Error Cancellation in an SU(2) Commutator

From [F1], left multiplication by a unitary $Q$ preserves norm because $\mathrm{\parallel }QXv\mathrm{\parallel }=\mathrm{\parallel }Xv\mathrm{\parallel }$ for each unit vector $v$; right multiplication preserves norm because $Q$ maps the unit sphere bijectively to itself. Taking $X=I$ also gives $\mathrm{\parallel }Q\mathrm{\parallel }=1$. Set $E=V^{\ast }\tilde{V}$ and $F=W^{\ast }\tilde{W}$. Then $E,F$ are unitary, $\mathrm{\parallel }E-I\mathrm{\parallel }=\mathrm{\parallel }{V}^{\ast }(\tilde{V}-V)\mathrm{\parallel }=\mathrm{\parallel }\tilde{V}-V\mathrm{\parallel }\le \eta$, and $\mathrm{\parallel }F-I\mathrm{\parallel }=\mathrm{\parallel }{W}^{\ast }(\tilde{W}-W)\mathrm{\parallel }=\mathrm{\parallel }\tilde{W}-W\mathrm{\parallel }\le \eta$.

Keeping $W$ fixed, the commutator difference caused by replacing $V$ by $\tilde{V}=VE$ is $V(EW{E}^{\ast }-W)V^{\ast }{W}^{\ast }$. Unitary left and right multiplication from [F1] leave its norm equal to $\mathrm{\parallel }EW{E}^{\ast }-W\mathrm{\parallel }$. Since $E{E}^{\ast }=I$, $EW{E}^{\ast }-W=E(W-I)E^{\ast }-(W-I)=(E-I)(W-I)E^{\ast }+(W-I)(E^{\ast }-I)$. Unitary norm invariance, $\mathrm{\parallel }{E}^{\ast }\mathrm{\parallel }=1$, and $\mathrm{\parallel }{E}^{\ast }-I\mathrm{\parallel }=\mathrm{\parallel }E-I\mathrm{\parallel }$ bound this by $2\mathrm{\parallel }E-I\mathrm{\parallel }\mathrm{\parallel }W-I\mathrm{\parallel }\le 2{\eta }^{3\mathrm{/}2}$; the equality $\mathrm{\parallel }{E}^{\ast }-I\mathrm{\parallel }=\mathrm{\parallel }E-I\mathrm{\parallel }$ follows from $E^{\ast }-I=-E^{\ast }(E-I)$.

After the first replacement, keep $\tilde{V}$ fixed and replace $W$ by $\tilde{W}=WF$. The commutator difference is $\tilde{V}W(F{\tilde{V}}^{\ast }{F}^{\ast }-{\tilde{V}}^{\ast })W^{\ast }$. The expansion in step 2.1, with ${\tilde{V}}^{\ast }-I$ in place of $W-I$, bounds its norm by $2\mathrm{\parallel }F-I\mathrm{\parallel }\mathrm{\parallel }\tilde{V}-I\mathrm{\parallel }\le 2\eta (\sqrt{\eta }+\eta )$, since $\mathrm{\parallel }\tilde{V}-I\mathrm{\parallel }\le \mathrm{\parallel }V-I\mathrm{\parallel }+\mathrm{\parallel }\tilde{V}-V\mathrm{\parallel }\le \sqrt{\eta }+\eta$. The operator-norm triangle inequality follows from [F1] and the vector triangle inequality by taking the maximum over unit vectors. Combining the two replacements gives $4{\eta }^{3\mathrm{/}2}+2{\eta }^2\le 5{\eta }^{3\mathrm{/}2}$ because $\eta \le 1\mathrm{/}16$.