" display="inline" xref="S0.Ex3.m2.1.cmml"><semantics xml:id="S0.Ex3.m2.1a" xref="S0.Ex3.m2.1.cmml"><mrow xml:id="S0.Ex3.m2.1.21" xref="S0.Ex3.m2.1.21.cmml"><none xml:id="S0.Ex3.m2.1.21.1" xref="S0.Ex3.m2.1.21.1.cmml"/><mo xml:id="S0.Ex3.m2.1.1" xref="S0.Ex3.m2.1.1.cmml">=</mo><mrow xml:id="S0.Ex3.m2.1.21.2" xref="S0.Ex3.m2.1.21.2.cmml"><mrow xml:id="S0.Ex3.m2.1.21.2.1" xref="S0.Ex3.m2.1.21.2.1.cmml"><mi xml:id="S0.Ex3.m2.1.2" xref="S0.Ex3.m2.1.2.cmml">𝔦</mi><mo xml:id="S0.Ex3.m2.1.21.2.1.1" xref="S0.Ex3.m2.1.21.2.1.1.cmml"></mo><mrow xml:id="S0.Ex3.m2.1.21.2.1.2" xref="S0.Ex3.m2.1.21.2.1.2.cmml"><mo xml:id="S0.Ex3.m2.1.21.2.1.2a" xref="S0.Ex3.m2.1.21.2.1.2.cmml">[</mo><mrow xml:id="S0.Ex3.m2.1.21.2.1.2b" xref="S0.Ex3.m2.1.21.2.1.2.cmml"><msub xml:id="S0.Ex3.m2.1.21.2.1.2.2" xref="S0.Ex3.m2.1.21.2.1.2.2.cmml"><mi xml:id="S0.Ex3.m2.1.4" xref="S0.Ex3.m2.1.4.cmml">H</mi><mn xml:id="S0.Ex3.m2.1.5.1" xref="S0.Ex3.m2.1.5.1.cmml">0</mn></msub><mo xml:id="S0.Ex3.m2.1.21.2.1.2c" xref="S0.Ex3.m2.1.21.2.1.2.cmml">,</mo><mover accent="true" xml:id="S0.Ex3.m2.1.7" xref="S0.Ex3.m2.1.7.cmml"><mi xml:id="S0.Ex3.m2.1.7.2" xref="S0.Ex3.m2.1.7.2.cmml">ρ</mi><mo xml:id="S0.Ex3.m2.1.7.1" xref="S0.Ex3.m2.1.7.1.cmml">~</mo></mover></mrow><mo xml:id="S0.Ex3.m2.1.21.2.1.2d" xref="S0.Ex3.m2.1.21.2.1.2.cmml">]</mo></mrow></mrow><mo xml:id="S0.Ex3.m2.1.9" xref="S0.Ex3.m2.1.9.cmml">-</mo><mrow xml:id="S0.Ex3.m2.1.21.2.2" xref="S0.Ex3.m2.1.21.2.2.cmml"><mi xml:id="S0.Ex3.m2.1.10" xref="S0.Ex3.m2.1.10.cmml">𝔦</mi><mo xml:id="S0.Ex3.m2.1.21.2.2.1" xref="S0.Ex3.m2.1.21.2.2.1.cmml"></mo><msubsup xml:id="S0.Ex3.m2.1.21.2.2.2" xref="S0.Ex3.m2.1.21.2.2.2.cmml"><mi xml:id="S0.Ex3.m2.1.11" xref="S0.Ex3.m2.1.11.cmml">U</mi><mn xml:id="S0.Ex3.m2.1.12.1" xref="S0.Ex3.m2.1.12.1.cmml">0</mn><mo xml:id="S0.Ex3.m2.1.13.1" xref="S0.Ex3.m2.1.13.1.cmml"></mo></msubsup><mo xml:id="S0.Ex3.m2.1.21.2.2.1a" xref="S0.Ex3.m2.1.21.2.2.1.cmml"></mo><mrow xml:id="S0.Ex3.m2.1.21.2.2.3" xref="S0.Ex3.m2.1.21.2.2.3.cmml"><mo xml:id="S0.Ex3.m2.1.21.2.2.3a" xref="S0.Ex3.m2.1.21.2.2.3.cmml">[</mo><mrow xml:id="S0.Ex3.m2.1.21.2.2.3b" xref="S0.Ex3.m2.1.21.2.2.3.cmml"><mi xml:id="S0.Ex3.m2.1.15" xref="S0.Ex3.m2.1.15.cmml">H</mi><mo xml:id="S0.Ex3.m2.1.21.2.2.3c" xref="S0.Ex3.m2.1.21.2.2.3.cmml">,</mo><mi xml:id="S0.Ex3.m2.1.17" xref="S0.Ex3.m2.1.17.cmml">ρ</mi></mrow><mo xml:id="S0.Ex3.m2.1.21.2.2.3d" xref="S0.Ex3.m2.1.21.2.2.3.cmml">]</mo></mrow><mo xml:id="S0.Ex3.m2.1.21.2.2.1b" xref="S0.Ex3.m2.1.21.2.2.1.cmml"></mo><msub xml:id="S0.Ex3.m2.1.21.2.2.4" xref="S0.Ex3.m2.1.21.2.2.4.cmml"><mi xml:id="S0.Ex3.m2.1.19" xref="S0.Ex3.m2.1.19.cmml">U</mi><mn xml:id="S0.Ex3.m2.1.20.1" xref="S0.Ex3.m2.1.20.1.cmml">0</mn></msub></mrow></mrow></mrow><annotation-xml xml:id="S0.Ex3.m2.1.cmml" encoding="MathML-Content" xref="S0.Ex3.m2.1"><apply xml:id="S0.Ex3.m2.1.21.cmml" xref="S0.Ex3.m2.1.21"><eq xml:id="S0.Ex3.m2.1.1.cmml" xref="S0.Ex3.m2.1.1"/><csymbol cd="latexml" xml:id="S0.Ex3.m2.1.21.1.cmml" xref="S0.Ex3.m2.1.21.1">absent</csymbol><apply xml:id="S0.Ex3.m2.1.21.2.cmml" xref="S0.Ex3.m2.1.21.2"><minus xml:id="S0.Ex3.m2.1.9.cmml" xref="S0.Ex3.m2.1.9"/><apply xml:id="S0.Ex3.m2.1.21.2.1.cmml" xref="S0.Ex3.m2.1.21.2.1"><times xml:id="S0.Ex3.m2.1.21.2.1.1.cmml" xref="S0.Ex3.m2.1.21.2.1.1"/><ci xml:id="S0.Ex3.m2.1.2.cmml" xref="S0.Ex3.m2.1.2">𝔦</ci><apply xml:id="S0.Ex3.m2.1.21.2.1.2.cmml" xref="S0.Ex3.m2.1.21.2.1.2"><interval closure="closed" xml:id="S0.Ex3.m2.1.21.2.1.2.1.cmml"/><apply xml:id="S0.Ex3.m2.1.21.2.1.2.2.cmml" xref="S0.Ex3.m2.1.21.2.1.2.2"><csymbol cd="ambiguous" xml:id="S0.Ex3.m2.1.21.2.1.2.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex3.m2.1.4.cmml" xref="S0.Ex3.m2.1.4">H</ci><cn type="integer" xml:id="S0.Ex3.m2.1.5.1.cmml" xref="S0.Ex3.m2.1.5.1">0</cn></apply><apply xml:id="S0.Ex3.m2.1.7.cmml" xref="S0.Ex3.m2.1.7"><ci xml:id="S0.Ex3.m2.1.7.1.cmml" xref="S0.Ex3.m2.1.7.1">normal-~</ci><ci xml:id="S0.Ex3.m2.1.7.2.cmml" xref="S0.Ex3.m2.1.7.2">ρ</ci></apply></apply></apply><apply xml:id="S0.Ex3.m2.1.21.2.2.cmml" xref="S0.Ex3.m2.1.21.2.2"><times xml:id="S0.Ex3.m2.1.21.2.2.1.cmml" xref="S0.Ex3.m2.1.21.2.2.1"/><ci xml:id="S0.Ex3.m2.1.10.cmml" xref="S0.Ex3.m2.1.10">𝔦</ci><apply xml:id="S0.Ex3.m2.1.21.2.2.2.cmml" xref="S0.Ex3.m2.1.21.2.2.2"><csymbol cd="ambiguous" xml:id="S0.Ex3.m2.1.21.2.2.2.1.cmml">superscript</csymbol><apply xml:id="S0.Ex3.m2.1.21.2.2.2.2.cmml"><csymbol cd="ambiguous" xml:id="S0.Ex3.m2.1.21.2.2.2.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex3.m2.1.11.cmml" xref="S0.Ex3.m2.1.11">U</ci><cn type="integer" xml:id="S0.Ex3.m2.1.12.1.cmml" xref="S0.Ex3.m2.1.12.1">0</cn></apply><ci xml:id="S0.Ex3.m2.1.13.1.cmml" xref="S0.Ex3.m2.1.13.1">normal-†</ci></apply><apply xml:id="S0.Ex3.m2.1.21.2.2.3.cmml" xref="S0.Ex3.m2.1.21.2.2.3"><interval closure="closed" xml:id="S0.Ex3.m2.1.21.2.2.3.1.cmml"/><ci xml:id="S0.Ex3.m2.1.15.cmml" xref="S0.Ex3.m2.1.15">H</ci><ci xml:id="S0.Ex3.m2.1.17.cmml" xref="S0.Ex3.m2.1.17">ρ</ci></apply><apply xml:id="S0.Ex3.m2.1.21.2.2.4.cmml" xref="S0.Ex3.m2.1.21.2.2.4"><csymbol cd="ambiguous" xml:id="S0.Ex3.m2.1.21.2.2.4.1.cmml">subscript</csymbol><ci xml:id="S0.Ex3.m2.1.19.cmml" xref="S0.Ex3.m2.1.19">U</ci><cn type="integer" xml:id="S0.Ex3.m2.1.20.1.cmml" xref="S0.Ex3.m2.1.20.1">0</cn></apply></apply></apply></apply></annotation-xml><annotation xml:id="S0.Ex3.m2.1b" encoding="application/x-tex" xref="S0.Ex3.m2.1.cmml">\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_%
{{0}}^{{\dagger}}\left[H,\rho\right]{{U}_{{0}}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/></tr>
<tr id="S0.Ex4" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_td ltx_align_right"/>
<td class="ltx_td ltx_align_left"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex4.m2" class="ltx_Math" alttext="\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_%&#10;{{0}}^{{\dagger}}\left[{{H}_{{0}}}+{{H}_{{I}}},\rho\right]{{U}_{{0}}}" xml:id="S0.Ex4.m2.1" display="inline" xref="S0.Ex4.m2.1.cmml"><semantics xml:id="S0.Ex4.m2.1a" xref="S0.Ex4.m2.1.cmml"><mrow xml:id="S0.Ex4.m2.1.25" xref="S0.Ex4.m2.1.25.cmml"><none xml:id="S0.Ex4.m2.1.25.1" xref="S0.Ex4.m2.1.25.1.cmml"/><mo xml:id="S0.Ex4.m2.1.1" xref="S0.Ex4.m2.1.1.cmml">=</mo><mrow xml:id="S0.Ex4.m2.1.25.2" xref="S0.Ex4.m2.1.25.2.cmml"><mrow xml:id="S0.Ex4.m2.1.25.2.1" xref="S0.Ex4.m2.1.25.2.1.cmml"><mi xml:id="S0.Ex4.m2.1.2" xref="S0.Ex4.m2.1.2.cmml">𝔦</mi><mo xml:id="S0.Ex4.m2.1.25.2.1.1" xref="S0.Ex4.m2.1.25.2.1.1.cmml"></mo><mrow xml:id="S0.Ex4.m2.1.25.2.1.2" xref="S0.Ex4.m2.1.25.2.1.2.cmml"><mo xml:id="S0.Ex4.m2.1.25.2.1.2a" xref="S0.Ex4.m2.1.25.2.1.2.cmml">[</mo><mrow 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xref="S0.Ex4.m2.1.cmml">\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_%
{{0}}^{{\dagger}}\left[{{H}_{{0}}}+{{H}_{{I}}},\rho\right]{{U}_{{0}}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/></tr>
<tr id="S0.Ex5" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_td ltx_align_right"/>
<td class="ltx_td ltx_align_left"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex5.m2" class="ltx_Math" alttext="\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}%&#10;\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_{{0}}^{{\dagger}}\left[{{H%&#10;}_{{I}}},\rho\right]{{U}_{{0}}}" xml:id="S0.Ex5.m2.1" display="inline" xref="S0.Ex5.m2.1.cmml"><semantics xml:id="S0.Ex5.m2.1a" xref="S0.Ex5.m2.1.cmml"><mrow xml:id="S0.Ex5.m2.1.30" xref="S0.Ex5.m2.1.30.cmml"><none xml:id="S0.Ex5.m2.1.30.1" xref="S0.Ex5.m2.1.30.1.cmml"/><mo xml:id="S0.Ex5.m2.1.1" xref="S0.Ex5.m2.1.1.cmml">=</mo><mrow xml:id="S0.Ex5.m2.1.30.2" xref="S0.Ex5.m2.1.30.2.cmml"><mrow xml:id="S0.Ex5.m2.1.30.2.1" xref="S0.Ex5.m2.1.30.2.1.cmml"><mi xml:id="S0.Ex5.m2.1.2" xref="S0.Ex5.m2.1.2.cmml">𝔦</mi><mo xml:id="S0.Ex5.m2.1.30.2.1.1" xref="S0.Ex5.m2.1.30.2.1.1.cmml"></mo><mrow xml:id="S0.Ex5.m2.1.30.2.1.2" xref="S0.Ex5.m2.1.30.2.1.2.cmml"><mo xml:id="S0.Ex5.m2.1.30.2.1.2a" 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xref="S0.Ex5.m2.1.30.2.1.2"><interval closure="closed" xml:id="S0.Ex5.m2.1.30.2.1.2.1.cmml"/><apply xml:id="S0.Ex5.m2.1.30.2.1.2.2.cmml" xref="S0.Ex5.m2.1.30.2.1.2.2"><csymbol cd="ambiguous" xml:id="S0.Ex5.m2.1.30.2.1.2.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex5.m2.1.4.cmml" xref="S0.Ex5.m2.1.4">H</ci><cn type="integer" xml:id="S0.Ex5.m2.1.5.1.cmml" xref="S0.Ex5.m2.1.5.1">0</cn></apply><apply xml:id="S0.Ex5.m2.1.7.cmml" xref="S0.Ex5.m2.1.7"><ci xml:id="S0.Ex5.m2.1.7.1.cmml" xref="S0.Ex5.m2.1.7.1">normal-~</ci><ci xml:id="S0.Ex5.m2.1.7.2.cmml" xref="S0.Ex5.m2.1.7.2">ρ</ci></apply></apply></apply><apply xml:id="S0.Ex5.m2.1.30.2.2.cmml" xref="S0.Ex5.m2.1.30.2.2"><times xml:id="S0.Ex5.m2.1.30.2.2.1.cmml" xref="S0.Ex5.m2.1.30.2.2.1"/><ci xml:id="S0.Ex5.m2.1.10.cmml" xref="S0.Ex5.m2.1.10">𝔦</ci><apply xml:id="S0.Ex5.m2.1.30.2.2.2.cmml" xref="S0.Ex5.m2.1.30.2.2.2"><interval closure="closed" xml:id="S0.Ex5.m2.1.30.2.2.2.1.cmml"/><apply xml:id="S0.Ex5.m2.1.30.2.2.2.2.cmml" xref="S0.Ex5.m2.1.30.2.2.2.2"><csymbol cd="ambiguous" xml:id="S0.Ex5.m2.1.30.2.2.2.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex5.m2.1.12.cmml" xref="S0.Ex5.m2.1.12">H</ci><cn type="integer" xml:id="S0.Ex5.m2.1.13.1.cmml" xref="S0.Ex5.m2.1.13.1">0</cn></apply><apply xml:id="S0.Ex5.m2.1.15.cmml" xref="S0.Ex5.m2.1.15"><ci xml:id="S0.Ex5.m2.1.15.1.cmml" xref="S0.Ex5.m2.1.15.1">normal-~</ci><ci xml:id="S0.Ex5.m2.1.15.2.cmml" xref="S0.Ex5.m2.1.15.2">ρ</ci></apply></apply></apply><apply xml:id="S0.Ex5.m2.1.30.2.3.cmml" xref="S0.Ex5.m2.1.30.2.3"><times xml:id="S0.Ex5.m2.1.30.2.3.1.cmml" xref="S0.Ex5.m2.1.30.2.3.1"/><ci xml:id="S0.Ex5.m2.1.18.cmml" xref="S0.Ex5.m2.1.18">𝔦</ci><apply xml:id="S0.Ex5.m2.1.30.2.3.2.cmml" xref="S0.Ex5.m2.1.30.2.3.2"><csymbol cd="ambiguous" xml:id="S0.Ex5.m2.1.30.2.3.2.1.cmml">superscript</csymbol><apply xml:id="S0.Ex5.m2.1.30.2.3.2.2.cmml"><csymbol cd="ambiguous" xml:id="S0.Ex5.m2.1.30.2.3.2.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex5.m2.1.19.cmml" xref="S0.Ex5.m2.1.19">U</ci><cn type="integer" xml:id="S0.Ex5.m2.1.20.1.cmml" xref="S0.Ex5.m2.1.20.1">0</cn></apply><ci xml:id="S0.Ex5.m2.1.21.1.cmml" xref="S0.Ex5.m2.1.21.1">normal-†</ci></apply><apply xml:id="S0.Ex5.m2.1.30.2.3.3.cmml" xref="S0.Ex5.m2.1.30.2.3.3"><interval closure="closed" xml:id="S0.Ex5.m2.1.30.2.3.3.1.cmml"/><apply xml:id="S0.Ex5.m2.1.30.2.3.3.2.cmml" xref="S0.Ex5.m2.1.30.2.3.3.2"><csymbol cd="ambiguous" xml:id="S0.Ex5.m2.1.30.2.3.3.2.1.cmml">subscript</csymbol><ci xml:id="S0.Ex5.m2.1.23.cmml" xref="S0.Ex5.m2.1.23">H</ci><ci xml:id="S0.Ex5.m2.1.24.1.cmml" xref="S0.Ex5.m2.1.24.1">I</ci></apply><ci xml:id="S0.Ex5.m2.1.26.cmml" xref="S0.Ex5.m2.1.26">ρ</ci></apply><apply xml:id="S0.Ex5.m2.1.30.2.3.4.cmml" xref="S0.Ex5.m2.1.30.2.3.4"><csymbol cd="ambiguous" xml:id="S0.Ex5.m2.1.30.2.3.4.1.cmml">subscript</csymbol><ci xml:id="S0.Ex5.m2.1.28.cmml" xref="S0.Ex5.m2.1.28">U</ci><cn type="integer" xml:id="S0.Ex5.m2.1.29.1.cmml" xref="S0.Ex5.m2.1.29.1">0</cn></apply></apply></apply></apply></annotation-xml><annotation xml:id="S0.Ex5.m2.1b" encoding="application/x-tex" xref="S0.Ex5.m2.1.cmml">\displaystyle=\mathfrak{i}\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}%
\left[{{H}_{{0}}},\tilde{\rho}\right]-\mathfrak{i}U_{{0}}^{{\dagger}}\left[{{H%
}_{{I}}},\rho\right]{{U}_{{0}}}</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/></tr>
<tr id="S0.Ex6" class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"/>
<td class="ltx_td ltx_align_right"/>
<td class="ltx_td ltx_align_left"><math xmlns="http://www.w3.org/1998/Math/MathML" id="S0.Ex6.m2" class="ltx_Math" alttext="\displaystyle=-\mathfrak{i}\left[{{{\tilde{H}}}_{{I}}},\tilde{\rho}\right]" xml:id="S0.Ex6.m2.1" display="inline" xref="S0.Ex6.m2.1.cmml"><semantics xml:id="S0.Ex6.m2.1a" xref="S0.Ex6.m2.1.cmml"><mrow xml:id="S0.Ex6.m2.1.10" xref="S0.Ex6.m2.1.10.cmml"><none xml:id="S0.Ex6.m2.1.10.1" xref="S0.Ex6.m2.1.10.1.cmml"/><mo xml:id="S0.Ex6.m2.1.1" xref="S0.Ex6.m2.1.1.cmml">=</mo><mrow xml:id="S0.Ex6.m2.1.10.2" xref="S0.Ex6.m2.1.10.2.cmml"><mo xml:id="S0.Ex6.m2.1.2" xref="S0.Ex6.m2.1.2.cmml">-</mo><mrow xml:id="S0.Ex6.m2.1.10.2.1" xref="S0.Ex6.m2.1.10.2.1.cmml"><mi xml:id="S0.Ex6.m2.1.3" xref="S0.Ex6.m2.1.3.cmml">𝔦</mi><mo xml:id="S0.Ex6.m2.1.10.2.1.1" xref="S0.Ex6.m2.1.10.2.1.1.cmml"></mo><mrow xml:id="S0.Ex6.m2.1.10.2.1.2" xref="S0.Ex6.m2.1.10.2.1.2.cmml"><mo xml:id="S0.Ex6.m2.1.10.2.1.2a" xref="S0.Ex6.m2.1.10.2.1.2.cmml">[</mo><mrow xml:id="S0.Ex6.m2.1.10.2.1.2b" xref="S0.Ex6.m2.1.10.2.1.2.cmml"><msub xml:id="S0.Ex6.m2.1.10.2.1.2.2" xref="S0.Ex6.m2.1.10.2.1.2.2.cmml"><mover accent="true" xml:id="S0.Ex6.m2.1.5" xref="S0.Ex6.m2.1.5.cmml"><mi xml:id="S0.Ex6.m2.1.5.2" xref="S0.Ex6.m2.1.5.2.cmml">H</mi><mo xml:id="S0.Ex6.m2.1.5.1" xref="S0.Ex6.m2.1.5.1.cmml">~</mo></mover><mi xml:id="S0.Ex6.m2.1.6.1" xref="S0.Ex6.m2.1.6.1.cmml">I</mi></msub><mo xml:id="S0.Ex6.m2.1.10.2.1.2c" xref="S0.Ex6.m2.1.10.2.1.2.cmml">,</mo><mover accent="true" xml:id="S0.Ex6.m2.1.8" xref="S0.Ex6.m2.1.8.cmml"><mi xml:id="S0.Ex6.m2.1.8.2" xref="S0.Ex6.m2.1.8.2.cmml">ρ</mi><mo xml:id="S0.Ex6.m2.1.8.1" xref="S0.Ex6.m2.1.8.1.cmml">~</mo></mover></mrow><mo xml:id="S0.Ex6.m2.1.10.2.1.2d" xref="S0.Ex6.m2.1.10.2.1.2.cmml">]</mo></mrow></mrow></mrow></mrow><annotation-xml xml:id="S0.Ex6.m2.1.cmml" encoding="MathML-Content" xref="S0.Ex6.m2.1"><apply xml:id="S0.Ex6.m2.1.10.cmml" xref="S0.Ex6.m2.1.10"><eq xml:id="S0.Ex6.m2.1.1.cmml" xref="S0.Ex6.m2.1.1"/><csymbol cd="latexml" xml:id="S0.Ex6.m2.1.10.1.cmml" xref="S0.Ex6.m2.1.10.1">absent</csymbol><apply xml:id="S0.Ex6.m2.1.10.2.cmml" xref="S0.Ex6.m2.1.10.2"><minus xml:id="S0.Ex6.m2.1.2.cmml" xref="S0.Ex6.m2.1.2"/><apply xml:id="S0.Ex6.m2.1.10.2.1.cmml" xref="S0.Ex6.m2.1.10.2.1"><times xml:id="S0.Ex6.m2.1.10.2.1.1.cmml" xref="S0.Ex6.m2.1.10.2.1.1"/><ci xml:id="S0.Ex6.m2.1.3.cmml" xref="S0.Ex6.m2.1.3">𝔦</ci><apply xml:id="S0.Ex6.m2.1.10.2.1.2.cmml" xref="S0.Ex6.m2.1.10.2.1.2"><interval closure="closed" xml:id="S0.Ex6.m2.1.10.2.1.2.1.cmml"/><apply xml:id="S0.Ex6.m2.1.10.2.1.2.2.cmml" xref="S0.Ex6.m2.1.10.2.1.2.2"><csymbol cd="ambiguous" xml:id="S0.Ex6.m2.1.10.2.1.2.2.1.cmml">subscript</csymbol><apply xml:id="S0.Ex6.m2.1.5.cmml" xref="S0.Ex6.m2.1.5"><ci xml:id="S0.Ex6.m2.1.5.1.cmml" xref="S0.Ex6.m2.1.5.1">normal-~</ci><ci xml:id="S0.Ex6.m2.1.5.2.cmml" xref="S0.Ex6.m2.1.5.2">H</ci></apply><ci xml:id="S0.Ex6.m2.1.6.1.cmml" xref="S0.Ex6.m2.1.6.1">I</ci></apply><apply xml:id="S0.Ex6.m2.1.8.cmml" xref="S0.Ex6.m2.1.8"><ci xml:id="S0.Ex6.m2.1.8.1.cmml" xref="S0.Ex6.m2.1.8.1">normal-~</ci><ci xml:id="S0.Ex6.m2.1.8.2.cmml" xref="S0.Ex6.m2.1.8.2">ρ</ci></apply></apply></apply></apply></apply></annotation-xml><annotation xml:id="S0.Ex6.m2.1b" encoding="application/x-tex" xref="S0.Ex6.m2.1.cmml">\displaystyle=-\mathfrak{i}\left[{{{\tilde{H}}}_{{I}}},\tilde{\rho}\right]</annotation></semantics></math></td>
<td class="ltx_eqn_pad"/></tr>
</table>